• Author / Uploaded
• Mario J. Kafati

• Categories
• Funciones trigonométricas
• Factorización
• Matriz (Matemáticas)
• Polinomio
• Función (Matemáticas)

Citation preview

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:27 AM

Page i

Precalculus EIGHTH EDITION

Graphical, Numerical, Algebraic Franklin D. Demana

The Ohio State University

Bert K. Waits

The Ohio State University

Gregory D. Foley Daniel Kennedy

Ohio University Baylor School

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:27 AM

Page ii

Executive Editor Senior Project Editor Editorial Assistant Senior Managing Editor Senior Production Supervisor Design Coordinator Photo Researcher Supplements Coordinator Media Producer Software Development Executive Marketing Manager Senior Marketing Manager Marketing Assistant Senior Author Support/ Technology Specialist Senior Prepress Supervisor Senior Manufacturing Buyer Developmental Editor Cover Design Text Design Project Management Production Coordination, Composition, and Illustrations Cover photo

Anne Kelly Joanne Dill Sarah Gibbons Karen Wernholm Peggy McMahon Christina Gleason Beth Anderson Kayla Smith-Tarbox Carl Cottrell John O’Brien and Mary Durnwald Becky Anderson Katherine Greig Katherine Minton Joe Vetere Caroline Fell Carol Melville Elka Block Christina Gleason Leslie Haimes Joanne Boehme Nesbitt Graphics, Inc. Blue Geometry, © Clara/Shutterstock images

For permission to use copyrighted material, grateful acknowledgment is made to the copyright holders listed on page xxx, which is hereby made part of this copyright page. Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and Addison-Wesley was aware of a trademark claim, the designations have been printed in initial caps or all caps. *Advanced Placement Program and AP are registered trademarks of The College Board, which was not involved in the production of, and does not endorse, this product. Library of Congress Cataloging-in-Publication Data Precalculus : graphical, numerical, algebraic / Franklin D. Demana . . . [et al.]. -- 8th ed. p. cm. Includes index. ISBN 0-13-136906-7 (student edition) -- ISBN 0-13-136907-5 (annotated teacher’s edition) 1. Algebra--Textbooks. 2. Trigonometry--Textbooks. I. Demana, Franklin D., 1938QA154.3.P74 2010 512'.13--dc22 2009039915 Copyright © 2011, 2007, 2004, 2001 Pearson Education, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116, fax your request to 617-848-7047, or e-mail at http://www.pearsoned.com/legal/permissions.htm. 1 2 3 4 5 6 7 8 9 10—CRK—12 11 10 ISBN-13: 978-0-13-136906-1 ISBN-10: 0-13-136906-7 (high school binding)

www.PearsonSchool.com/Advanced

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:27 AM

Page iii

Foreword Although much attention has been paid since 1990 to reforming calculus courses, precalculus textbooks have remained surprisingly traditional. Now that The College Board’s AP* Calculus curriculum is accepted as a model for a twenty-first century calculus course, the path is cleared for a new precalculus course to match the AP* goals and objectives. With this edition of Precalculus: Graphical, Numerical, Algebraic, the authors of Calculus: Graphical, Numerical, Algebraic, the best-selling textbook in the AP* Calculus market, have designed such a precalculus course. For those students continuing to a calculus course, this precalculus textbook concludes with a chapter that prepares students for the two central themes of calculus: instantaneous rate of change and continuous accumulation. This intuitively appealing preview of calculus is both more useful and more reasonable than the traditional, unmotivated foray into the computation of limits, and it is more in keeping with the stated goals and objectives of the AP* courses and their emphasis on depth of knowledge. Recognizing that precalculus is a capstone course for many students, we include quantitative literacy topics such as probability, statistics, and the mathematics of finance and integrate the use of data and modeling throughout the text. Our goal is to provide students with the critical-thinking skills and mathematical know-how needed to succeed in college or any endeavor. Continuing in the spirit of two earlier editions, we have integrated graphing technology throughout the course, not as an additional topic but as an essential tool for both mathematical discovery and effective problem solving. Graphing technology enables students to study a full catalog of basic functions at the beginning of the course, thereby giving them insights into function properties that are not seen in many books until later chapters. By connecting the algebra of functions to the visualization of their graphs, we are even able to introduce students to parametric equations, piecewise-defined functions, limit notation, and an intuitive understanding of continuity as early as Chapter 1. However, the advances in technology and increased familiarity with calculators have blurred some of the distinctions between solving problems and supporting solutions that we had once assumed to be apparent. Therefore, we are asking that some exercises be solved without calculators. (See the “Technology and Exercises” section.) Once students are comfortable with the language of functions, the text guides them through a more traditional exploration of twelve basic functions and their algebraic properties, always reinforcing the connections among their algebraic, graphical, and numerical representations. This book uses a consistent approach to modeling, emphasizing in every chapter the use of particular types of functions to model behavior in the real world. This textbook has faithfully incorporated not only the teaching strategies that have made Calculus: Graphical, Numerical, Algebraic so popular, but also some of the strategies from the popular Prentice Hall high school algebra series, and thus has produced a seamless pedagogical transition from prealgebra through calculus for *AP is a registered trademark of The College Board, which was not involved in the production of, and does not endorse, this product.

Foreword

iii

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:27 AM

Page iv

students. Although this book can certainly be appreciated on its own merits, teachers who seek continuity and vertical alignment in their mathematics sequence might consider this pedagogical approach to be an additional asset of Precalculus: Graphical, Numerical, Algebraic. This textbook is written to address current and emerging state curriculum standards. In addition, we embrace NCTM’s Guiding Principles for Mathematics Curriculum and Assessment and agree that a curriculum “must be coherent, focused on important mathematics, and well articulated across the grades.” As statistics is increasingly used in college coursework, the workplace, and everyday life, we have added a “Statistical Literacy” section in Chapter 9 to help students see that statistical analysis is an investigative process that turns loosely formed ideas into scientific studies. Our three sections on data analysis and statistics are aligned with the GAISE Report published by the American Statistical Association; however, they are not intended as a course in statistics but rather as an introduction to set the stage for possible further study in this area of growing importance.

References Franklin, C., Kader, G., Mewborn, D., Moreno, J., Peck, R., Perry, M., and Scheaffer, R. (2007). Guidelines for assessment and instruction in statistics education (GAISE) report: A pre K-12 curriculum framework. Alexandria, VA: American Statistical Association. National Council of Teachers of Mathematics. (2009, June). Guiding principles for mathematics curriculum and assessment. Reston, VA: Author. Retrieved August 13, 2009, from http://www.nctm.org/standards/content.aspx?id=23273

iv

Foreword

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:27 AM

Page v

Contents CHAPTER P

Prerequisites P.1

Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Representing Real Numbers ~ Order and Interval Notation ~ Basic Properties of Algebra ~ Integer Exponents ~ Scientific Notation

P.2

Cartesian Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . 12 Cartesian Plane ~ Absolute Value of a Real Number ~ Distance Formulas ~ Midpoint Formulas ~ Equations of Circles ~ Applications

P.3

Linear Equations and Inequalities . . . . . . . . . . . . . . . . . . . 21 Equations ~ Solving Equations ~ Linear Equations in One Variable ~ Linear Inequalities in One Variable

P.4

Lines in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Slope of a Line ~ Point-Slope Form Equation of a Line ~ SlopeIntercept Form Equation of a Line ~ Graphing Linear Equations in Two Variables ~ Parallel and Perpendicular Lines ~ Applying Linear Equations in Two Variables

P.5

Solving Equations Graphically, Numerically, and Algebraically . . . . . . . . . . . . . . . . . . . . . . 40 Solving Equations Graphically ~ Solving Quadratic Equations ~ Approximating Solutions of Equations Graphically ~ Approximating Solutions of Equations Numerically with Tables ~ Solving Equations by Finding Intersections

P.6

Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49 Complex Numbers ~ Operations with Complex Numbers ~ Complex Conjugates and Division ~ Complex Solutions of Quadratic Equations

P.7

Solving Inequalities Algebraically and Graphically . . . . . 54 Solving Absolute Value Inequalities ~ Solving Quadratic Inequalities ~ Approximating Solutions to Inequalities ~ Projectile Motion

Key Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

CHAPTER 1

Functions and Graphs 1.1

Modeling and Equation Solving . . . . . . . . . . . . . . . . . . . . . . .64 Numerical Models ~ Algebraic Models ~ Graphical Models ~ The Zero Factor Property ~ Problem Solving ~ Grapher Failure and Hidden Behavior ~ A Word About Proof

Contents

v

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:27 AM

1.2

Page vi

Functions and Their Properties . . . . . . . . . . . . . . . . . . . . . 80 Function Definition and Notation ~ Domain and Range ~ Continuity ~ Increasing and Decreasing Functions ~ Boundedness ~ Local and Absolute Extrema ~ Symmetry ~ Asymptotes ~ End Behavior

1.3

Twelve Basic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 What Graphs Can Tell Us ~ Twelve Basic Functions ~ Analyzing Functions Graphically

1.4

Building Functions from Functions . . . . . . . . . . . . . . . . . 110 Combining Functions Algebraically ~ Composition of Functions ~ Relations and Implicitly Defined Functions

1.5

Parametric Relations and Inverses . . . . . . . . . . . . . . . . . . 119 Relations Defined Parametrically ~ Inverse Relations and Inverse Functions

1.6

Graphical Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 129 Transformations ~ Vertical and Horizontal Translations ~ Reflections Across Axes ~ Vertical and Horizontal Stretches and Shrinks ~ Combining Transformations

1.7

Modeling with Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Functions from Formulas ~ Functions from Graphs ~ Functions from Verbal Descriptions ~ Functions from Data

Key Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Chapter Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

CHAPTER 2

Polynomial, Power, and Rational Functions 2.1

Linear and Quadratic Functions and Modeling . . . . . . . 158 Polynomial Functions ~ Linear Functions and Their Graphs ~ Average Rate of Change ~ Linear Correlation and Modeling ~ Quadratic Functions and Their Graphs ~ Applications of Quadratic Functions

2.2

Power Functions with Modeling . . . . . . . . . . . . . . . . . . . . 174 Power Functions and Variation ~ Monomial Functions and Their Graphs ~ Graphs of Power Functions ~ Modeling with Power Functions

2.3

Polynomial Functions of Higher Degree with Modeling . . . . . . . . . . . . . . . . . . . . . . 185 Graphs of Polynomial Functions ~ End Behavior of Polynomial Functions ~ Zeros of Polynomial Functions ~ Intermediate Value Theorem ~ Modeling

vi

Contents

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page vii

2.4

Real Zeros of Polynomial Functions . . . . . . . . . . . . . . . . 197 Long Division and the Division Algorithm ~ Remainder and Factor Theorems ~ Synthetic Division ~ Rational Zeros Theorem ~ Upper and Lower Bounds

2.5

Complex Zeros and the Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . 210 Two Major Theorems ~ Complex Conjugate Zeros ~ Factoring with Real Number Coefficients

2.6

Graphs of Rational Functions . . . . . . . . . . . . . . . . . . . . . . 218 Rational Functions ~ Transformations of the Reciprocal Function ~ Limits and Asymptotes ~ Analyzing Graphs of Rational Functions ~ Exploring Relative Humidity

2.7

Solving Equations in One Variable . . . . . . . . . . . . . . . . . . 228 Solving Rational Equations ~ Extraneous Solutions ~ Applications

2.8

Solving Inequalities in One Variable . . . . . . . . . . . . . . . . . 236 Polynomial Inequalities ~ Rational Inequalities ~ Other Inequalities ~ Applications

Key Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 Chapter Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

CHAPTER 3

Exponential, Logistic, and Logarithmic Functions 3.1

Exponential and Logistic Functions . . . . . . . . . . . . . . . . . 252 Exponential Functions and Their Graphs ~ The Natural Base e ~ Logistic Functions and Their Graphs ~ Population Models

3.2

Exponential and Logistic Modeling . . . . . . . . . . . . . . . . . . 265 Constant Percentage Rate and Exponential Functions ~ Exponential Growth and Decay Models ~ Using Regression to Model Population ~ Other Logistic Models

3.3

Logarithmic Functions and Their Graphs . . . . . . . . . . . . 274 Inverses of Exponential Functions ~ Common Logarithms—Base 10 ~ Natural Logarithms—Base e ~ Graphs of Logarithmic Functions ~ Measuring Sound Using Decibels

3.4

Properties of Logarithmic Functions . . . . . . . . . . . . . . . . 283 Properties of Logarithms ~ Change of Base ~ Graphs of Logarithmic Functions with Base b ~ Re-expressing Data

3.5

Equation Solving and Modeling . . . . . . . . . . . . . . . . . . . . . 292 Solving Exponential Equations ~ Solving Logarithmic Equations ~ Orders of Magnitude and Logarithmic Models ~ Newton’s Law of Cooling ~ Logarithmic Re-expression

Contents

vii

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

3.6

Page viii

Mathematics of Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 Interest Compounded Annually ~ Interest Compounded k Times per Year ~ Interest Compounded Continuously ~ Annual Percentage Yield ~ Annuities—Future Value ~ Loans and Mortgages—Present Value

Key Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Chapter Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

CHAPTER 4

Trigonometric Functions 4.1

Angles and Their Measures . . . . . . . . . . . . . . . . . . . . . . . . 320 The Problem of Angular Measure ~ Degrees and Radians ~ Circular Arc Length ~ Angular and Linear Motion

4.2

Trigonometric Functions of Acute Angles . . . . . . . . . . . . 329 Right Triangle Trigonometry ~ Two Famous Triangles ~ Evaluating Trigonometric Functions with a Calculator ~ Common Calculator Errors When Evaluating Trig Functions ~ Applications of Right Triangle Trigonometry

4.3

Trigonometry Extended: The Circular Functions . . . . . 338 Trigonometric Functions of Any Angle ~ Trigonometric Functions of Real Numbers ~ Periodic Functions ~ The 16-Point Unit Circle

4.4

Graphs of Sine and Cosine: Sinusoids . . . . . . . . . . . . . . . 350 The Basic Waves Revisited ~ Sinusoids and Transformations ~ Modeling Periodic Behavior with Sinusoids

4.5

Graphs of Tangent, Cotangent, Secant, and Cosecant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 The Tangent Function ~ The Cotangent Function ~ The Secant Function ~ The Cosecant Function

4.6

Graphs of Composite Trigonometric Functions . . . . . . . 369 Combining Trigonometric and Algebraic Functions ~ Sums and Differences of Sinusoids ~ Damped Oscillation

4.7

Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . 378 Inverse Sine Function ~ Inverse Cosine and Tangent Functions ~ Composing Trigonometric and Inverse Trigonometric Functions ~ Applications of Inverse Trigonometric Functions

4.8

Solving Problems with Trigonometry . . . . . . . . . . . . . . . . 388 More Right Triangle Problems ~ Simple Harmonic Motion

Key Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Chapter Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

viii

Contents

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

CHAPTER 5

Page ix

Analytic Trigonometry 5.1

Fundamental Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 Identities ~ Basic Trigonometric Identities ~ Pythagorean Identities ~ Cofunction Identities ~ Odd-Even Identities ~ Simplifying Trigonometric Expressions ~ Solving Trigonometric Equations

5.2

Proving Trigonometric Identities . . . . . . . . . . . . . . . . . . . 413 A Proof Strategy ~ Proving Identities ~ Disproving Non-Identities ~ Identities in Calculus

5.3

Sum and Difference Identities . . . . . . . . . . . . . . . . . . . . . . 421 Cosine of a Difference ~ Cosine of a Sum ~ Sine of a Difference or Sum ~ Tangent of a Difference or Sum ~ Verifying a Sinusoid Algebraically

5.4

Multiple-Angle Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 Double-Angle Identities ~ Power-Reducing Identities ~ Half-Angle Identities ~ Solving Trigonometric Equations

5.5

The Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 Deriving the Law of Sines ~ Solving Triangles (AAS, ASA) ~ The Ambiguous Case (SSA) ~ Applications

5.6

The Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 Deriving the Law of Cosines ~ Solving Triangles (SAS, SSS) ~ Triangle Area and Heron’s Formula ~ Applications

Key Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 Chapter Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

CHAPTER 6

Applications of Trigonometry 6.1

Vectors in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 Two-Dimensional Vectors ~ Vector Operations ~ Unit Vectors ~ Direction Angles ~ Applications of Vectors

6.2

Dot Product of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 The Dot Product ~ Angle Between Vectors ~ Projecting One Vector onto Another ~ Work

6.3

Parametric Equations and Motion . . . . . . . . . . . . . . . . . . 475 Parametric Equations ~ Parametric Curves ~ Eliminating the Parameter ~ Lines and Line Segments ~ Simulating Motion with a Grapher

6.4

Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 Polar Coordinate System ~ Coordinate Conversion ~ Equation Conversion ~ Finding Distance Using Polar Coordinates

Contents

ix

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

6.5

Page x

Graphs of Polar Equations . . . . . . . . . . . . . . . . . . . . . . . . . 494 Polar Curves and Parametric Curves ~ Symmetry ~ Analyzing Polar Graphs ~ Rose Curves ~ Limaçon Curves ~ Other Polar Curves

6.6

De Moivre’s Theorem and nth Roots . . . . . . . . . . . . . . . . 503 The Complex Plane ~ Trigonometric Form of Complex Numbers ~ Multiplication and Division of Complex Numbers ~ Powers of Complex Numbers ~ Roots of Complex Numbers

Key Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 Chapter Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

CHAPTER 7

Systems and Matrices 7.1

Solving Systems of Two Equations . . . . . . . . . . . . . . . . . . 520 The Method of Substitution ~ Solving Systems Graphically ~ The Method of Elimination ~ Applications

7.2

Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 Matrices ~ Matrix Addition and Subtraction ~ Matrix Multiplication ~ Identity and Inverse Matrices ~ Determinant of a Square Matrix ~ Applications

7.3

Multivariate Linear Systems and Row Operations . . . . . 544 Triangular Form for Linear Systems ~ Gaussian Elimination ~ Elementary Row Operations and Row Echelon Form ~ Reduced Row Echelon Form ~ Solving Systems with Inverse Matrices ~ Applications

7.4

Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 Partial Fraction Decomposition ~ Denominators with Linear Factors ~ Denominators with Irreducible Quadratic Factors ~ Applications

7.5

Systems of Inequalities in Two Variables . . . . . . . . . . . . . 565 Graph of an Inequality ~ Systems of Inequalities ~ Linear Programming

Key Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 Chapter Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577

CHAPTER 8

Analytic Geometry in Two and Three Dimensions 8.1

Conic Sections and Parabolas . . . . . . . . . . . . . . . . . . . . . . 580 Conic Sections ~ Geometry of a Parabola ~ Translations of Parabolas ~ Reflective Property of a Parabola

x

Contents

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page xi

8.2

Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 Geometry of an Ellipse ~ Translations of Ellipses ~ Orbits and Eccentricity ~ Reflective Property of an Ellipse

8.3

Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 Geometry of a Hyperbola ~ Translations of Hyperbolas ~ Eccentricity and Orbits ~ Reflective Property of a Hyperbola ~ Long-Range Navigation

8.4

Translation and Rotation of Axes . . . . . . . . . . . . . . . . . . . 612 Second-Degree Equations in Two Variables ~ Translating Axes Versus Translating Graphs ~ Rotation of Axes ~ Discriminant Test

8.5

Polar Equations of Conics . . . . . . . . . . . . . . . . . . . . . . . . . 620 Eccentricity Revisited ~ Writing Polar Equations for Conics ~ Analyzing Polar Equations of Conics ~ Orbits Revisited

8.6

Three-Dimensional Cartesian Coordinate System . . . . . 629 Three-Dimensional Cartesian Coordinates ~ Distance and Midpoint Formulas ~ Equation of a Sphere ~ Planes and Other Surfaces ~ Vectors in Space ~ Lines in Space

Key Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 Chapter Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640

CHAPTER 9

Discrete Mathematics 9.1

Basic Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 Discrete Versus Continuous ~ The Importance of Counting ~ The Multiplication Principle of Counting ~ Permutations ~ Combinations ~ Subsets of an n-Set

9.2

The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 Powers of Binomials ~ Pascal’s Triangle ~ The Binomial Theorem ~ Factorial Identities

9.3

Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 Sample Spaces and Probability Functions ~ Determining Probabilities ~ Venn Diagrams and Tree Diagrams ~ Conditional Probability ~ Binomial Distributions

9.4

Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 Infinite Sequences ~ Limits of Infinite Sequences ~ Arithmetic and Geometric Sequences ~ Sequences and Graphing Calculators

9.5

Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 Summation Notation ~ Sums of Arithmetic and Geometric Sequences ~ Infinite Series ~ Convergence of Geometric Series

Contents

xi

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

9.6

Page xii

Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 The Tower of Hanoi Problem ~ Principle of Mathematical Induction ~ Induction and Deduction

9.7

Statistics and Data (Graphical) . . . . . . . . . . . . . . . . . . . . . 693 Statistics ~ Displaying Categorical Data ~ Stemplots ~ Frequency Tables ~ Histograms ~ Time Plots

9.8

Statistics and Data (Algebraic) . . . . . . . . . . . . . . . . . . . . . 704 Parameters and Statistics ~ Mean, Median, and Mode ~ The FiveNumber Summary ~ Boxplots ~ Variance and Standard Deviation ~ Normal Distributions

9.9

Statistical Literacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 The Many Misuses of Statistics ~ Correlation Revisited ~ The Importance of Randomness ~ Surveys and Observational Studies ~ Experimental Design ~ Using Randomness ~ Probability Simulations

Key Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 Chapter Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733

CHAPTER 10

An Introduction to Calculus: Limits, Derivatives, and Integrals 10.1 Limits and Motion: The Tangent Problem . . . . . . . . . . . 736 Average Velocity ~ Instantaneous Velocity ~ Limits Revisited ~ The Connection to Tangent Lines ~ The Derivative

10.2 Limits and Motion: The Area Problem . . . . . . . . . . . . . . 747 Distance from a Constant Velocity ~ Distance from a Changing Velocity ~ Limits at Infinity ~ The Connection to Areas ~ The Definite Integral

10.3 More on Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 A Little History ~ Defining a Limit Informally ~ Properties of Limits ~ Limits of Continuous Functions ~ One-Sided and Two-Sided Limits ~ Limits Involving Infinity

10.4 Numerical Derivatives and Integrals . . . . . . . . . . . . . . . . 766 Derivatives on a Calculator ~ Definite Integrals on a Calculator ~ Computing a Derivative from Data ~ Computing a Definite Integral from Data

Key Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775 Chapter Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777

xii

Contents

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

APPENDIX A

Page xiii

Algebra Review A.1 Radicals and Rational Exponents . . . . . . . . . . . . . . . . . . . 779 Radicals ~ Simplifying Radical Expressions ~ Rationalizing the Denominator ~ Rational Exponents

A.2 Polynomials and Factoring . . . . . . . . . . . . . . . . . . . . . . . . . 784 Adding, Subtracting, and Multiplying Polynomials ~ Special Products ~ Factoring Polynomials Using Special Products ~ Factoring Trinomials ~ Factoring by Grouping

A.3 Fractional Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791 Domain of an Algebraic Expression ~ Reducing Rational Expressions ~ Operations with Rational Expressions ~ Compound Rational Expressions

APPENDIX B

Key Formulas B.1 Formulas from Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796 Exponents ~ Radicals and Rational Exponents ~ Special Products ~ Factoring Polynomials ~ Inequalities ~ Quadratic Formula ~ Logarithms ~ Determinants ~ Arithmetic Sequences and Series ~ Geometric Sequences and Series ~ Factorial ~ Binomial Coefficient ~ Binomial Theorem

B.2 Formulas from Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 797 Triangle ~ Trapezoid ~ Circle ~ Sector of Circle ~ Right Circular Cone ~ Right Circular Cylinder ~ Right Triangle ~ Parallelogram ~ Circular Ring ~ Ellipse ~ Cone ~ Sphere

B.3 Formulas from Trigonometry . . . . . . . . . . . . . . . . . . . . . . 797 Angular Measure ~ Reciprocal Identities ~ Quotient Identities ~ Pythagorean Identities ~ Odd-Even Identities ~ Sum and Difference Identities ~ Cofunction Identities ~ Double-Angle Identities ~ Power-Reducing Identities ~ Half-Angle Identities ~ Triangles ~ Trigonometric Form of a Complex Number ~ De Moivre’s Theorem

B.4 Formulas from Analytic Geometry . . . . . . . . . . . . . . . . . . 799 Basic Formulas ~ Equations of a Line ~ Equation of a Circle ~ Parabolas with Vertex (h, k) ~ Ellipses with Center (h, k) and a > b > 0 ~ Hyperbolas with Center (h, k)

B.5 Gallery of Basic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 800

Contents

xiii

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

APPENDIX C

11:28 AM

Page xiv

Logic C.1

Logic: An Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801 Statements ~ Compound Statements

C.2

Conditionals and Biconditionals . . . . . . . . . . . . . . . . . . . . 807 Forms of Statements ~ Valid Reasoning

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiv

Contents

814 816 833 935 939

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page xv

About the Authors Franklin D. Demana Frank Demana received his master’s degree in mathematics and his Ph.D. from Michigan State University. Currently, he is Professor Emeritus of Mathematics at The Ohio State University. As an active supporter of the use of technology to teach and learn mathematics, he is cofounder of the national Teachers Teaching with Technology (T3) professional development program. He has been the director and codirector of more than $10 million of National Science Foundation (NSF) and foundational grant activities. He is currently a co–principal investigator on a $3 million grant from the U.S. Department of Education Mathematics and Science Educational Research program awarded to The Ohio State University. Along with frequent presentations at professional meetings, he has published a variety of articles in the areas of computer- and calculator-enhanced mathematics instruction. Dr. Demana is also cofounder (with Bert Waits) of the annual International Conference on Technology in Collegiate Mathematics (ICTCM). He is co-recipient of the 1997 Glenn Gilbert National Leadership Award presented by the National Council of Supervisors of Mathematics, and co-recipient of the 1998 Christofferson-Fawcett Mathematics Education Award presented by the Ohio Council of Teachers of Mathematics. Dr. Demana coauthored Calculus: Graphical, Numerical, Algebraic; Essential Algebra: A Calculator Approach; Transition to College Mathematics; College Algebra and Trigonometry: A Graphing Approach; College Algebra: A Graphing Approach; Precalculus: Functions and Graphs; and Intermediate Algebra: A Graphing Approach.

Bert K. Waits Bert Waits received his Ph.D. from The Ohio State University and is currently Professor Emeritus of Mathematics there. Dr. Waits is cofounder of the national Teachers Teaching with Technology (T3) professional development program, and has been codirector or principal investigator on several large National Science Foundation projects. Dr. Waits has published articles in more than 50 nationally recognized professional journals. He frequently gives invited lectures, workshops, and minicourses at national meetings of the MAA and the National Council of Teachers of Mathematics (NCTM) on how to use computer technology to enhance the teaching and learning of mathematics. He has given invited presentations at the International Congress on Mathematical Education (ICME-6, -7, and -8) in Budapest (1988), Quebec (1992), and Seville (1996). Dr. Waits is co-recipient of the 1997 Glenn Gilbert National Leadership Award presented by the National Council of Supervisors of Mathematics, and is the cofounder (with Frank Demana) of the ICTCM. He is also co-recipient of the 1998 Christofferson-Fawcett Mathematics Education Award presented by the Ohio Council of Teachers of Mathematics. Dr. Waits coauthored Calculus: Graphical, Numerical, Algebraic; College Algebra and Trigonometry: A Graphing Approach; College Algebra: A Graphing Approach; Precalculus: Functions and Graphs; and Intermediate Algebra: A Graphing Approach.

About the Authors

xv

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page xvi

Gregory D. Foley Greg Foley received B.A. and M.A. degrees in mathematics and a Ph.D. in mathematics education from The University of Texas at Austin. He is the Robert L. Morton Professor of Mathematics Education at Ohio University. Foley has taught elementary arithmetic through graduate-level mathematics, as well as upper division and graduate-level mathematics education classes. He has held full-time faculty positions at North Harris County College, Austin Community College, The Ohio State University, Sam Houston State University, and Appalachian State University, and served as Director of the Liberal Arts and Science Academy and as Senior Scientist for Secondary School Mathematics Improvement for the Austin Independent School District in Austin, Texas. Dr. Foley has presented over 250 lectures, workshops, and institutes throughout the United States and internationally, has directed or codirected more than 40 funded projects totaling some $5 million, and has published over 30 scholarly works. In 1998, he received the biennial American Mathematical Association of Two-Year Colleges (AMATYC) Award for Mathematics Excellence, and in 2005, the annual Teachers Teaching with Technology (T3) Leadership Award. Dr. Foley coauthored Precalculus: A Graphing Approach and Precalculus: Functions and Graphs.

Daniel Kennedy Dan Kennedy received his undergraduate degree from the College of the Holy Cross and his master’s degree and Ph.D. in mathematics from the University of North Carolina at Chapel Hill. Since 1973 he has taught mathematics at the Baylor School in Chattanooga, Tennessee, where he holds the Cartter Lupton Distinguished Professorship. Dr. Kennedy became an Advanced Placement Calculus reader in 1978, which led to an increasing level of involvement with the program as workshop consultant, table leader, and exam leader. He joined the Advanced Placement Calculus Test Development Committee in 1986, then in 1990 became the first high school teacher in 35 years to chair that committee. It was during his tenure as chair that the program moved to require graphing calculators and laid the early groundwork for the 1998 reform of the Advanced Placement Calculus curriculum. The author of the 1997 Teacher’s Guide—-AP* Calculus, Dr. Kennedy has conducted more than 50 workshops and institutes for high school calculus teachers. His articles on mathematics teaching have appeared in the Mathematics Teacher and the American Mathematical Monthly, and he is a frequent speaker on education reform at professional and civic meetings. Dr. Kennedy was named a Tandy Technology Scholar in 1992 and a Presidential Award winner in 1995. Dr. Kennedy coauthored Calculus: Graphical, Numerical, Algebraic; Prentice Hall Algebra I; Prentice Hall Geometry; and Prentice Hall Algebra 2.

xvi

About the Authors

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page xvii

Preface Our Approach The Rule of Four—A Balanced Approach A principal feature of this edition is the balance among the algebraic, numerical, graphical, and verbal methods of representing problems: the rule of four. For instance, we obtain solutions algebraically when that is the most appropriate technique to use, and we obtain solutions graphically or numerically when algebra is difficult to use. We urge students to solve problems by one method and then support or confirm their solutions by using another method. We believe that students must learn the value of each of these methods or representations and must learn to choose the one most appropriate for solving the particular problem under consideration. This approach reinforces the idea that to understand a problem fully, students need to understand it algebraically as well as graphically and numerically. Problem-Solving Approach Systematic problem solving is emphasized in the examples throughout the text, using the following variation of Polya’s problem-solving process: • understand the problem, • develop a mathematical model, • solve the mathematical model and support or confirm the solutions, and • interpret the solution. Students are encouraged to use this process throughout the text. Twelve Basic Functions Twelve basic functions are emphasized throughout the book as a major theme and focus. These functions are: • The Identity Function • The Natural Logarithm Function • The Squaring Function

• The Sine Function

• The Cubing Function

• The Cosine Function

• The Reciprocal Function

• The Absolute Value Function

• The Square Root Function

• The Greatest Integer Function

• The Exponential Function

• The Logistic Function

One of the most distinctive features of this textbook is that it introduces students to the full vocabulary of functions early in the course. Students meet the twelve basic functions graphically in Chapter 1 and are able to compare and contrast them as they learn about concepts like domain, range, symmetry, continuity, end behavior, asymptotes, extrema, and even periodicity—concepts that are difficult to appreciate when the only examples a teacher can refer to are polynomials. With this book, students are able to characterize functions by their behavior within the first month of classes. (For example, thanks to graphing technology, it is no longer necessary to understand radians before one can learn that the sine function is bounded, periodic, odd, and continuous, with domain 1- q, q2 and range 3- 1, 14.) Once students have a comfortable understanding of functions in general, the rest of the course consists of studying the various types of functions in greater depth, particularly with respect to their algebraic properties and modeling applications.

Preface

xvii

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page xviii

These functions are used to develop the fundamental analysis skills that are needed in calculus and advanced mathematics courses. Section 3.1 provides an overview of these functions by examining their graphs. A complete gallery of basic functions is included in Appendix B for easy reference. Each basic function is revisited later in the book with a deeper analysis that includes investigation of the algebraic properties. General characteristics of families of functions are also summarized.

268

CHAPTER 3 Exponential, Logistic, and Logarithmic Functions

Table 3.9 U.S. Population (in millions) Year

Population

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2007

76.2 92.2 106.0 123.2 132.2 151.3 179.3 203.3 226.5 248.7 281.4 301.6

Source: World Almanac and Book of Facts 2009.

EXAMPLE 6 Modeling U.S. Population Using Exponential Regression Use the 1900–2000 data in Table 3.9 and exponential regression to predict the U.S. population for 2007. Compare the result with the listed value for 2007. SOLUTION

Model Let P1t2 be the population (in millions) of the United States t years after 1900. Figure 3.15a shows a scatter plot of the data. Using exponential regression, we find a model for the 1990–2000 data: P1t2 = 80.5514 # 1.01289 t Figure 3.15b shows the scatter plot of the data with a graph of the population model just found. You can see that the curve fits the data fairly well. The coefficient of determination is r 2 L 0.995, indicating a close fit and supporting the visual evidence. Solve Graphically To predict the 2007 U.S. population we substitute t = 107 into the regression model. Figure 3.15c reports that P(1072 = 80.5514 # 1.01289 107 L 317.1. Interpret The model predicts the U.S. population was 317.1 million in 2007. The actual population was 301.6 million. We overestimated by 15.5 million, a 5.1% error. Now try Exercise 43.

Y1=80.5514*1.01289^X

X=107 [–10, 120] by [0, 400] (a)

[–10, 120] by [0, 400] (b)

Y=317.13007

Exponential Functions ƒ1x2 ! bx y

y

Domain: All reals Range:10, q2 Continuous No symmetry: neither even nor odd Bounded below, but not above No local extrema Horizontal asymptote: y = 0 No vertical asymptotes

f (x) = bx

f (x) = bx b> 1

0 < b< 1

(1, b) (0, 1)

(1, b)

(0, 1) x

x

If b 7 1 (see Figure 3.3a), then • ƒ is an increasing function, • lim ƒ1x2 = 0 and lim ƒ1x2 = q . x: -q

x: q

If 0 6 b 6 1 (see Figure 3.3b), then (a)

(b) x

FIGURE 3.3 Graphs of ƒ1x2 = b for (a) b 7 1 and (b) 0 6 b 6 1.

• ƒ is a decreasing function, • lim ƒ1x2 = q and lim ƒ1x2 = 0. x: -q

x: q

BASIC FUNCTION The Natural Logarithmic

Function

[–2, 6] by [–3, 3]

FIGURE 3.22

ƒ1x2 = ln x Domain: 10, q 2 Range: All reals Continuous on 10, q 2 Increasing on 10, q 2 No symmetry Not bounded above or below No local extrema No horizontal asymptotes Vertical asymptote: x = 0 End behavior: lim ln x = q x: q

Applications and Real Data The majority of the applications in the text are based on real data from cited sources, and their presentations are self-contained; students will not need any experience in the fields from which the applications are drawn. As they work through the applications, students are exposed to functions as mechanisms for modeling data and are motivated to learn about how various functions can help model real-life problems. They learn to analyze and model data, represent data graphically, interpret from graphs, and fit curves. Additionally, the tabular representation of data presented in this text highlights the concept that a function is a correspondence between numerical variables. This helps students build the connection between the numbers and graphs and recognize the importance of a full graphical, numerical, and algebraic understanding of a problem. For a complete listing of applications, please see the Applications Index on page 935.

[–10, 120] by [0, 400] (c)

FIGURE 3.15 Scatter plots and graphs for Example 6. The red “ + ” depicts the data point for 2007. The blue “x” in (c) represents the model’s prediction for 2007.

Technology and Exercises The authors of this textbook have encouraged the use of technology, particularly graphing calculator technology, in mathematics education for two decades. Longtime users of this textbook are well acquainted with our approach to problem solving (pages 69–70), which distinguishes between solving the problem and supporting or confirming the solution, and how technology figures into each of those processes. We have come to realize, however, that advances in technology and increased familiarity with calculators have gradually blurred some of the distinctions between solving and supporting that we had once assumed to be apparent. Textbook exercises that we had designed for a particular pedagogical purpose are now being solved with technology in ways that either circumvent or obscure the learning we had hoped might take place. For example, students will find an equation of the line through two points by using linear regression, or they will match a set of equations to their graphs by simply graphing each equation. Now

xviii

Preface

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page xix

that calculators with computer algebra have arrived on the scene, exercises meant for practicing algebraic manipulations are being solved without the benefit of the practice. We do not want to retreat in any way from our support of modern technology, but we feel that the time has come to provide more guidance about the intent of the various exercises in our textbook. Therefore, as a service to teachers and students alike, exercises in this textbook that should be solved without calculators will be identified with gray ovals around the exercise numbers. These will usually be exercises that demonstrate how various functions behave algebraically or how algebraic representations reflect graphical behavior and vice versa. Application problems will usually have no restrictions, in keeping with our emphasis on modeling and on bringing all representations to bear when confronting real-world problems. Incidentally, we continue to encourage the use of calculators to support answers graphically or numerically after the problems have been solved with pencil and paper. Any time students can make those connections among the graphical, analytical, and numerical representations, they are doing good mathematics. We just don’t want them to miss something along the way because they brought in their calculators too soon. As a final note, we will freely admit that different teachers use our textbook in different ways, and some will probably override our no-calculator recommendations to fit with their pedagogical strategies. In the end, the teachers know what is best for their students, and we are just here to help. That’s the kind of textbook authors we strive to be.

CHAPTER 3

Exponential, Logistic, and Logarithmic Functions 3.1 Exponential and Logistic

Functions

3.2 Exponential and Logistic

Modeling

3.3 Logarithmic Functions

and Their Graphs

3.4 Properties of Logarithmic

Functions

3.5 Equation Solving

and Modeling

3.6 Mathematics of Finance

The loudness of a sound we hear is based on the intensity of the associated sound wave. This sound intensity is the energy per unit time of the wave over a given area, measured in watts per square meter 1W/m22. The intensity is greatest near the source and decreases as you move away, whether the sound is rustling leaves or rock music. Because of the wide range of audible sound intensities, they are generally converted into decibels, which are based on logarithms. See pages 279–280.

Features Chapter Openers include a motivating photograph and a general description of an application that can be solved with the concepts learned in the chapter. The application is revisited later in the chapter via a specific problem that is solved. These problems enable students to explore realistic situations using graphical, numerical, and algebraic methods. Students are also asked to model problem situations using the functions studied in the chapter. In addition, the chapter sections are listed here. A Chapter Overview begins each chapter to give students a sense of what they are going to learn. This overview provides a roadmap of the chapter, as well as tells how the different topics in the chapter are connected under one big idea. It is always helpful to remember that mathematics isn’t modular, but interconnected, and that the skills and concepts learned throughout the course build on one another to help students understand more complicated processes and relationships.

Chapter Opener Problem (from page 251) Problem: How loud is a train inside a subway tunnel? Solution: Based on the data in Table 3.17, 251

b = = = =

10 log1I/I02 10 log110 -2/10 -122 10 log110 102 10 # 10 = 100

So the sound intensity level inside the subway tunnel is 100 dB.

Preface

xix

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page xx

9.9 Statistical Literacy What you’ll learn about • The Many Misuses of Statistics • Correlation revisited • The Importance of Randomness • Surveys and Observational Studies • Experimental Design • Using Randomness

The Many Misuses of Statistics Just as knowing a little bit about edible wild mushrooms can get you into trouble, so can knowing a little about statistics. This book has not ventured too far into the realm of inferential statistics, the methods of using statistics to draw conclusions about real-world phenomena, because that is rightfully another course. Unfortunately, a lack of true understanding does not stop people from misusing statistics every day to draw conclusions, many of them totally unjustified, and then inflicting those conclusions on you. We will therefore end this chapter with a brief “consumer’s guide” to the most common misuses of statistics.

Similarly, the What you’ll learn about ...and why feature presents the big ideas in each section and explains their purpose. Students should read this as they begin the section and always review it after they have completed the section to make sure they understand all of the key topics that have just been studied.

• Probability Simulations

... and why Statistical literacy is important in today’s data-driven world.

EXPLORATION 1

Test Your Statistical Savvy

Each one of the following scenarios contains at least one common misuse of statistics. How many can you catch? 1. A researcher reported finding a high correlation between aggression in chil-

dren and gender.

Vocabulary is highlighted in yellow for easy reference.

Common Logarithms—Base 10 Logarithms with base 10 are called common logarithms. Because of their connection to our base-ten number system, the metric system, and scientific notation, common logarithms are especially useful. We often drop the subscript of 10 for the base when using common logarithms. The common logarithmic function log10 x = log x is the inverse of the exponential function ƒ1x2 = 10 x. So y = log x

if and only if

10 y = x.

Applying this relationship, we can obtain other relationships for logarithms with base 10.

Basic Properties of Common Logarithms Let x and y be real numbers with x 7 0.

Properties are boxed in green so that they can be easily found.

• log 1 = 0 because 10 0 = 1. • log 10 = 1 because 10 1 = 10. • log 10 y = y because 10 y = 10 y. • 10 log x = x because log x = log x.

Each example ends with a suggestion to Now Try a related exercise. Working the suggested exercise is an easy way for students to check their comprehension of the material while reading each section, instead of waiting until the end of each section or chapter to see if they “got it.” In the Annotated Teacher’s Edition, various examples are marked for the teacher with the icon. Alternates are provided for these examples in the PowerPoint Slides. EXPLORATION 1

Test Your Statistical Savvy

Each one of the following scenarios contains at least one common misuse of statistics. How many can you catch? 1. A researcher reported finding a high correlation between aggression in chil-

dren and gender. 2. Based on a survey of shoppers at the city’s busiest mall on two consecutive

weekday afternoons, the mayor’s staff concluded that 68% of the voters would support his re-election. 3. A doctor recommended vanilla chewing gum to headache sufferers, noting that

he had tested it himself on 100 of his patients, 87 of whom reported feeling better within two hours. 4. A school system studied absenteeism in its secondary schools and found a neg-

ative correlation between student GPA and student absences. They concluded that absences cause a student’s grade to go down.

xx

Preface

Explorations appear throughout the text and provide students with the perfect opportunity to become active learners and to discover mathematics on their own. This will help hone critical thinking and problem-solving skills. Some are technologybased and others involve exploring mathematical ideas and connections. Margin Notes and Tips on various topics appear throughout the text. Tips offer practical advice on using the grapher to obtain the best, most accurate results. Margin notes include historical information and hints about examples, and provide additional insight to help students avoid common pitfalls and errors.

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page xxi

The Looking Ahead to Calculus icon is found throughout the text next to many examples and topics to point out concepts that students will encounter again in calculus. Ideas that foreshadow calculus, such as limits, maximum and minimum, asymptotes, and continuity, are highlighted. Early in the text, the idea of the limit, using an intuitive and conceptual approach, is introduced. Some calculus notation and language is introduced in the early chapters and used throughout the text to establish familiarity. The Web/Real Data

CHAPTER 3 Key Ideas Properties, Theorems, and Formulas Exponential Growth and Decay 254 Exponential Functions ƒ1x2 = bx 255 Exponential Functions and the Base e 257 Exponential Population Model 265 Changing Between Logarithmic and Exponential Form 274 Basic Properties of Logarithms 274 Basic Properties of Common Logarithms 276 Basic Properties of Natural Logarithms 277 Properties of Logarithms 283 Change-of-Base Formula for Logarithms 285 Logarithmic Functions ƒ1x2 = logb x, with b 7 1 287 One-to-One Properties 292 Newton’s Law of Cooling 296 Interest Compounded Annually 304 Interest Compounded k Times per Year 304, 306 Interest Compounded Continuously 306 Future Value of an Annuity 308 Present Value of an Annuity 309

Procedures Re-expression of Data 287–288 Logarithmic Re-expression of Data 298–300

Gallery of Functions Basic Logistic

Exponential

[–4, 4] by [–1, 5]

ƒ1x2 = e x

[–4.7, 4.7] by [–0.5, 1.5]

1 1 + e -x

ƒ1x2 =

Natural Logarithmic

[–2, 6] by [–3, 3]

ƒ1x2 = ln x

CHAPTER 3

1

icon marks the examples and exercises that use real cited data.

The Chapter Review material at the end of each chapter consists of sections dedicated to helping students review the chapter concepts. Key Ideas are broken into parts: Properties, Theorems, and Formulas; Procedures; and Gallery of Functions. The Review Exercises represent the full range of exercises covered in the chapter and give additional practice with the ideas developed in the chapter. The exercises with red numbers indicate problems that would make up a good chapter test. Chapter Projects conclude each chapter and require students to analyze data. They can be assigned as either individual or group work. Each project expands upon concepts and ideas taught in the chapter, and many projects refer to the Web for further investigation of real data.

CHAPTER 3 Review Exercises

Project

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator.

When a ball bounces up and down on a flat surface, the maximum height of the ball decreases with each bounce. Each rebound is a percentage of the previous height. For most balls, the percentage is a constant. In this project, you will use a motion detection device to collect height data for a ball bouncing underneath a motion detector, then find a mathematical model that describes the maximum bounce height as a function of bounce number.

Collecting the Data

Height (ft)

Analyzing a Bouncing Ball

Time (sec) [0, 4.25] by [0, 3]

Bounce Number 0 1 2 3 4 5

Set up the Calculator Based Laboratory (CBL™) system with a motion detector or a Calculator Based Ranger (CBR™) system to collect ball bounce data using a ball bounce program for the CBL or the Ball Bounce Application for the CBR. See the CBL/CBR guidebook for specific setup instruction. Hold the ball at least 2 feet below the detector and release it so that it bounces straight up and down beneath the detector. These programs convert distance versus time data to height from the ground versus time. The graph shows a plot of sample data collected with a racquetball and CBR. The data table below shows each maximum height collected.

The collection of exercises marked in red could be used as a chapter test.

Maximum Height (feet) 2.7188 2.1426 1.6565 1.2640 0.98309 0.77783

In Exercises 1 and 2, compute the exact value of the function for the given x-value without using a calculator.

23. Initial value = 12, limit to growth = 30, passing through 12, 202. In Exercises 25 and 26, determine a formula for the logistic function whose graph is shown in the figure.

In Exercises 3 and 4, determine a formula for the exponential function whose graph is shown in the figure. y

In Exercises 23 and 24, find the logistic function that satisfies the given conditions.

24. Initial height = 6, limit to growth = 20, passing through 13, 152.

1 1. ƒ1x2 = - 3 # 4x for x = 3 3 x # 2. ƒ1x2 = 6 3 for x = 2

3.

21. Initial height = 18 cm, doubling every 3 weeks 22. Initial mass = 117 g, halving once every 262 hours

25.

(0, 5)

x

7. ƒ1x2 = - 8-x - 3 9. ƒ1x2 = e2x - 3

6. ƒ1x2 = - 4-x 8. ƒ1x2 = 8-x + 3 10. ƒ1x2 = e3x - 4

100 5 + 3e -0.05x

12. ƒ1x2 =

50 5 + 2e -0.04x

14. ƒ1x2 = 215x - 32 + 1

In Exercises 15–18, graph the function, and analyze it for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. 15. ƒ1x2 = e 17. ƒ1x2 =

3-x

+ 1

6 1 + 3 # 0.4x

x+1

16. g1x2 = 314 18. g1x2 =

In Exercises 27–30, evaluate the logarithmic expression without using a calculator. 27. log2 32 29. log 2 3 10

2 - 2

100

4 + 2e -0.01x

In Exercises 19–22, find the exponential function that satisfies the given conditions. 19. Initial value = 24, increasing at a rate of 5.3% per day 20. Initial population = 67,000, increasing at a rate of 1.67% per year

28. log3 81 1 30. ln 2e7

In Exercises 31–34, rewrite the equation in exponential form. 31. log3 x = 5

In Exercises 13 and 14, state whether the function is an exponential growth function or an exponential decay function, and describe its end behavior using limits. 13. ƒ1x2 = e4 - x + 2

x

x

In Exercises 11 and 12, find the y-intercept and the horizontal asymptotes. 11. ƒ1x2 =

(0, 11)

(5, 22)

x

In Exercises 5–10, describe how to transform the graph of ƒ into the graph of g1x2 = 2x or h1x2 = ex. Sketch the graph by hand and support your answer with a grapher. 5. ƒ1x2 = 4-x + 3

y = 44

(3, 10)

(3, 1)

(0, 2)

1. If you collected motion data using a CBL or CBR, a plot

of height versus time should be shown on your graphing calculator or computer screen. Trace to the maximum height for each bounce and record your data in a table

y

y = 20

(2, 6) (0, 3)

Explorations

26.

y

y

4.

33. ln

x = -2 y

32. log2 x = y 34. log

a = -3 b

In Exercises 35–38, describe how to transform the graph of y = log2 x into the graph of the given function. Sketch the graph by hand and support with a grapher. 35. ƒ1x2 = log2 1x + 42 36. g1x2 = log2 14 - x2

37. h1x2 = - log2 1x - 12 + 2 38. h1x2 = - log2 1x + 12 + 4

In Exercises 39–42, graph the function, and analyze it for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. 40. ƒ1x2 = x 2 ln x ln x 42. ƒ1x2 = x In Exercises 43–54, solve the equation. 39. ƒ1x2 = x ln x

41. ƒ1x2 = x 2 ln ƒ x ƒ 43. 10 x = 4

44. ex = 0.25

45. 1.05x = 3

46. ln x = 5.4

Preface

xxi

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page xxii

Exercise Sets Each exercise set begins with a Quick Review to help students review skills needed in the exercise set, thus reminding them again that mathematics is not modular. There are also directions that give a section to go to for help so that students are prepared to do the Section Exercises. Some exercises are also designed to be solved without a calculator; the numbers of these exercises are printed within a gray oval. Students are urged to support the answers to these (and all) exercises graphically or numerically, but only after they have solved them with pencil and paper. Real-world application problems will rarely be designated with gray ovals.

SECTION 3.5 EXERCISES In Exercises 1–10, find the exact solution algebraically, and check it by substituting into the original equation. 1 x/5 1. 36a b = 4 3 3. 2 # 5x/4 = 250

1 x/3 2. 32 a b = 2 4

5. 2110 -x/32 = 20

6. 315-x/42 = 15

7. log x = 4

9. log4 1x - 52 = - 1

4. 3 # 4x/2 = 96 8. log2 x = 5

10. log4 11 - x2 = 1

In Exercises 11–18, solve each equation algebraically. Obtain a numerical approximation for your solution and check it by substituting into the original equation. 11. 1.06x = 4.1

12. 0.98x = 1.6

13. 50e0.035x = 200

14. 80e0.045x = 240

15. 3 + 2e -x = 6

16. 7 - 3e -x = 2

17. 3 ln (x - 3) + 4 = 5

18. 3 - log 1x + 22 = 5

In Exercises 19–24, state the domain of each function. Then match the function with its graph. (Each graph shown has a window of 3 -4.7, 4.74 by 3 - 3.1, 3.14). 19. ƒ1x2 = log 3x1x + 124

20. g1x2 = log x + log 1x + 12

23. ƒ1x2 = 2 ln x

24. g1x2 = ln x 2

21. ƒ1x2 = ln

x x + 1

(a)

22. g1x2 = ln x - ln 1x + 12

(b)

27. log x 4 = 2

28. ln x 6 = 12

2x - 2-x 2x + 2-x = 4 = 3 30. 3 2 ex + e -x = 4 31. 32. 2e2x + 5ex - 3 = 0 2 500 400 33. 34. = 200 = 150 1 + 25e0.3x 1 + 95e -0.6x 1 35. ln 1x + 32 - ln x = 0 2 1 36. log x - log 1x + 42 = 1 2 29.

37. ln 1x - 32 + ln 1x + 42 = 3 ln 2

38. log 1x - 22 + log 1x + 52 = 2 log 3

In Exercises 39–44, determine by how many orders of magnitude the quantities differ. 39. A $100 bill and a dime 40. A canary weighing 20 g and a hen weighing 2 kg 41. An earthquake rated 7 on the Richter scale and one rated 5.5 42. Lemon juice with pH = 2.3 and beer with pH = 4.1 43. The sound intensities of a riveter at 95 dB and ordinary conversation at 65 dB 44. The sound intensities of city traffic at 70 dB and rustling leaves at 10 dB

QUICK REVIEW 3.5

(For help, go to Sections P.1 and 1.4.)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator.

6. An atomic nucleus has a diameter of about 0.000000000000001 m.

In Exercises 1–4, prove that each function in the given pair is the inverse of the other.

In Exercises 7 and 8, write the number in decimal form.

1. ƒ1x2 = e2x and g1x2 = ln 1x 1/22

2. ƒ1x2 = 10 x/2 and g1x2 = log x 2, x 7 0 3. ƒ1x2 = 11/3) ln x and g1x2 = e3x

4. ƒ1x2 = 3 log x 2, x 7 0 and g1x2 = 10 x/6 In Exercises 5 and 6, write the number in scientific notation. 5. The mean distance from Jupiter to the Sun is about 778,300,000 km.

7. Avogadro’s number is about 6.02 * 10 23. 8. The atomic mass unit is about 1.66 * 10 -27 kg. In Exercises 9 and 10, use scientific notation to simplify the expression; leave your answer in scientific notation. 9. 1186,0002131,000,0002

10.

0.0000008 0.000005

There are over 6000 exercises, including 680 Quick Review Exercises. Following the Quick Review are exercises that allow practice on the algebraic skills learned in that section. These exercises have been carefully graded from routine to challenging. The following types of skills are tested in each exercise set: • Algebraic and analytic manipulation • Connecting algebra to geometry • Interpretation of graphs • Graphical and numerical representations of functions • Data analysis

45. Comparing Earthquakes How many times more severe was the 1978 Mexico City earthquake 1R = 7.92 than the 1994 Los Angeles earthquake 1R = 6.62?

46. Comparing Earthquakes How many times more severe was the 1995 Kobe, Japan, earthquake 1R = 7.22 than the 1994 Los Angeles earthquake 1R = 6.62? 47. Chemical Acidity The pH of carbonated water is 3.9 and the pH of household ammonia is 11.9. (a) What are their hydrogen-ion concentrations? (b) How many times greater is the hydrogen-ion concentration of carbonated water than that of ammonia? (c) By how many orders of magnitude do the concentrations differ?

(c)

(d)

48. Chemical Acidity Stomach acid has a pH of about 2.0, and blood has a pH of 7.4. (a) What are their hydrogen-ion concentrations? (b) How many times greater is the hydrogen-ion concentration of stomach acid than that of blood?

(e)

(f)

In Exercises 25–38, solve each equation by the method of your choice. Support your solution by a second method. 25. log x 2 = 6 26. ln x 2 = 4

(c) By how many orders of magnitude do the concentrations differ? 49. Newton’s Law of Cooling A cup of coffee has cooled from 92°C to 50°C after 12 min in a room at 22°C. How long will the cup take to cool to 30°C? 50. Newton’s Law of Cooling A cake is removed from an oven at 350°F and cools to 120°F after 20 min in a room at 65°F. How long will the cake take to cool to 90°F?

Also included in the exercise sets are thought-provoking exercises: • Standardized Test Questions include two true-false problems with justifications and four multiple-choice questions.

Standardized Test Questions 59. True or False The order of magnitude of a positive number is its natural logarithm. Justify your answer. 60. True or False According to Newton’s Law of Cooling, an object will approach the temperature of the medium that surrounds it. Justify your answer. In Exercises 61–64, solve the problem without using a calculator. 61. Multiple Choice Solve 23x - 1 = 32. (A) x = 1

(B) x = 2

(D) x = 11

(E) x = 13

(C) x = 4

62. Multiple Choice Solve ln x = - 1.

xxii

Preface

(A) x = - 1

(B) x = 1/e

(D) x = e

(E) No solution is possible.

(C) x = 1

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page xxiii

SECTION 3.5

63. Multiple Choice How many times more severe was the 2001 earthquake in Arequipa, Peru 1R1 = 8.12, than the 1998 double earthquake in Takhar province, Afghanistan 1R2 = 6.12? (A) 2

(B) 6.1

(D) 14.2

(E) 100

64. Multiple Choice

Equation Solving and Modeling

303

(a) Graph ƒ for c = 1 and k = 0.1, 0.5, 1, 2, 10. Explain the effect of changing k. (b) Graph ƒ for k = 1 and c = 0.1, 0.5, 1, 2, 10. Explain the effect of changing c.

(C) 8.1

Newton’s Law of Cooling is

(A) an exponential model.

(B) a linear model.

(C) a logarithmic model.

(D) a logistic model.

Extending the Ideas 68. Writing to Learn Prove if u/v = 10 n for u 7 0 and v 7 0, then log u - log v = n. Explain how this result relates to powers of ten and orders of magnitude. 69. Potential Energy The potential energy E (the energy stored for use at a later time) between two ions in a certain molecular structure is modeled by the function

(E) a power model.

Explorations In Exercises 65 and 66, use the data in Table 3.26. Determine whether a linear, logarithmic, exponential, power, or logistic regression equation is the best model for the data. Explain your choice. Support your writing with tables and graphs as needed.

Table 3.26 Populations of Two U.S. States (in thousands) Year

Alaska

Hawaii

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

63.6 64.4 55.0 59.2 72.5 128.6 226.2 302.6 401.9 550.0 626.9

154 192 256 368 423 500 633 770 965 1108 1212

Source: U.S. Census Bureau.

E = -

5.6 + 10e -r/3 r

where r is the distance separating the nuclei. (a) Writing to Learn Graph this function in the window 3-10, 104 by 3- 10, 304, and explain which portion of the graph does not represent this potential energy situation. (b) Identify a viewing window that shows that portion of the graph (with r … 10) which represents this situation, and find the maximum value for E. 70. In Example 8, the Newton’s Law of Cooling model was T1t2 - Tm = 1T0 - Tm2e -kt = 61.656 * 0.92770 t.

• Explorations are opportunities for students to discover mathematics on their own or in groups. These exercises often require the use of critical thinking to explore the ideas. • Writing to Learn exercises give students practice at communicating about mathematics and opportunities to demonstrate understanding of important ideas. • Group Activity exercises ask students to work on the problems in groups or solve them as individual or group projects. • Extending the Ideas exercises go beyond what is presented in the textbook. These exercises are challenging extensions of the book’s material.

Determine the value of k.

71. Justify the conclusion made about natural logarithmic regression on page 299. 72. Justify the conclusion made about power regression on page 299.

This variety of exercises provides sufficient flexibility to emphasize the skills most needed for each student or class.

In Exercises 73–78, solve the equation or inequality. 73. ex + x = 5 74. e2x - 8x + 1 = 0

65. Writing to Learn Modeling Population Which regression equation is the best model for Alaska’s population?

75. ex 6 5 + ln x

66. Writing to Learn Modeling Population Which regression equation is the best model for Hawaii’s population?

77. 2 log x - 4 log 3 7 0

67. Group Activity Normal Distribution The function

76. ln ƒ x ƒ - e2x Ú 3 78. 2 log 1x + 12 - 2 log 6 6 0

ƒ1x2 = k # e -cx , 2

where c and k are positive constants, is a bell-shaped curve that is useful in probability and statistics.

Content Changes to This Edition Mindful of the need to keep the applications of mathematics relevant to our students, we have changed many of the examples and exercises throughout the book to include the most current data available to us at the time of publication. We also looked carefully at the pedagogy of each section and added features to clarify (for students and teachers) where technology might interfere with the intended learning experience. In some cases (as with the section on solving simultaneous linear equations), this led us to reconsider how some topics were introduced. We hope that the current edition retains our commitment to graphical, numerical, and algebraic representations, while reviving some of the algebraic emphasis that we never intended to lose. Chapter P The example on scientific notation was improved to further emphasize its advantage in mental arithmetic. Chapter 1 The chapter opener on the consumer price index for housing was updated to include a real-world caution against extrapolation, exemplified by the mortgage meltdown of 2008. The section on grapher failure was updated to reflect the changing capabilities of the technology. Limit notation was introduced a little more carefully (although still quite informally).

Preface

xxiii

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page xxiv

Chapter 2 The introduction of linear correlation was revised to include a caution against unwarranted conclusions, a theme continued in the new Section 9.9, “Statistical Literacy.” A clarification of approximate answers was added. Data problems were updated throughout. Chapter 3 Some real-world applications were given supporting margin notes. The introduction of logarithmic functions was improved. The section on financial mathematics was updated to include an introduction to the Finance menu on the graphing calculator. Chapter 4 Throughout the chapter, the pedagogical focus was clarified so that students would know when to practice trigonometric calculations without a calculator and why. A margin note was added to explain the display of radical fractions. Chapter 5 Some exercises were modified so that they could be solved algebraically (when that was the intent of the exercise). Data tables were updated throughout the chapter, including the chapter project on the illumination of the moon, for which we used data for the year after the book will be published . . . because it was available. Chapter 6 Examples and exercises were altered to include mental estimation and to specify noncalculator solutions where trigonometric skills with the special angles could be practiced. Chapter 7 Section 7.1 (“Solving Systems of Two Equations”) was restructured to devote a little more attention to the algebraic means of solution in preparation for the matrix methods to follow. Chapter 8 Examples and exercises were clarified and updated to include (for example) the new classification of planets in the solar system. Chapter 9 An entire section, “Statistical Literacy,” has been added, expanding our introduction to statistics to three sections. Although statistical topics are not usually part of a classical “precalculus” course, their importance in today’s world has led many states to require that they be part of the curriculum, so we include them in our book as a service to our readers. Sections 9.7 and 9.8 deal with the mathematics used in descriptive statistics and data analysis, while Section 9.9 deals more with how statistics are used (or misused) in applications. Even if students read this new section on their own, it should make them more savvy consumers. Chapter 10 We have retained the balance of the last edition, which added an enhanced limit section to our precalculus-level introduction to the two central problems of the calculus: the tangent line problem and the area problem. Our intention is to “set the scene” for calculus, not to cover the first two weeks of the course.

xxiv

Preface

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page xxv

Technology Resources The following supplements are available for purchase: MathXL® for School (optional, for purchase only—access code required), www.MathXLforSchool.com MathXL® for School is a powerful online homework, tutorial, and assessment program designed specifically for Pearson Education mathematics textbooks. With MathXL for School, students: • Do their homework and receive immediate feedback • Get self-paced assistance on problems through interactive learning aids (guided solutions, step-by-step examples, video clips, animations) • Have a large number of practice problems to choose from—helping them master a topic • Receive personalized study plans based on quiz and test results With MathXL for School, teachers: • Quickly and easily create quizzes, tests, and homework assignments • Utilize automatic grading to rapidly assess student understanding • Prepare students for high-stakes testing • Deliver quality instruction regardless of experience level The new Flash-based, platform- and browser-independent MathXL Player now supports Firefox on Windows (XP and Vista), Safari and Firefox on Macintosh, as well as Internet Explorer. For more information, visit our Web site at www.MathXLforSchool.com, or contact your Pearson sales representative. MathXL® Tutorials on CD This interactive tutorial CD-ROM provides algorithmically generated practice exercises that are correlated at the chapter, section, and objective level to the exercises in the textbook. Every practice exercise is accompanied by an example and a guided solution designed to involve students in the solution process. Selected exercises may also include a video clip to help students visualize concepts. The software provides helpful feedback for incorrect answers and can generate printed summaries of students’ progress. It is available for purchase separately, using ISBN-13: 978-0-13-137636-6; ISBN-10: 0-13-137636-5.

Preface

xxv

6965_Demana_SE_FM_ppi-xxx.qxd

2/2/10

11:18 AM

Page xxvi

Additional Teacher Resources Most of the teacher supplements and resources available for this text are available electronically for download at the Instructor Resource Center (IRC). Please go to www.PearsonSchool.com/Access_Request and select “access to online instructor resources.” You will be required to complete a one-time registration subject to verification before being emailed access information for download materials. The following supplements are available to qualified adopters: Annotated Teacher’s Edition • Provides answers in the margins next to the corresponding problem for almost all exercises, including sample answers for writing exercises. • Various examples marked with the provided in the PowerPoint Slides.

icon indicate that alternative examples are

• Provides notes written specifically for the teacher. These notes include chapter and section objectives, suggested assignments, lesson guides, and teaching tips. • ISBN-13: 978-0-13-136907-8; ISBN-10: 0-13-136907-5 Solutions Manual Provides complete solutions to all exercises, including Quick Reviews, Exercises, Explorations, and Chapter Reviews. ISBN-13: 978-0-13-137641-0; ISBN-10: 0-13-137641-1 Online Resource Manual (Download Only) Provides Major Concepts Review, Group Activity Worksheets, Sample Chapter Tests, Standardized Test Preparation Questions, Contest Problems. Online Tests and Quizzes (Download Only) Provides two parallel tests per chapter, two quizzes for every three to four sections, two parallel midterm tests covering Chapters P–5, and two parallel end-of-year tests, covering Chapters 6–10. TestGen® TestGen enables teachers to build, edit, print, and administer tests using a computerized bank of questions developed to cover all the objectives of the text. TestGen is algorithmically based, allowing teachers to create multiple but equivalent versions of the same question or test with the click of a button. Teachers can also modify test bank questions or add new questions. Tests can be printed or administered online. ISBN-13: 978-0-13-137640-3; ISBN-10: 0-13-137640-3 PowerPoint Slides Features presentations written and designed specifically for this text, including figures, alternate examples, definitions, and key concepts. Web Site Our Web site, www.awl.com/demana, provides dynamic resources for teachers and students. Some of the resources include TI graphing calculator downloads, online quizzing, teaching tips, study tips, Explorations, end-of-chapter projects, and more.

xxvi

Preface

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page xxvii

Acknowledgments We wish to express our gratitude to the reviewers of this and previous editions who provided such invaluable insight and comment. Judy Ackerman Montgomery College

Daniel Harned Michigan State University

Ignacio Alarcon Santa Barbara City College

Vahack Haroutunian Fresno City College

Ray Barton Olympus High School

Celeste Hernandez Richland College

Nicholas G. Belloit Florida Community College at Jacksonville

Rich Hoelter Raritan Valley Community College

Margaret A. Blumberg University of Southwestern Louisiana

Dwight H. Horan Wentworth Institute of Technology

Ray Cannon Baylor University

Margaret Hovde Grossmont College

Marilyn P. Carlson Arizona State University

Miles Hubbard Saint Cloud State University

Edward Champy Northern Essex Community College

Sally Jackman Richland College

Janis M. Cimperman Saint Cloud State University

T. J. Johnson Hendrickson High School

Wil Clarke La Sierra University

Stephen C. King University of South Carolina—Aiken

Marilyn Cobb Lake Travis High School

Jeanne Kirk William Howard Taft High School

Donna Costello Plano Senior High School

Georgianna Klein Grand Valley State University

Gerry Cox Lake Michigan College

Deborah L. Kruschwitz-List University of Wisconsin—Stout

Deborah A. Crocker Appalachian State University

Carlton A. Lane Hillsborough Community College

Marian J. Ellison University of Wisconsin—Stout

James Larson Lake Michigan University

Donna H. Foss University of Central Arkansas

Edward D. Laughbaum Columbus State Community College

Betty Givan Eastern Kentucky University

Ron Marshall Western Carolina University

Brian Gray Howard Community College

Janet Martin Lubbock High School

Acknowledgments

xxvii

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page xxviii

Beverly K. Michael University of Pittsburgh

Mary Margaret Shoaf-Grubbs College of New Rochelle

Paul Mlakar St. Mark’s School of Texas

Malcolm Soule California State University, Northridge

John W. Petro Western Michigan University

Sandy Spears Jefferson Community College

Cynthia M. Piez University of Idaho

Shirley R. Stavros Saint Cloud State University

Debra Poese Montgomery College

Stuart Thomas University of Oregon

Jack Porter University of Kansas

Janina Udrys Schoolcraft College

Antonio R. Quesada The University of Akron

Mary Voxman University of Idaho

Hilary Risser Plano West Senior High

Eddie Warren University of Texas at Arlington

Thomas H. Rousseau Siena College

Steven J. Wilson Johnson County Community College

David K. Ruch Sam Houston State University

Gordon Woodward University of Nebraska

Sid Saks Cuyahoga Community College

Cathleen Zucco-Teveloff Trinity College

Consultants We would like to extend a special thank you to the following consultants for their guidance and invaluable insight in the development of this edition. Dave Bock Cornell University

Laura Reddington Forest Hill High School, Florida

Jane Nordquist Ida S. Baker High School, Florida

James Timmons Heide Trask High School, North Carolina

Sudeepa Pathak Williamston High School, North Carolina

Jill Weitz The G-Star School of the Arts, Florida

We express special thanks to Chris Brueningsen, Linda Antinone, and Bill Bower for their work on the Chapter Projects. We would also like to thank Frank Purcell and David Alger for their meticulous accuracy checking of the text. We are grateful to Nesbitt Graphics, who pulled off an amazing job on composition and proofreading, and specifically to Joanne Boehme and Harry Druding for expertly managing the entire production process. Finally, our thanks as well are extended to the professional and remarkable staff at Pearson AddisonWesley, for their advice and support in revising this text, particularly Anne Kelly, Becky Anderson, Katherine Greig, Greg Tobin, Rich Williams, Joanne Dill, Karen Wernholm, Peggy McMahon, Christina Gleason, Carol Melville, Katherine Minton, Sarah Gibbons, and

xxviii

Acknowledgments

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page xxix

Carl Cottrell. Particular recognition is due Elka Block, who tirelessly helped us through the development and production of this book. —F. D. D. —B. K. W. —G. D. F. —D. K.

Acknowledgments

xxix

6965_Demana_SE_FM_ppi-xxx.qxd

1/25/10

11:28 AM

Page xxx

Credits Photographs Page 1, © European Southern Observatory; Page 8, © PhotoDisc; Page 27, © PhotoDisc; Page 33, © DAJ (Getty Royalty Free); Page 38, © Image Source/Getty Royalty Free; Page 63, © Shutterstock; Page 65, © Great American Stock; Page 108, © PhotoDisc Red; Page 117, © Corbis; Page 138, © PhotoDisc; Page 157, © iStockphoto; Page 162, © Photos.com; Page 179, © NASA; Page 244, © BrandX Pictures; Page 251, © iStockphoto; Page 261, © PhotoDisc; Page 270, © PhotoDisc; Page 281, © NASA; Page 288, © Corbis; Page 291, © PhotoDisc; Page 295, © PhotoDisc; Page 312, © Digital Vision; Page 319, © Shutterstock; Page 356, © iStockphoto; Page 368, © BrandX Pictures; Page 376, © Stockdisc Premium Royalty Free; Page 403, © iStockphoto; Page 412, © Digital Vision; Page 427, © NASA; Page 450, © Digital Vision; Page 455, © Shutterstock; Page 462, © Digital Vision; Page 480, © Shutterstock and iStockphoto; Page 491, © PhotoDisc/Getty Royalty Free; Page 497, © Getty Royalty Free; Page 519, © Shutterstock; Page 521, © Getty Royalty Free; Page 524, © Getty Royalty Free; Page 531, © Getty Royalty Free; Page 555, © iStockphoto; Page 571, © Getty Royalty Free; Page 576, © Getty Royalty Free; Page 579, © Shutterstock; Page 588, © Tony Roberts/Corbis; Page 607, © NASA; Page 626, © National Optical Astronomy Observatories; Page 639, Photo by Jill Britton; table by Dan Bergerud, Camosun College, Victoria, BC; Page 641, © Shutterstock; Page 649, © Shutterstock; Page 663, © PhotoDisc/ Getty Royalty Free; Page 676, © PhotoDisc/Getty Royalty Free; Page 702, © Photos.com; Page 716, © Shutterstock; Page 735, © iStockphoto; Page 744, © iStockphoto; Page 752, © iStockphoto; Page 765, © PhotoDisc; Page 768, © iStockphoto

xxx

Credits

6965_CH0P_pp001-062.qxd

1/14/10

12:43 PM

Page 1

CHAPTER P

Prerequisites P.1 Real Numbers P.2 Cartesian Coordinate

System

P.3 Linear Equations and

Inequalities

P.4 Lines in the Plane P.5 Solving Equations

Graphically, Numerically, and Algebraically

P.6 Complex Numbers P.7 Solving Inequalities Alge-

braically and Graphically

Large distances are measured in light years, the distance light travels in one year. Astronomers use the speed of light, approximately 186,000 miles per second, to approximate distances between planets. See page 35 for examples.

1

6965_CH0P_pp001-062.qxd

2

1/14/10

12:43 PM

Page 2

CHAPTER P Prerequisites

Bibliography

Chapter P Overview

For students: Great Jobs for Math Majors, Stephen Lambert, Ruth J. DeCotis. Mathematical Association of America, 1998. For teachers: Algebra in a Technological World, Addenda Series, Grades 9–12. National Council of Teachers of Mathematics, 1995. Why Numbers Count—Quantitative Literacy for Tommorrow’s America, Lynn Arthur Steen (Ed.). National Council of Teachers of Mathematics, 1997.

Historically, algebra was used to represent problems with symbols (algebraic models) and solve them by reducing the solution to algebraic manipulation of symbols. This technique is still important today. Graphing calculators are used today to approach problems by representing them with graphs (graphical models) and solve them with numerical and graphical techniques of the technology. We begin with basic properties of real numbers and introduce absolute value, distance formulas, midpoint formulas, and equations of circles. Slope of a line is used to write standard equations for lines, and applications involving linear equations are discussed. Equations and inequalities are solved using both algebraic and graphical techniques.

P.1 Real Numbers What you’ll learn about • Representing Real Numbers • Order and Interval Notation • Basic Properties of Algebra • Integer Exponents

Representing Real Numbers A real number is any number that can be written as a decimal. Real numbers 3 16, e, are represented by symbols such as -8, 0, 1.75, 2.333 Á , 0.36, 8/5, 23, 2 and p. The set of real numbers contains several important subsets:

• Scientific Notation

The natural (or counting) numbers:

... and why

The whole numbers:

These topics are fundamental in the study of mathematics and science.

Objective Students will be able to convert between decimals and fractions, write inequalities, apply the basic properties of algebra, and work with exponents and scientific notation.

The integers:

51, 2, 3, Á 6

50, 1, 2, 3, Á 6

5 Á , -3, - 2, -1, 0, 1, 2, 3, Á 6

The braces 5 6 are used to enclose the elements, or objects, of the set. The rational numbers are another important subset of the real numbers. A rational number is any number that can be written as a ratio a/b of two integers, where b Z 0. We can use set-builder notation to describe the rational numbers: a e ` a, b are integers, and b Z 0 f b The vertical bar that follows a/b is read “such that.”

Motivate Ask students how real numbers can be classified. Have students discuss ways to display very large or very small numbers without using a lot of zeros.

The decimal form of a rational number either terminates like 7/4 = 1.75, or is infinitely repeating like 4/11 = 0.363636 Á = 0.36. The bar over the 36 indicates the block of digits that repeats. A real number is irrational if it is not rational. The decimal form of an irrational number is infinitely nonrepeating. For example, 23 = 1.7320508 Á and p = 3.14159265Á . Real numbers are approximated with calculators by giving a few of its digits. Sometimes we can find the decimal form of rational numbers with calculators, but not very often.

6965_CH0P_pp001-062.qxd

1/14/10

12:43 PM

Page 3

SECTION P.1

1/17 N

3

EXAMPLE 1 Examining Decimal Forms of Rational Numbers

1/16 55/27

Real Numbers

.0625 2.037037037 .0588235294

FIGURE P.1 Calculator decimal representations of 1/16, 55/27, and 1/17 with the calculator set in floating decimal mode. (Example 1)

Determine the decimal form of 1/16, 55/27, and 1/17. SOLUTION Figure P.1 suggests that the decimal form of 1/16 terminates and that

of 55/27 repeats in blocks of 037. 1 55 = 0.0625 = 2.037 and 16 27 We cannot predict the exact decimal form of 1/17 from Figure P.1; however, we can say that 1/17 L 0.0588235294. The symbol L is read “is approximately equal to.” We can use long division (see Exercise 66) to show that 1 = 0.0588235294117647. 17

Now try Exercise 3.

The real numbers and the points of a line can be matched one-to-one to form a real number line. We start with a horizontal line and match the real number zero with a point O, the origin. Positive numbers are assigned to the right of the origin, and negative numbers to the left, as shown in Figure P.2. – 3

π

O

–5 –4 –3 –2 –1 Negative real numbers

0

1

2 3 4 Positive real numbers

5

FIGURE P.2 The real number line. Every real number corresponds to one and only one point on the real number line, and every point on the real number line corresponds to one and only one real number. Between every pair of real numbers on the number line there are infinitely many more real numbers. The number associated with a point is the coordinate of the point. As long as the context is clear, we will follow the standard convention of using the real number for both the name of the point and its coordinate.

Order and Interval Notation The set of real numbers is ordered. This means that we can use inequalities to compare any two real numbers that are not equal and say that one is “less than” or “greater than” the other.

Order of Real Numbers Let a and b be any two real numbers. Symbol

Unordered Systems Not all number systems are ordered. For example, the complex number system, to be introduced in Section P.6, has no natural ordering.

a a a a

7 6 Ú …

b b b b

Definition

a a a a

-

b is positive b is negative b is positive or zero b is negative or zero

Read

a is greater than b a is less than b a is greater than or equal to b a is less than or equal to b

The symbols 7 , 6 , Ú , nd a … are inequality symbols.

6965_CH0P_pp001-062.qxd

4

1/14/10

12:43 PM

Page 4

CHAPTER P Prerequisites

Opposites and Number Line

a 6 0 Q -a 7 0 If a 6 0, then a is to the left of 0 on the real number line, and its opposite, - a, is to the right of 0. Thus, - a 7 0.

Geometrically, a 7 b means that a is to the right of b (equivalently b is to the left of a) on the real number line. For example, since 6 7 3, 6 is to the right of 3 on the real number line. Note also that a 7 0 means that a - 0, or simply a, is positive and a 6 0 means that a is negative. We are able to compare any two real numbers because of the following important property of the real numbers.

Trichotomy Property Let a and b be any two real numbers. Exactly one of the following is true: a 6 b,

a = b,

or

a 7 b.

Inequalities can be used to describe intervals of real numbers, as illustrated in Example 2.

EXAMPLE 2 Interpreting Inequalities x –3 –2 –1

0

1

2

3

4

5

SOLUTION

(a) x –3 –2 –1

0

1

2

3

4

5

(b) –0.5 x –5 –4 –3 –2 –1

0

1

2

3

2

3

4

5

(c) x –3 –2 –1

0

1

Describe and graph the interval of real numbers for the inequality. (a) x 6 3 (b) -1 6 x … 4

(d)

FIGURE P.3 In graphs of inequalities, parentheses correspond to 6 and 7 and brackets to … and Ú . (Examples 2 and 3)

(a) The inequality x 6 3 describes all real numbers less than 3 (Figure P.3a). (b) The double inequality - 1 6 x … 4 represents all real numbers between -1 and 4, excluding -1 and including 4 (Figure P.3b). Now try Exercise 5.

EXAMPLE 3 Writing Inequalities Write an interval of real numbers using an inequality and draw its graph. (a) The real numbers between -4 and - 0.5 (b) The real numbers greater than or equal to zero SOLUTION

(a) -4 6 x 6 - 0.5 (Figure P.3c) (b) x Ú 0 (Figure P.3d)

Now try Exercise 15.

As shown in Example 2, inequalities define intervals on the real number line. We often use 32, 54 to describe the bounded interval determined by 2 … x … 5. This interval is closed because it contains its endpoints 2 and 5. There are four types of bounded intervals.

Bounded Intervals of Real Numbers Let a and b be real numbers with a 6 b. Interval Notation

Interval Type

Inequality Notation

3a, b4

Closed

a … x … b

Open

a 6 x 6 b

3a, b2

Half-open

a … x 6 b

1a, b2 1a, b4

Half-open

a 6 x … b

Graph a

b

a

b

a

b

a

b

The numbers a and b are the endpoints of each interval.

6965_CH0P_pp001-062.qxd

1/14/10

12:43 PM

Page 5

SECTION P.1

Interval Notation at !ˆ Because - q is not a real number, we use 1 - q , 22 instead of 3- q , 22 to describe x 6 2. Similarly, we use 3-1, q 2 instead of 3- 1, q 4 to describe x Ú -1.

Real Numbers

5

The interval of real numbers determined by the inequality x 6 2 can be described by the unbounded interval 1 - q , 22. This interval is open because it does not contain its endpoint 2.

We use the interval notation 1 - q , q 2 to represent the entire set of real numbers. The symbols - q (negative infinity) and q (positive infinity) allow us to use interval notation for unbounded intervals and are not real numbers. There are four types of unbounded intervals.

Unbounded Intervals of Real Numbers Let a and b be real numbers. Interval Notation

Interval Type

Inequality Notation

3a, q 2

Closed

x Ú a

1a, q 2

1 - q , b4

1 - q , b2

Graph a

x 7 a

Open

a

x … b

Closed

b

x 6 b

Open

b

Each of these intervals has exactly one endpoint, namely a or b.

EXAMPLE 4 Converting Between Intervals and Inequalities Convert interval notation to inequality notation or vice versa. Find the endpoints and state whether the interval is bounded, its type, and graph the interval. (a) 3-6, 32 (b) 1 - q , -12 (c) -2 … x … 3 SOLUTION

(a) The interval 3-6, 32 corresponds to - 6 … x 6 3 and is bounded and half-open (see Figure P.4a). The endpoints are -6 and 3. (b) The interval 1 - q , -12 corresponds to x 6 - 1 and is unbounded and open (see Figure P.4b). The only endpoint is - 1. (c) The inequality -2 … x … 3 corresponds to the closed, bounded interval 3-2, 34 (see Figure P.4c). The endpoints are -2 and 3. Now try Exercise 29. (a)

x –6 –5 –4 –3 –2 –1

0

1

2

3

4

–5 –4 –3 –2 –1

0

1

2

3

4

5

–5 –4 –3 –2 –1

0

1

2

3

4

5

(b)

x

(c)

x

FIGURE P.4 Graphs of the intervals of real numbers in Example 4.

Basic Properties of Algebra Algebra involves the use of letters and other symbols to represent real numbers. A variable is a letter or symbol 1for example, x, y, t, u2 that represents an unspecified real number. A constant is a letter or symbol 1for example, - 2, 0, 23, p2 that represents a specific real number. An algebraic expression is a combination of variables and constants involving addition, subtraction, multiplication, division, powers, and roots.

6965_CH0P_pp001-062.qxd

6

1/14/10

12:43 PM

Page 6

CHAPTER P Prerequisites

We state some of the properties of the arithmetic operations of addition, subtraction, multiplication, and division, represented by the symbols +, - , * (or # ) and , (or /), respectively. Addition and multiplication are the primary operations. Subtraction and division are defined in terms of addition and multiplication. Subtraction: Subtraction vs. Negative Numbers On many calculators, there are two “ - ” keys, one for subtraction and one for negative numbers or opposites. Be sure you know how to use both keys correctly. Misuse can lead to incorrect results.

Division:

a - b = a + 1-b2 a 1 = aa b, b Z 0 b b

In the above definitions, - b is the additive inverse or opposite of b, and 1/b is the multiplicative inverse or reciprocal of b. Perhaps surprisingly, additive inverses are not always negative numbers. The additive inverse of 5 is the negative number -5. However, the additive inverse of -3 is the positive number 3. The following properties hold for real numbers, variables, and algebraic expressions.

Properties of Algebra Let u, v, and w be real numbers, variables, or algebraic expressions. 1. Commutative property Addition: u + v = v + u Multiplication: uv = vu

4. Inverse property Addition: u + 1-u2 = 0 1 Multiplication: u # = 1, u Z 0 u 5. Distributive property Multiplication over addition: u1v + w2 = uv + uw 1u + v2w = uw + vw

2. Associative property Addition: 1u + v2 + w = u + 1v + w2 Multiplication: 1uv2w = u1vw2 3. Identity property Addition: u + 0 = u Multiplication: u # 1 = u

Multiplication over subtraction: u1v - w2 = uv - uw 1u - v2w = uw - vw

The left-hand sides of the equations for the distributive property show the factored form of the algebraic expressions, and the right-hand sides show the expanded form.

EXAMPLE 5 Using the Distributive Property (a) Write the expanded form of 1a + 22x. (b) Write the factored form of 3y - by. SOLUTION

(a) 1a + 22x = ax + 2x (b) 3y - by = 13 - b2y

Now try Exercise 37.

Here are some properties of the additive inverse together with examples that help illustrate their meanings.

Properties of the Additive Inverse Let u and v be real numbers, variables, or algebraic expressions. Property

Example

1. 2. 3. 4. 5.

-1- 32 = 3 1-423 = 41-32 = - 14 # 32 = -12 1-621-72 = 6 # 7 = 42

-1-u2 = u 1- u2v = u1-v2 = - 1uv2 1- u21-v2 = uv

1-12u = - u -1u + v2 = 1- u2 + 1-v2

1-125 = - 5 -17 + 92 = 1-72 + 1-92 = -16

6965_CH0P_pp001-062.qxd

1/14/10

12:43 PM

Page 7

SECTION P.1

Real Numbers

7

Integer Exponents Exponential notation is used to shorten products of factors that repeat. For example, 1 -321-321-321-3) = 1- 324

and

12x + 1212x + 12 = 12x + 122.

Exponential Notation Let a be a real number, variable, or algebraic expression and n a positive integer. Then a n = a # a # Á # a, n factors where n is the exponent, a is the base, and a n is the nth power of a, read as “a to the nth power.”

The two exponential expressions in Example 6 have the same value but have different bases. Be sure you understand the difference. Understanding Notation

EXAMPLE 6 Identifying the Base

1-322 = 9

(a) In 1 -325, the base is - 3. (b) In - 35, the base is 3.

- 32 = - 9

Be careful!

Now try Exercise 43.

Here are the basic properties of exponents together with examples that help illustrate their meanings.

Properties of Exponents Let u and v be real numbers, variables, or algebraic expressions and m and n be integers. All bases are assumed to be nonzero. Property

Example

1. u mu n = u m + n um 2. n = u m - n u 3. u 0 = 1 1 4. u -n = n u

53 # 54 = 53 + 4 = 57 x9 = x9-4 = x5 x4 80 = 1 1 y -3 = 3 y

5. 1uv2m = u mv m

12z25 = 25z 5 = 32z 5

u m um 7. a b = m v v

a 7 a7 a b = 7 b b

6. 1u m2n = u mn

#

1x 223 = x 2 3 = x 6

To simplify an expression involving powers means to rewrite it so that each factor appears only once, all exponents are positive, and exponents and constants are combined as much as possible.

6965_CH0P_pp001-062.qxd

8

1/14/10

12:43 PM

Page 8

CHAPTER P Prerequisites

Moving Factors

EXAMPLE 7 Simplifying Expressions Involving Powers

Be sure you understand how exponent property 4 permits us to move factors from the numerator to the denominator and vice versa:

(a) 12ab 3215a 2b 52 = 101aa 221b 3b 52 = 10a 3b 8

v -m un -n = m u v

(b)

u 2v -2 u 2u 1 u3 = = u -1v 3 v 2v 3 v5

(c) a

1x 22-3 x 2 -3 x -6 23 8 b = = = = 6 -3 -3 6 2 2 2 x x

Now try Exercise 47.

Scientific Notation Any positive number can be written in scientific notation, c * 10 m, where 1 … c 6 10 and m is an integer. This notation provides a way to work with very large and very small numbers. For example, the distance between the Earth and the Sun is about 93,000,000 miles. In scientific notation, 93,000,000 mi = 9.3 * 10 7 mi. The positive exponent 7 indicates that moving the decimal point in 9.3 to the right 7 places produces the decimal form of the number. The mass of an oxygen molecule is about 0.000 000 000 000 000 000 000 053 gram. In scientific notation, 0.000 000 000 000 000 000 000 053 g = 5.3 * 10 -23 g. The negative exponent -23 indicates that moving the decimal point in 5.3 to the left 23 places produces the decimal form of the number.

EXAMPLE 8 Converting to and from Scientific Notation (a) 2.375 * 10 8 = 237,500,000 (b) 0.000000349 = 3.49 * 10 -7

Now try Exercises 57 and 59.

EXAMPLE 9 Using Scientific Notation Simplify

1360,000214,500,000,0002 18,000

, without using a calculator.

SOLUTION

1360,000214,500,000,0002 18,000

=

13.6 * 10 5214.5 * 10 92

1.8 * 10 4 13.6214.52 = * 10 5 + 9 - 4 1.8 = 9 * 10 10 = 90,000,000,000 Now try Exercise 63.

Using a Calculator Figure P.5 shows two ways to perform the computation. In the first, the numbers are entered in decimal form. In the second, the numbers are entered in scientific notation. The calculator uses “9E10” to stand for 9 * 10 10.

6965_CH0P_pp001-062.qxd

1/14/10

12:43 PM

Page 9

SECTION P.1

(360000)(4500000 000)/(18000) (3.6E5)(4.5E9)/( 1.8E4)

Real Numbers

9

9E10 9E10

N

FIGURE P.5 Be sure you understand how your calculator displays scientific notation. (Example 9)

QUICK REVIEW P.1 1. List the positive integers between -3 and 7. 2. List the integers between -3 and 7.

In Exercises 7 and 8, evaluate the algebraic expression for the given values of the variables. 7. x 3 - 2x + 1, x = - 2, 1.5

3. List all negative integers greater than -4.

8. a 2 + ab + b 2, a = - 3, b = 2

4. List all positive integers less than 5. In Exercises 5 and 6, use a calculator to evaluate the expression. Round the value to two decimal places. 21 - 5.52 - 6 5. (a) 41- 3.123 - 1-4.225 (b) 7.4 - 3.8

In Exercises 9 and 10, list the possible remainders. 9. When the positive integer n is divided by 7 10. When the positive integer n is divided by 13

(b) 5-2 + 2-4

6. (a) 5331-1.122 - 41 - 0.5234

SECTION P.1 EXERCISES Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 1–4, find the decimal form for the rational number. State whether it repeats or terminates.

15. x is between -1 and 2. 16. x is greater than or equal to 5. In Exercises 17–22, use interval notation to describe the interval of real numbers.

1. -37/8

2. 15/99

17. x 7 - 3

3. -13/6

4. 5/37

19.

In Exercises 5–10, describe and graph the interval of real numbers.

18. - 7 6 x 6 -2 x

–5 –4 –3 –2 –1

0

1

2

3

4

5

–5 –4 –3 –2 –1

0

1

2

3

4

5

5. x … 2

6. - 2 … x 6 5

20.

7. 1- q , 72

8. 3 - 3, 34

21. x is greater than -3 and less than or equal to 4.

9. x is negative.

10. x is greater than or equal to 2 and less than or equal to 6. In Exercises 11–16, use an inequality to describe the interval of real numbers. 11. 3 - 1, 12

13.

–5 –4 –3 –2 –1

0

1

12. 1 - q , 44 2

3

4

–5 –4 –3 –2 –1

0

1

2

3

4

22. x is positive. In Exercises 23–28, use words to describe the interval of real numbers. 23. 4 6 x … 9

24. x Ú -1

x

25. 3- 3, q 2

26. 1 - 5, 72

x

28.

27.

5

14. 5

x

–5 –4 –3 –2 –1

0

1

–5 –4 –3 –2 –1

0

1

x

2

3

4

5

2

3

4

5

x

6965_CH0P_pp001-062.qxd

1/14/10

12:43 PM

Page 10

CHAPTER P Prerequisites

10

In Exercises 29–32, convert to inequality notation. Find the endpoints and state whether the interval is bounded or unbounded and its type. 29. 1 -3, 44

30. 1 -3, - 12

31. 1 - q , 52

51.

1x -3y 22-4

1y 6x -42-2

52. a

4a 3b

3b 2

a 2b

2a 2b 4

ba 3

b

The data in Table P.1 give the expenditures in millions of dollars for U.S. public schools for the 2005–2006 school year.

32. 3 - 6, q 2

In Exercises 33–36, use both inequality and interval notation to describe the set of numbers. State the meaning of any variables you use.

Table P.1 U.S. Public Schools

33. Writing to Learn Bill is at least 29 years old.

Category

Amount (in millions)

34. Writing to Learn No item at Sarah’s Variety Store costs more than $2.00.

Current expenditures Capital outlay Interest on school debt Total

449,595 57,375 14,347 528,735

35. Writing to Learn The price of a gallon of gasoline varies from $1.099 to $1.399.

Source: National Center for Education Statistics, U.S. Department of Education, as reported in The World Almanac and Book of Facts 2009.

36. Writing to Learn Salary raises at the State University of California at Chico will average between 2% and 6.5%. In Exercises 37–40, use the distributive property to write the factored form or the expanded form of the given expression. 37. a1x 2 + b2

38. 1y - z 32c

39. ax 2 + dx 2

53. Current expenditures

40. a 3z + a 3w

54. Capital outlay

In Exercises 41 and 42, find the additive inverse of the number. 41. 6 - p

55. Interest on school debt

42. -7

56. Total

In Exercises 43 and 44, identify the base of the exponential expression. 2

43. - 5

44. 1 -22

7

45. Group Activity Discuss which algebraic property or properties are illustrated by the equation. Try to reach a consensus. (a) 13x2y = 31xy2 2

2

(c) a b + 1- a b2 = 0

(e) a1x + y2 = ax + ay

(b) a 2b = ba 2 2

1 = 1 x + 2

(c) 21x - y2 = 2x - 2y

(d) 1x + 32 + 0 = 1x + 32

(b) 1 # 1x + y2 = x + y

(d) 2x + 1y - z2 = 2x + 1y + 1-z22 = 12x + y2 + 1- z2 = 12x + y2 - z

49. a

4 x

2

b

50. a

2 b xy

58. The electric charge, in coulombs, of an electron is about -0.000 000 000 000 000 000 16. 59. 3.33 * 10 -8

60. 6.73 * 10 11

61. The distance that light travels in 1 year (one light year) is about 5.87 * 10 12 mi. 62. The mass of a neutron is about 1.6747 * 10 -24 g. In Exercises 63 and 64, use scientific notation to simplify. 11.3 * 10 -7212.4 * 10 82 63. without using a calculator 1.3 * 10 9 13.7 * 10 -7214.3 * 10 62 2.5 * 10 7

Explorations

In Exercises 47–52, simplify the expression. Assume that the variables in the denominators are nonzero. 13x 222y 4 x 4y 3 47. 2 5 48. x y 3y 2 2

57. The mean distance from Jupiter to the Sun is about 483,900,000 miles.

64.

1 1 1ab2 = a ab b = 1 # b = b a a

(e)

In Exercises 57 and 58, write the number in scientific notation.

In Exercises 59–62, write the number in decimal form. 2

46. Group Activity Discuss which algebraic property or properties are illustrated by the equation. Try to reach a consensus. (a) 1x + 22

In Exercises 53–56, write the amount of expenditures in dollars obtained from the category in scientific notation.

-3

65. Investigating Exponents For positive integers m and n, we can use the definition to show that a ma n = a m + n. (a) Examine the equation a ma n = a m + n for n = 0 and explain why it is reasonable to define a 0 = 1 for a Z 0. (b) Examine the equation a ma n = a m + n for n = - m and explain why it is reasonable to define a -m = 1/a m for a Z 0.

6965_CH0P_pp001-062.qxd

1/14/10

12:43 PM

Page 11

SECTION P.1

66. Decimal Forms of Rational Numbers Here is the third step when we divide 1 by 17. (The first two steps are not shown, because the quotient is 0 in both cases.) 0.05 17 ! 1.00 85 15 By convention we say that 1 is the first remainder in the long division process, 10 is the second, and 15 is the third remainder. (a) Continue this long division process until a remainder is repeated, and complete the following table: Step 1 2 3 o

Quotient 0 0 5 o

Remainder 1 10 15 o

(b) Explain why the digits that occur in the quotient between the pair of repeating remainders determine the infinitely repeating portion of the decimal representation. In this case 1 = 0.0588235294117647. 17 (c) Explain why this procedure will always determine the infinitely repeating portion of a rational number whose decimal representation does not terminate.

Standardized Test Questions 67. True or False The additive inverse of a real number must be negative. Justify your answer. 68. True or False The reciprocal of a positive real number must be less than 1. Justify your answer.

Real Numbers

11

In Exercises 69–72, solve these problems without using a calculator. 69. Multiple Choice Which of the following inequalities corresponds to the interval 3-2, 12? (A) x … -2

(B) - 2 … x … 1

(C) -2 6 x 6 1

(D) - 2 6 x … 1

(E) -2 … x 6 1 70. Multiple Choice What is the value of 1- 224? (A) 16

(B) 8

(C) 6

(D) - 8

(E) -16 71. Multiple Choice What is the base of the exponential expression -72? (A) -7

(B) 7

(C) -2

(D) 2

(E) 1 72. Multiple Choice Which of the following is the simplix6 fied form of 2 , x Z 0? x -4 (A) x (B) x 2 (C) x 3

(D) x 4

(E) x 8

Extending the Ideas The magnitude of a real number is its distance from the origin. 73. List the whole numbers whose magnitudes are less than 7. 74. List the natural numbers whose magnitudes are less than 7. 75. List the integers whose magnitudes are less than 7.

6965_CH0P_pp001-062.qxd

2/3/10

2:59 PM

Page 12

CHAPTER P Prerequisites

12

P.2 Cartesian Coordinate System Cartesian Plane

What you’ll learn about • Cartesian Plane • Absolute Value of a Real Number • Distance Formulas • Midpoint Formulas • Equations of Circles • Applications

... and why These topics provide the foundation for the material that will be covered in this textbook.

y

The points in a plane correspond to ordered pairs of real numbers, just as the points on a line can be associated with individual real numbers. This correspondence creates the Cartesian plane, or the rectangular coordinate system in the plane. To construct a rectangular coordinate system, or a Cartesian plane, draw a pair of perpendicular real number lines, one horizontal and the other vertical, with the lines intersecting at their respective 0-points (Figure P.6). The horizontal line is usually the x-axis and the vertical line is usually the y-axis. The positive direction on the x-axis is to the right, and the positive direction on the y-axis is up. Their point of intersection, O, is the origin of the Cartesian plane. Each point P of the plane is associated with an ordered pair (x, y) of real numbers, the (Cartesian) coordinates of the point. The x-coordinate represents the intersection of the x-axis with the perpendicular from P, and the y-coordinate represents the intersection of the y-axis with the perpendicular from P. Figure P.6 shows the points P and Q with coordinates 14, 22 and 1-6, - 42, respectively. As with real numbers and a number line, we use the ordered pair 1a, b2 for both the name of the point and its coordinates. The coordinate axes divide the Cartesian plane into four quadrants, as shown in Figure P.7.

6 4

–8

–4

–2

EXAMPLE 1 Plotting Data on U.S. Exports to Mexico

P(4, 2)

2 2

O

4

6

x

–2

The value in billions of dollars of U.S. exports to Mexico from 2000 to 2007 is given in Table P.2. Plot the (year, export value) ordered pairs in a rectangular coordinate system. Table P.2 U.S. Exports to Mexico

Q(–6, –4) –6

FIGURE P.6 The Cartesian coordinate plane. y First quadrant P(x, y)

y Second quadrant

–3

3 2 1

–1 O

Year

U.S. Exports (billions of dollars)

2000 2001 2002 2003 2004 2005 2006 2007

111.3 101.3 97.5 97.4 110.8 120.4 134.0 136.0

Source: U.S. Census Bureau, The World Almanac and Book of Facts 2009.

1

3

x

x

SOLUTION The points are plotted in Figure P.8 on page 13.

Now try Exercise 31.

–2 –3 Third quadrant

Fourth quadrant

FIGURE P.7 The four quadrants. Points on the x- or y-axis are not in any quadrant.

A scatter plot is a plotting of the 1x, y2 data pairs on a Cartesian plane. Figure P.8 shows a scatter plot of the data from Table P.2.

Absolute Value of a Real Number The absolute value of a real number suggests its magnitude (size). For example, the absolute value of 3 is 3 and the absolute value of -5 is 5.

6965_CH0P_pp001-062.qxd

1/14/10

12:43 PM

Page 13

SECTION P.2

U.S. Exports to Mexico

y

13

DEFINITION Absolute Value of a Real Number

140

The absolute value of a real number a is

130 120

a, if a 7 0 ƒ a ƒ = c -a, if a 6 0 0, if a = 0.

110

Value (billions of dollars)

Cartesian Coordinate System

100 90 80 70 60 50 40

EXAMPLE 2 Using the Definition of Absolute Value

30 20 10 2000

2004 Year

2008

x

FIGURE P.8 The graph for Example 1.

Evaluate: (a) ƒ -4 ƒ

(b) ƒ p - 6 ƒ

SOLUTION

(a) Because -4 6 0, ƒ -4 ƒ = - 1-42 = 4. (b) Because p L 3.14, p - 6 is negative, so p - 6 6 0. Thus, Now try Exercise 9. ƒ p - 6 ƒ = - 1p - 62 = 6 - p L 2.858. Here is a summary of some important properties of absolute value.

Properties of Absolute Value Let a and b be real numbers. 1. ƒ a ƒ Ú 0

2. ƒ -a ƒ = ƒ a ƒ

3. ƒ ab ƒ = ƒ a ƒ ƒ b ƒ

ƒaƒ a ,b Z 0 4. ` ` = b ƒbƒ

Distance Formulas |4 – (–1)| = |–1 – 4| = 5 x –3 –2 –1

0

1

2

3

4

5

FIGURE P.9 Finding the distance between

The distance between - 1 and 4 on the number line is 5 (see Figure P.9). This distance may be found by subtracting the smaller number from the larger: 4 - 1-12 = 5. If we use absolute value, the order of subtraction does not matter: ƒ 4 - 1- 12 ƒ = ƒ -1 - 4 ƒ = 5.

-1 and 4.

Distance Formula (Number Line)

Absolute Value and Distance

Let a and b be real numbers. The distance between a and b is

If we let b = 0 in the distance formula, we see that the distance between a and 0 is ƒ a ƒ . Thus, the absolute value of a number is its distance from zero.

ƒa - bƒ. Note that ƒ a - b ƒ = ƒ b - a ƒ .

To find the distance between two points that lie on the same horizontal or vertical line in the Cartesian plane, we use the distance formula for points on a number line. For example, the distance between points x 1 and x 2 on the x-axis is ƒ x 1 - x 2 ƒ = ƒ x 2 - x 1 ƒ and the distance between points y1 and y2 on the y-axis is ƒ y1 - y2 ƒ = ƒ y2 - y1 ƒ . To find the distance between two points P1x 1, y12 and Q1x 2, y22 that do not lie on the same horizontal or vertical line, we form the right triangle determined by P, Q, and R1x 2, y12, (Figure P.10).

6965_CH0P_pp001-062.qxd

14

1/14/10

12:43 PM

Page 14

CHAPTER P Prerequisites

y y2

Q(x2, y2)

d

y1 O

! y1 – y2 !

R(x2, y1)

P(x1, y1) x1

x

x2 ! x1 – x2 !

FIGURE P.10 Forming a right triangle with hypotenuse PQ.

c

a

The distance from P to R is ƒ x 1 - x 2 ƒ , and the distance from R to Q is ƒ y1 - y2 ƒ . By the Pythagorean Theorem (see Figure P.11), the distance d between P and Q is d = 2 ƒ x 1 - x 2 ƒ 2 + ƒ y1 - y2 ƒ 2.

b

FIGURE P.11 The Pythagorean Theorem: c 2 = a 2 + b 2.

Because ƒ x 1 - x 2 ƒ 2 = 1x 1 - x 222 and ƒ y1 - y2 ƒ 2 = 1y1 - y222, we obtain the following formula.

Distance Formula (Coordinate Plane) The distance d between points P1x1, y12 and Q1x2, y22 in the coordinate plane is d = 21x 1 - x 222 + 1y1 - y222.

EXAMPLE 3 Finding the Distance Between Two Points Find the distance d between the points 11, 52 and 16, 22. SOLUTION

d = 211 - 622 + 15 - 222 2

The distance formula

2

= 21- 52 + 3

= 225 + 9

= 234 L 5.83

Using a calculator

Now try Exercise 11.

Midpoint Formulas When the endpoints of a segment in a number line are known, we take the average of their coordinates to find the midpoint of the segment.

Midpoint Formula (Number Line) The midpoint of the line segment with endpoints a and b is a + b . 2

6965_CH0P_pp001-062.qxd

1/14/10

12:43 PM

Page 15

SECTION P.2

Cartesian Coordinate System

15

EXAMPLE 4 Finding the Midpoint of a Line Segment The midpoint of the line segment with endpoints - 9 and 3 on a number line is 1 -92 + 3 2

=

-6 = -3. 2 Now try Exercise 23.

See Figure P.12.

Midpoint 6

6 x

–9

–3

0

3

FIGURE P.12 Notice that the distance from the midpoint, -3, to 3 or to - 9 is 6.

(Example 4)

Just as with number lines, the midpoint of a line segment in the coordinate plane is determined by its endpoints. Each coordinate of the midpoint is the average of the corresponding coordinates of its endpoints.

Midpoint Formula (Coordinate Plane) The midpoint of the line segment with endpoints (a, b) and (c, d) is a

y (3, 7)

Midpoint

a + c b + d , b. 2 2

EXAMPLE 5 Finding the Midpoint of a Line Segment The midpoint of the line segment with endpoints 1-5, 22 and 13, 72 is

(–1, 4.5) (–5, 2) 1

x

1

See Figure P.13.

1x, y2 = a

-5 + 3 2 + 7 , b = 1-1, 4.52. 2 2

Now try Exercise 25.

FIGURE P.13 (Example 5.)

Equations of Circles

y

A circle is the set of points in a plane at a fixed distance 1radius2 from a fixed point (center). Figure P.14 shows the circle with center 1h, k2 and radius r. If 1x, y2 is any point on the circle, the distance formula gives

(x, y) r (h, k)

x

21x - h22 + 1 y - k22 = r.

Squaring both sides, we obtain the following equation for a circle.

DEFINITION Standard Form Equation of a Circle FIGURE P.14 The circle with center

1h, k2 and radius r.

The standard form equation of a circle with center 1h, k2 and radius r is 1x - h22 + 1y - k22 = r 2.

6965_CH0P_pp001-062.qxd

16

1/14/10

12:43 PM

Page 16

CHAPTER P Prerequisites

EXAMPLE 6 Finding Standard Form Equations of Circles Find the standard form equation of the circle. (a) Center 1- 4, 12, radius 8 (b) Center 10, 02, radius 5 SOLUTION

(a) 1x - h22 + 1y - k22 = r 2 1x - 1-4222 + 1y - 122 = 82 1x + 422 + 1y - 122 = 64 (b) 1x - h22 + 1y - k22 = r 2 1x - 022 + 1y - 022 = 52 x 2 + y 2 = 25

Standard form equation Substitute h = - 4, k = 1, r = 8.

Standard form equation Substitute h = 0, k = 0, r = 5.

Now try Exercise 41.

Applications EXAMPLE 7 Using an Inequality to Express Distance We can state that “the distance between x and - 3 is less than 9” using the inequality

ƒ x - 1- 32 ƒ 6 9 or ƒ x + 3 ƒ 6 9.

Now try Exercise 51.

The converse of the Pythagorean Theorem is true. That is, if the sum of squares of the lengths of the two sides of a triangle equals the square of the length of the third side, then the triangle is a right triangle. y

EXAMPLE 8 Verifying Right Triangles c

(–3, 4) a

Use the converse of the Pythagorean Theorem and the distance formula to show that the points 1-3, 42, 11, 02, and 15, 42 determine a right triangle.

(5, 4)

SOLUTION The three points are plotted in Figure P.15. We need to show that the

b (1, 0)

x

FIGURE P.15 The triangle in Example 8.

lengths of the sides of the triangle satisfy the Pythagorean relationship a 2 + b 2 = c2. Applying the distance formula we find that a = 21- 3 - 122 + 14 - 022 = 232, b = 211 - 522 + 10 - 422 = 232,

c = 21-3 - 522 + 14 - 422 = 264.

The triangle is a right triangle because

a 2 + b 2 = 123222 + 123222 = 32 + 32 = 64 = c2.

Now try Exercise 39.

Properties of geometric figures can sometimes be confirmed using analytic methods such as the midpoint formulas.

EXAMPLE 9 Using the Midpoint Formula It is a fact from geometry that the diagonals of a parallelogram bisect each other. Prove this with a midpoint formula.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 17

SECTION P.2

y

Cartesian Coordinate System

17

SOLUTION We can position a parallelogram in the rectangular coordinate plane as shown in Figure P.16. Applying the midpoint formula for the coordinate plane to segments OB and AC, we find that B(a + c, b)

A(a, b)

midpoint of segment OB = a

D

O O(0, 0)

midpoint of segment AC = a

x

C(c, 0)

0 + a + c 0 + b a + c b , b = a , b, 2 2 2 2

a + c b + 0 a + c b , b = a , b. 2 2 2 2

The midpoints of segments OA and AC are the same, so the diagonals of the parallelogram OABC meet at their midpoints and thus bisect each other. Now try Exercise 37.

FIGURE P.16 The coordinates of B must be 1a + c, b2 in order for CB to be parallel to OA. (Example 9)

QUICK REVIEW P.2 In Exercises 1 and 2, plot the two numbers on a number line. Then find the distance between them. 5 9 2. - , 3 5

1. 27, 22

5. A13, 52, B1- 2, 42, C13, 02, D10, - 32 6. A1-3, -52, B12, -42, C10, 52, D1- 4, 02 In Exercises 7–10, use a calculator to evaluate the expression. Round your answer to two decimal places.

In Exercises 3 and 4, plot the real numbers on a number line. 5 1 2 4. - , - , , 0, -1 2 2 3

3. - 3, 4, 2.5, 0, -1.5

In Exercises 5 and 6, plot the points.

7.

-17 + 28 2

8. 2132 + 172

9. 262 + 82

10. 2117 - 322 + 1-4 - 822

SECTION P.2 EXERCISES Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 1 and 2, estimate the coordinates of the points. y

1.

B A

2 C

D

B

A 2

x

11. -9.3, 10.6

12. - 5, -17

13. 1-3, -12, 15, -12

14. 1- 4, -32, 11, 12

17. 1 -2, 02, 15, 02

18. 10, - 82, 10, - 12

15. 10, 02, 13, 42

y

2.

In Exercises 11–18, find the distance between the points.

16. 1- 1, 22, 12, - 32

In Exercises 19–22, find the area and perimeter of the figure determined by the points.

C 1 1

x D

19. 1 -5, 32, 10, -12, 14, 42

20. 1- 2, -22, 1 -2, 22, 12, 22, 12, -22 In Exercises 3 and 4, name the quadrants containing the points. 3. (a) 12, 42

1 3 4. (a) a , b 2 2

(b) 10, 32

(b) 1 - 2, 02

(c) 1- 2, 32

(c) 1 - 1, -22

In Exercises 5–8, evaluate the expression. 5. 3 + ƒ - 3 ƒ 7. ƒ 1 - 223 ƒ

(d) 1-1, -42

3 7 (d) a - , - b 2 3

6. 2 - ƒ -2 ƒ -2 8. ƒ -2ƒ

In Exercises 9 and 10, rewrite the expression without using absolute value symbols. 9. ƒ p - 4 ƒ

10. ƒ 25 - 5/2 ƒ

21. 1- 3, - 12, 1 -1, 32, 17, 32, 15, -12 22. 1- 2, 12, 1-2, 62, 14, 62, 14, 12

In Exercises 23–28, find the midpoint of the line segment with the given endpoints. 23. -9.3, 10.6 25. 1- 1, 32, 15, 92

26. 13, 222, 16, 22

27. 1 - 7/3, 3/42, 15/3, - 9/42 28. 15, - 22, 1- 1, -42

24. -5, - 17

6965_CH0P_pp001-062.qxd

12:44 PM

Page 18

CHAPTER P Prerequisites

In Exercises 29–34, draw a scatter plot of the data given in the table. 29. U.S. Motor Vehicle Production The total number of motor vehicles in thousands 1y2 produced by the United States each year from 2001 to 2007 is given in the table. (Source: Automotive News Data Center and R. L. Polk Marketing Systems as reported in The World Almanac and Book of Facts 2009.) x

2001

2002

2003

2004

2005

2006

33. U.S. Agricultural Trade Surplus The total in billions of dollars of U.S. agricultural trade surplus from 2000 to 2007 is given in Table P.5.

Table P.5 U.S. Agricultural Trade Surplus

2007

y 11,518 12,328 12,145 12,021 12,018 11,351 10,611 30. World Motor Vehicle Production The total number of motor vehicles in thousands 1y2 produced in the world each year from 2001 to 2007 is given in the table. (Source: American Automobile Manufacturers Association as reported in The World Almanac and Book of Facts 2009.) x

2001

2002

2003

2004

2005

2006

2007

y 57,705 59,587 61,562 65,654 67,892 70,992 74,647

2000 2001 2002 2003 2004 2005 2006 2007

135.0 131.3 134.6 138.1 155.9 170.1 188.2 210.7

Source: U.S. Census Bureau, The World Almanac and Book of Facts 2009.

32. U.S. Agricultural Exports The total in billions of dollars of U.S. agricultural exports from 2000 to 2007 is given in Table P.4.

2000 2001 2002 2003 2004 2005 2006 2007

12.2 14.3 11.2 10.3 9.7 4.8 4.7 12.1

Table P.6 U.S. Exports to Canada

Table P.3 U.S. Imports from Mexico U.S. Imports (billions of dollars)

U.S. Agricultural Trade Surplus (billions of dollars)

34. U.S. Exports to Canada The total in billions of dollars of U.S. exports to Canada from 2000 to 2007 is given in Table P.6.

31. U.S. Imports from Mexico The total in billions of dollars of U.S. imports from Mexico from 2000 to 2007 is given in Table P.3.

Year

Year

Source: U.S. Department of Agriculture, The World Almanac and Book of Facts 2009.

Year

U.S. Exports (billions of dollars)

2000 2001 2002 2003 2004 2005 2006 2007

178.9 163.4 160.9 169.9 189.9 211.9 230.6 248.9

Source: U.S. Census Bureau, The World Almanac and Book of Facts 2009.

In Exercises 35 and 36, use the graph of the investment value of a $10,000 investment made in 1978 in Fundamental Investors™ of the American Funds™. The value as of January is shown for a few recent years in the graph below. (Source: Annual report of Fundamental Investors for the year ending December 31, 2004.)

300

Table P.4 U.S. Agricultural Exports Year

U.S. Agricultural Exports (billions of dollars)

2000 2001 2002 2003 2004 2005 2006 2007

51.2 53.7 53.1 56.0 62.4 62.5 68.7 89.2

Source: U.S. Department of Agriculture, The World Almanac and Book of Facts 2009.

Investment Value (in thousands of dollars)

18

1/14/10

260

220

180

140

100 ’95

’96

’97

’98

’99

’00 ’01 Year

’02

’03

’04

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 19

SECTION P.2

35. Reading from Graphs Use the graph to estimate the value of the investment as of (a) January 1997 and (b) January 2000. 36. Percent Increase Estimate the percent increase in the value of the $10,000 investment from (a) January 1996 to January 1997.

19

Cartesian Coordinate System

47. x 2 + y 2 = 5 48. 1x - 222 + 1y + 622 = 25

In Exercises 49–52, write the statement using absolute value notation. 49. The distance between x and 4 is 3. 50. The distance between y and -2 is greater than or equal to 4. 51. The distance between x and c is less than d units.

(b) January 2000 to January 2001.

52. y is more than d units from c.

(c) January 1995 to January 2004. 37. Prove that the figure determined by the points is an isosceles triangle: 11, 32, 14, 72, 18, 42 38. Group Activity Prove that the diagonals of the figure determined by the points bisect each other. (a) Square 1 -7, -12, 1- 2, 42, 13, -12, 1- 2, -62 (b) Parallelogram 1- 2, - 32, 10, 12, 16, 72, 14, 32

39. (a) Find the lengths of the sides of the triangle in the figure. y

53. Determining a Line Segment with Given Midpoint Let 14, 42 be the midpoint of the line segment determined by the points 11, 22 and 1a, b2. Determine a and b. 54. Writing to Learn Isosceles but Not Equilateral Triangle Prove that the triangle determined by the points 13, 02, 1- 1, 22, and 15, 42 is isosceles but not equilateral.

55. Writing to Learn Equidistant Point from Vertices of a Right Triangle Prove that the midpoint of the hypotenuse of the right triangle with vertices 10, 02, 15, 02, and 10, 72 is equidistant from the three vertices. 56. Writing to Learn Describe the set of real numbers that satisfy ƒ x - 2 ƒ 6 3.

(3, 6)

57. Writing to Learn Describe the set of real numbers that satisfy ƒ x + 3 ƒ Ú 5.

x (–2, –2)

(3, –2)

(b) Writing to Learn Show that the triangle is a right triangle. 40. (a) Find the lengths of the sides of the triangle in the figure. y

(3, 3)

x

(b) Writing to Learn Show that the triangle is a right triangle. In Exercises 41–44, find the standard form equation for the circle. 41. Center 11, 22, radius 5

42. Center 1- 3, 22, radius 1

43. Center 1- 1, -42, radius 3 44. Center 10, 02, radius 23

In Exercises 45–48, find the center and radius of the circle. 45. 1x - 322 + 1y - 122 = 36

46. 1x + 422 + 1y - 222 = 121

B

58. True or False If a is a real number, then ƒ a ƒ Ú 0. Justify your answer. 59. True or False Consider the right triangle ABC shown at the right. If M is the midpoint of the segment AB, then M' is the midpoint of the segment AC. Justify your answer. In Exercises 60–63, solve these problems without using a calculator. 60. Multiple Choice Which of the following is equal to ƒ 1 - 23 ƒ ? (A) 1 - 23 (B) 23 - 1

(–4, 4)

(0, 0)

Standardized Test Questions

(C) 11 - 2322 (E) 21/3

M

A

M'

C

(D) 22

61. Multiple Choice Which of the following is the midpoint of the line segment with endpoints - 3 and 2? (A) 5/2

(B) 1

(C) -1/2

(D) - 1

(E) -5/2

6965_CH0P_pp001-062.qxd

20

1/14/10

12:44 PM

Page 20

CHAPTER P Prerequisites

62. Multiple Choice Which of the following is the center of the circle 1x - 322 + 1y + 422 = 2? (A) 13, - 42 (C) 14, -32

(E) 13/2, -22

(B) 1 - 3, 42

(D) 1-4, 32

63. Multiple Choice Which of the following points is in the third quadrant? (A) 10, -32 (C) 12, - 12

(E) 1 -2, - 32

(B) 1 -1, 02

(D) 1- 1, 22

Explorations 64. Dividing a Line Segment into Thirds (a) Find the coordinates of the points one-third and two-thirds of the way from a = 2 to b = 8 on a number line. (b) Repeat 1a2 for a = - 3 and b = 7.

(c) Find the coordinates of the points one-third and two-thirds of the way from a to b on a number line. (d) Find the coordinates of the points one-third and two-thirds of the way from the point 11, 22 to the point 17, 112 in the coordinate plane. (e) Find the coordinates of the points one-third and two-thirds of the way from the point 1a, b2 to the point 1c, d2 in the coordinate plane.

Extending the Ideas 65. Writing to Learn Equidistant Point from Vertices of a Right Triangle Prove that the midpoint of the hypotenuse of any right triangle is equidistant from the three vertices. 66. Comparing Areas Consider the four points A10, 02, B10, a2, C1a, a2, and D1a, 02. Let P be the midpoint of the line segment CD and Q the point one-fourth of the way from A to D on segment AD. (a) Find the area of triangle BPQ. (b) Compare the area of triangle BPQ with the area of square ABCD. In Exercises 67–69, let P1a, b2 be a point in the first quadrant. 67. Find the coordinates of the point Q in the fourth quadrant so that PQ is perpendicular to the x-axis. 68. Find the coordinates of the point Q in the second quadrant so that PQ is perpendicular to the y-axis. 69. Find the coordinates of the point Q in the third quadrant so that the origin is the midpoint of the segment PQ. 70. Writing to Learn Prove that the distance formula for the number line is a special case of the distance formula for the Cartesian plane.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 21

SECTION P.3

What you’ll learn about

Linear Equations and Inequalities

P.3 Linear Equations and Inequalities

• Equations

Equations

• Solving Equations

An equation is a statement of equality between two expressions. Here are some properties of equality that we use to solve equations algebraically.

• Linear Equations in One Variable • Linear Inequalities in One Variable

... and why These topics provide the foundation for algebraic techniques needed throughout this textbook.

21

Properties of Equality Let u, v, w, and z be real numbers, variables, or algebraic expressions. 1. 2. 3. 4. 5.

Reflexive Symmetric Transitive Addition Multiplication

u = If u If u If u If u

u = = = =

v, then v = u. v, and v = w, then u = w. v and w = z, then u + w = v + z. v and w = z, then uw = vz.

Solving Equations A solution of an equation in x is a value of x for which the equation is true. To solve an equation in x means to find all values of x for which the equation is true, that is, to find all solutions of the equation.

EXAMPLE 1 Confirming a Solution Prove that x = - 2 is a solution of the equation x 3 - x + 6 = 0. SOLUTION

1-223 - 1-22 + 6 ! 0 -8 + 2 + 6 ! 0 0 = 0

Now try Exercise 1.

Linear Equations in One Variable The most basic equation in algebra is a linear equation.

DEFINITION Linear Equation in x A linear equation in x is one that can be written in the form ax + b = 0, where a and b are real numbers with a Z 0. The equation 2z - 4 = 0 is linear in the variable z. The equation 3u 2 - 12 = 0 is not linear in the variable u. A linear equation in one variable has exactly one solution. We solve such an equation by transforming it into an equivalent equation whose solution is obvious. Two or more equations are equivalent if they have the same solutions. For example, the equations 2z - 4 = 0, 2z = 4, and z = 2 are all equivalent. Here are operations that produce equivalent equations.

6965_CH0P_pp001-062.qxd

22

1/14/10

12:44 PM

Page 22

CHAPTER P Prerequisites

Operations for Equivalent Equations An equivalent equation is obtained if one or more of the following operations are performed. Given Equation

Operation

1. Combine like terms, reduce fractions, and remove grouping symbols.

2x + x =

2. Perform the same operation on both sides. (a) Add 1- 32. (b) Subtract 12x2. (c) Multiply by a nonzero constant 11/32. (d) Divide by a nonzero constant 132.

Equivalent Equation

3 9

3x =

1 3

x + 3 = 7 5x = 2x + 4

x = 4 3x = 4

3x = 12 3x = 12

x = 4 x = 4

The next two examples illustrate how to use equivalent equations to solve linear equations.

EXAMPLE 2 Solving a Linear Equation Solve 212x - 32 + 31x + 12 = 5x + 2. Support the result with a calculator. SOLUTION

212x - 32 + 31x + 12 4x - 6 + 3x + 3 7x - 3 2x x

= = = = =

5x + 2 5x + 2 5x + 2 5 2.5

Distributive property Combine like terms. Add 3, and subtract 5x. Divide by 2.

To support our algebraic work we can use a calculator to evaluate the original equation for x = 2.5. Figure P.17 shows that each side of the original equation is equal to 14.5 if x = 2.5.

2.5 X 2(2X–3)+3(X+1) 5X+2

2.5 14.5 14.5

FIGURE P.17 The top line stores the number 2.5 into the variable x. (Example 2) Now try Exercise 23.

If an equation involves fractions, find the least common denominator (LCD) of the fractions and multiply both sides by the LCD. This is sometimes referred to as clearing the equation of fractions. Example 3 illustrates.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 23

SECTION P.3

Linear Equations and Inequalities

Integers and Fractions

EXAMPLE 3 Solving a Linear Equation Involving Fractions

2 Notice in Example 3 that 2 = . 1

Solve

23

5y - 2 y = 2 + . 8 4 SOLUTION The denominators are 8, 1, and 4. The LCD of the fractions is 8. (See

Appendix A.3 if necessary.) 5y - 2 8 5y - 2 8a b 8 5y - 2 8# 8 5y - 2 5y 3y y

= 2 +

y 4

= 8a2 +

y b 4

= 8#2 + 8# = = = =

Multiply by the LCD 8.

y 4

16 + 2y 18 + 2y 18 6

Distributive property Simplify. Add 2. Subtract 2y. Divide by 3.

We leave it to you to check the solution using either paper and pencil or a calculator. Now try Exercise 25.

Linear Inequalities in One Variable We used inequalities to describe order on the number line in Section P.1. For example, if x is to the left of 2 on the number line, or if x is any real number less than 2, we write x 6 2. The most basic inequality in algebra is a linear inequality.

DEFINITION Linear Inequality in x A linear inequality in x is one that can be written in the form ax + b 6 0,

ax + b … 0,

ax + b 7 0,

or

ax + b Ú 0,

where a and b are real numbers with a Z 0. To solve an inequality in x means to find all values of x for which the inequality is true. A solution of an inequality in x is a value of x for which the inequality is true. The set of all solutions of an inequality is the solution set of the inequality. We solve an inequality by finding its solution set. Here is a list of properties we use to solve inequalities. Direction of an Inequality Multiplying (or dividing) an inequality by a positive number preserves the direction of the inequality. Multiplying (or dividing) an inequality by a negative number reverses the direction.

Properties of Inequalities Let u, v, w, and z be real numbers, variables, or algebraic expressions, and c a real number. 1. Transitive 2. Addition 3. Multiplication

If u If u If u If u If u

6 6 6 6 6

v and v 6 w, then u 6 w. v, then u + w 6 v + w. v and w 6 z, then u + w 6 v + z. v and c 7 0, then uc 6 vc. v and c 6 0, then uc 7 vc.

The above properties are true if 6 is replaced by … . There are similar properties for 7 and Ú .

6965_CH0P_pp001-062.qxd

24

1/14/10

12:44 PM

Page 24

CHAPTER P Prerequisites

The set of solutions of a linear inequality in one variable forms an interval of real numbers. Just as with linear equations, we solve a linear inequality by transforming it into an equivalent inequality whose solutions are obvious. Two or more inequalities are equivalent if they have the same set of solutions. The properties of inequalities listed above describe operations that transform an inequality into an equivalent one.

EXAMPLE 4 Solving a Linear Inequality Solve 3(x - 1) + 2 … 5x + 6. SOLUTION

3(x - 1) + 2 … 3x - 3 + 2 … 3x - 1 … 3x … -2x … 1 # a - b -2x Ú 2 x Ú

5x 5x 5x 5x 7

+ + + +

6 6 6 7

Distributive property Simplify. Add 1. Subtract 5x.

1 a- b #7 2 - 3.5

Multiply by - 1/2. (The inequality reverses.)

The solution set of the inequality is the set of all real numbers greater than or equal to -3.5. In interval notation, the solution set is 3-3.5, q 2. Now try Exercise 41. Because the solution set of a linear inequality is an interval of real numbers, we can display the solution set with a number line graph as illustrated in Example 5.

EXAMPLE 5 Solving a Linear Inequality Involving Fractions Solve the inequality and graph its solution set. x 1 x 1 + 7 + 3 2 4 3 SOLUTION The LCD of the fractions is 12.

1 x 1 x + 7 + 3 2 4 3 12 # a

x 1 + b 3 2 4x + 6 x + 6 x

The original inequality

7 12 # a

x 1 + b 4 3 7 3x + 4 7 4 7 -2

Multiply by the LCD 12. Simplify. Subtract 3x. Subtract 6.

The solution set is the interval 1-2, q 2. Its graph is shown in Figure P.18. x

–5 –4 –3 –2 –1

0

1

2

3

4

5

FIGURE P.18 The graph of the solution set of the inequality in Example 5. Now try Exercise 43.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 25

SECTION P.3

Linear Equations and Inequalities

25

Sometimes two inequalities are combined in a double inequality, whose solution set is a double inequality with x isolated as the middle term. Example 6 illustrates.

EXAMPLE 6 Solving a Double Inequality Solve the inequality and graph its solution set. -3 6

2x + 5 … 5 3

SOLUTION

2x + 5 … 5 3 -9 6 2x + 5 … 15 -14 6 2x … 10 -7 6 x … 5 -3 6

x –10 –8 –6 –4 –2

0

2

4

6

8

Multiply by 3. Subtract 5. Divide by 2.

The solution set is the set of all real numbers greater than - 7 and less than or equal to 5. In interval notation, the solution is set 1- 7, 54. Its graph is shown in Figure P.19. Now try Exercise 47.

FIGURE P.19 The graph of the solution set of the double inequality in Example 6.

QUICK REVIEW P.3 In Exercises 1 and 2, simplify the expression by combining like terms. 1. 2x + 5x + 7 + y - 3x + 4y + 2

In Exercises 5–10, use the LCD to combine the fractions. Simplify the resulting fraction. 5.

2. 4 + 2x - 3z + 5y - x + 2y - z - 2 In Exercises 3 and 4, use the distributive property to expand the products. Simplify the resulting expression by combining like terms. 3. 312x - y2 + 41y - x2 + x + y

6.

1 3 + y - 1 y - 2

1 x

8.

1 1 + - x x y

7. 2 + 9.

4. 512x + y - 12 + 41 y - 3x + 22 + 1

2 3 + y y

x + 4 3x - 1 + 2 5

10.

x x + 3 4

SECTION P.3 EXERCISES Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 1–4, find which values of x are solutions of the equation. 2

1. 2x + 5x = 3 (a) x = - 3 2.

(b) x = -

x 1 x + = 2 6 3 (a) x = - 1

(b) x = 0

1 2

(c) x =

1 2

(c) x = 1

2

3. 21 - x + 2 = 3 (a) x = - 2

(b) x = 0

(c) x = 2

4. 1x - 221/3 = 2

(b) x = 8

(c) x = 10

(a) x = - 6

In Exercises 5–10, determine whether the equation is linear in x. 5. 5 - 3x = 0 7. x + 3 = x - 5 9. 22x + 5 = 10

6. 5 = 10/2 8. x - 3 = x 2 1 10. x + = 1 x

In Exercises 11–24, solve the equation without using a calculator. 11. 3x = 24

12. 4x = - 16

13. 3t - 4 = 8

14. 2t - 9 = 3

15. 2x - 3 = 4x - 5

16. 4 - 2x = 3x - 6

17. 4 - 3y = 21y + 42

18. 41y - 22 = 5y

19.

1 7 x = 2 8

20.

4 2 x = 3 5

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 26

CHAPTER P Prerequisites

26

21.

1 1 x + = 1 2 3

1 1 x + = 1 3 4

22.

24. 315z - 32 - 412z + 12 = 5z - 2 In Exercises 25–28, solve the equation. Support your answer with a calculator. 25.

2x - 3 + 5 = 3x 4

26. 2x - 4 =

27.

t + 5 t - 2 1 = 8 2 3

28.

4x - 5 3

–2

(b) 3/2 X 2X2+X–6

0

1.5 0

30. Writing to Learn Write a statement about solutions of equations suggested by the computations in the figure. (a) 2 X

2

7X+5

(b) –4 X 7X+5

19

4X–7

4X–7

1

–4 –23 –23

In Exercises 31–34, find which values of x are solutions of the inequality. 31. 2x - 3 6 7 (a) x = 0

(b) x = 5

(c) x = 6

(b) x = 3

(c) x = 4

32. 3x - 4 Ú 5 (a) x = 0

33. - 1 6 4x - 1 … 11 (a) x = 0

(b) x = 2

(c) x = 3

34. -3 … 1 - 2x … 3 (a) x = - 1

(c) x = 2

In Exercises 35–42, solve the inequality, and draw a number line graph of the solution set. 35. x - 4 6 2

36. x + 3 7 5

37. 2x - 1 … 4x + 3

38. 3x - 1 Ú 6x + 8

39. 2 … x + 6 6 9

40. -1 … 3x - 2 6 7

41. 215 - 3x2 + 312x - 12 … 2x + 1 42. 411 - x2 + 511 + x2 7 3x - 1 In Exercises 43–54, solve the inequality. 43.

5x + 7 … -3 4

48. - 6 6 5t - 1 6 0

3 - x x - 5 3 - 2x 5x - 2 + 6 - 2 50. + 6 -1 4 3 2 3

51.

2y - 3 3y - 1 + 6 y - 1 2 5

52.

3 - 4y 2y - 3 Ú 2 - y 6 8

54.

1 1x - 42 - 2x … 513 - x2 2

1 1 1x + 32 + 21x - 42 6 1x - 32 2 3

In Exercises 55–58, find the solutions of the equation or inequality displayed in Figure P.20. 55. x 2 - 2x 6 0

56. x 2 - 2x = 0

57. x 2 - 2x 7 0

58. x 2 - 2x … 0

X 0 1 2 3 4 5 6

Y1 0 –1 0 3 8 15 24

Y1 = X2–2X FIGURE P.20 The second column gives values of y1 = x 2 - 2x for x = 0, 1, 2, 3, 4, 5, and 6. 59. Writing to Learn Explain how the second equation was obtained from the first. x - 3 = 2x + 3,

2x - 6 = 4x + 6

60. Writing to Learn Explain how the second equation was obtained from the first. 2x - 1 = 2x - 4,

(b) x = 0

44.

3x - 2 7 -1 5

3y - 1 7 -1 4

49.

53.

t + 5 1 t - 1 + = 3 4 2

29. Writing to Learn Write a statement about solutions of equations suggested by the computations in the figure.

2X2+X–6

46. 1 7

47. 0 … 2z + 5 6 8

23. 213 - 4z2 - 512z + 32 = z - 17

(a) –2 X

2y - 5 Ú -2 3

45. 4 Ú

x -

1 = x - 2 2

61. Group Activity Determine whether the two equations are equivalent. (a) 3x = 6x + 9,

x = 2x + 9

(b) 6x + 2 = 4x + 10,

3x + 1 = 2x + 5

62. Group Activity Determine whether the two equations are equivalent. (a) 3x + 2 = 5x - 7, (b) 2x + 5 = x - 7,

-2x + 2 = -7 2x = x - 7

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 27

SECTION P.3

Standardized Test Questions 63. True or False

- 6 7 - 2. Justify your answer.

64. True or False

2 …

6 . Justify your answer. 3

In Exercises 65–68, you may use a graphing calculator to solve these problems. 65. Multiple Choice Which of the following equations is equivalent to the equation 3x + 5 = 2x + 1? (A) 3x = 2x (C)

3 5 x + = x + 1 2 2

Explorations 69. Testing Inequalities on a Calculator (a) The calculator we use indicates that the statement 2 6 3 is true by returning the value 1 (for true) when 2 6 3 is entered. Try it with your calculator. (b) The calculator we use indicates that the statement 2 6 1 is false by returning the value 0 (for false) when 2 6 1 is entered. Try it with your calculator. (c) Use your calculator to test which of these two numbers is larger: 799/800, 800/801.

(D) 3x + 6 = 2x

(d) Use your calculator to test which of these two numbers is larger: -102/101, -103/102.

66. Multiple Choice Which of the following inequalities is equivalent to the inequality - 3x 6 6? (A) 3x 6 - 6

(B) x 6 10

(C) x 7 - 2

(D) x 7 2

(E) x 7 3 67. Multiple Choice Which of the following is the solution to the equation x1x + 12 = 0? (A) x = 0 or x = - 1

(B) x = 0 or x = 1

(C) Only x = - 1

(D) Only x = 0

(E) Only x = 1

(e) If your calculator returns 0 when you enter 2x + 1 6 4, what can you conclude about the value stored in x?

Extending the Ideas 70. Perimeter of a Rectangle The formula for the perimeter P of a rectangle is P = 21L + W2. Solve this equation for W. 71. Area of a Trapezoid The formula for the area A of a trapezoid is

68. Multiple Choice Which of the following represents an equation equivalent to the equation 2x 1 x 1 + = 3 2 4 3

(C) 4x + 3 =

3 x - 2 2

(E) 4x + 6 = 3x - 4

27

(B) 3x = 2x + 4

(E) 3x = 2x - 4

that is cleared of fractions? (A) 2x + 1 = x - 1

Linear Equations and Inequalities

(B) 8x + 6 = 3x - 4 (D) 4x + 3 = 3x - 4

A =

1 h1b1 + b22. 2

Solve this equation for b1.

72. Volume of a Sphere The formula for the volume V of a sphere is V =

4 3 pr . 3

Solve this equation for r. 73. Celsius and Fahrenheit The formula for Celsius temperature in terms of Fahrenheit temperature is C =

5 1F - 322. 9

Solve the equation for F.

6965_CH0P_pp001-062.qxd

28

1/14/10

12:44 PM

Page 28

CHAPTER P Prerequisites

P.4 Lines in the Plane What you’ll learn about • Slope of a Line • Point-Slope Form Equation of a Line

Slope of a Line The slope of a nonvertical line is the ratio of the amount of vertical change to the amount of horizontal change between two points. For the points 1x 1, y12 and 1x 2, y22, the vertical change is ¢y = y2 - y1 and the horizontal change is ¢x = x 2 - x 1. ¢y is read “delta” y. See Figure P.21.

• Slope-Intercept Form Equation of a Line

y

• Graphing Linear Equations in Two Variables • Parallel and Perpendicular Lines • Applying Linear Equations in Two Variables

... and why Linear equations are used extensively in applications involving business and behavioral science.

(x2, y2)

y2

!y = y2 – y1 y1

0

(x1, y1) !x = x2 – x1

x1

x

x2

FIGURE P.21 The slope of a nonvertical line can be found from the coordinates of any two points of the line.

DEFINITION Slope of a Line The slope of the nonvertical line through the points 1x 1, y12 and 1x 2, y22 is m =

¢y y2 - y1 = . x2 - x1 ¢x

If the line is vertical, then x 1 = x 2 and the slope is undefined.

EXAMPLE 1 Finding the Slope of a Line Find the slope of the line through the two points. Sketch a graph of the line. (a) 1-1, 22 and 14, - 22 (b) 11, 12 and 13, 42 SOLUTION

(a) The two points are 1x 1, y12 = 1- 1, 22 and 1x 2, y22 = 14, -22. Thus, m =

1- 22 - 2 y2 - y1 4 = = - . x2 - x1 4 - 1-12 5

(b) The two points are 1x 1, y12 = 11, 12 and 1x 2, y22 = 13, 42. Thus, Slope Formula The slope does not depend on the order of the points. We could use 1x 1, y12 = 14, - 22 and 1x 2, y22 = 1-1, 22 in Example 1a. Check it out.

m =

y2 - y1 3 4 - 1 = . = x2 - x1 3 - 1 2

The graphs of these two lines are shown in Figure P.22. Now try Exercise 3.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 29

SECTION P.4

Lines in the Plane

29

y

y

(3, 4) (–1, 2)

(1, 1) x

x (4, –2) y (3, 7)

FIGURE P.22 The graphs of the two lines in Example 1. Figure P.23 shows a vertical line through the points 13, 22 and 13, 72. If we try to calculate its slope using the slope formula 1y2 - y12/1x 2 - x 12, we get zero in the denominator. So, it makes sense to say that a vertical line does not have a slope, or that its slope is undefined.

(3, 2) x

Point-Slope Form Equation of a Line

FIGURE P.23 Applying the slope formula

to this vertical line gives m = 5/0, which is not defined. Thus, the slope of a vertical line is undefined. y

If we know the coordinates of one point on a line and the slope of the line, then we can find an equation for that line. For example, the line in Figure P.24 passes through the point 1x 1, y12 and has slope m. If 1x, y2 is any other point on this line, the definition of the slope yields the equation m =

y - y1 x - x1

or

y - y1 = m1x - x 12.

An equation written this way is in the point-slope form.

(x, y)

DEFINITION Point-Slope Form of an Equation of a Line (x1, y1) x Slope = m

FIGURE P.24 The line through 1x 1, y12

with slope m.

The point-slope form of an equation of a line that passes through the point 1x 1, y12 and has slope m is y - y1 = m1x - x 12.

EXAMPLE 2 Using the Point-Slope Form Use the point-slope form to find an equation of the line that passes through the point 1-3, - 42 and has slope 2.

SOLUTION Substitute x 1 = - 3, y1 = - 4, and m = 2 into the point-slope form, and simplify the resulting equation.

y-Intercept The b in y = mx + b is often referred to as “the y-intercept” instead of “the y-coordinate of the y-intercept.”

y - y1 y - 1-42 y + 4 y + 4 y

= = = = =

m1x - x 12 21x - 1-322 2x - 21-32 2x + 6 2x + 2

Point-slope form x1 = -3, y1 = -4, m = 2 Distributive property A common simplified form

Now try Exercise 11.

Slope-Intercept Form Equation of a Line The y-intercept of a nonvertical line is the point where the line intersects the y-axis. If we know the y-intercept and the slope of the line, we can apply the point-slope form to find an equation of the line.

6965_CH0P_pp001-062.qxd

2/2/10

11:19 AM

Page 30

CHAPTER P Prerequisites

30

y

Figure P.25 shows a line with slope m and y-intercept 10, b2. A point-slope form equation for this line is y - b = m1x - 02. By rewriting this equation we obtain the form known as the slope-intercept form.

(x, y)

DEFINITION Slope-Intercept Form of an Equation of a Line

Slope = m (0, b)

x

The slope-intercept form of an equation of a line with slope m and y-intercept 10, b2 is y = mx + b.

EXAMPLE 3 Using the Slope-Intercept Form FIGURE P.25 The line with slope m and

y-intercept 10, b2.

Write an equation of the line with slope 3 that passes through the point 1-1, 62 using the slope-intercept form. SOLUTION

y y 6 b

= = = =

mx + b 3x + b 31- 12 + b 9

Slope-intercept form m = 3 y = 6 when x = -1

The slope-intercept form of the equation is y = 3x + 9.

Now try Exercise 21.

We cannot use the phrase “the equation of a line” because each line has many different equations. Every line has an equation that can be written in the form Ax + By + C = 0 where A and B are not both zero. This form is the general form for an equation of a line. If B Z 0, the general form can be changed to the slope-intercept form as follows: Ax + By + C = 0 By = -Ax - C y = -

A C x + a- b B B

slope

y-intercept

Forms of Equations of Lines General form: Slope-intercept form: Point-slope form: Vertical line: Horizontal line:

Ax + By + C = 0, A and B not both zero y = mx + b y - y1 = m1x - x 12 x = a y = b

Graphing Linear Equations in Two Variables A linear equation in x and y is one that can be written in the form Ax + By = C, where A and B are not both zero. Rewriting this equation in the form Ax + By - C = 0 we see that it is the general form of an equation of a line. If B = 0, the line is vertical, and if A = 0, the line is horizontal.

6965_CH0P_pp001-062.qxd

1/21/10

7:17 AM

Page 31

SECTION P.4

Lines in the Plane

31

The graph of an equation in x and y consists of all pairs 1x, y2 that are solutions of the equation. For example, 11, 22 is a solution of the equation 2x + 3y = 8 because substituting x = 1 and y = 2 into the equation leads to the true statement 8 = 8. The pairs 1-2, 42 and 12, 4/32 are also solutions.

WINDOW Xmin=–10 Xmax=10 Xscl=1 Ymin=–10 Ymax=10 Yscl=1 Xres=1 FIGURE P.26 The window dimensions for the standard window. The notation “3 -10, 104 by 3 - 10, 104” is used to represent window dimensions like these.

Because the graph of a linear equation in x and y is a straight line, we need only to find two solutions and then connect them with a straight line to draw its graph. If a line is neither horizontal nor vertical, then two easy points to find are its x-intercept and the y-intercept. The x-intercept is the point 1x', 02 where the graph intersects the x-axis. Set y = 0 and solve for x to find the x-intercept. The coordinates of the y-intercept are 10, y'2. Set x = 0 and solve for y to find the y-intercept. Graphing with a Graphing Utility

To draw a graph of an equation using a grapher: 1. Rewrite the equation in the form y = 1an expression in x2. 2. Enter the equation into the grapher.

3. Select an appropriate viewing window (see Figure P.26). 4. Press the “graph” key. A graphing utility, often referred to as a grapher, computes y-values for a select set of x-values between Xmin and Xmax and plots the corresponding 1x, y2 points.

EXAMPLE 4 Use a Graphing Utility Draw the graph of 2x + 3y = 6. SOLUTION First we solve for y.

2x + 3y = 6 3y = - 2x + 6 2 y = - x + 2 3

[–4, 6] by [–3, 5]

FIGURE P.27 The graph of 2x + 3y = 6. The points 10, 22 1y-intercept2 and 13, 02 1x-intercept2 appear to lie on the graph and, as pairs, are solutions of the equation, providing visual support that the graph is correct. (Example 4) Viewing Window The viewing window 3-4, 64 by 3- 3, 54 in Figure P.27 means - 4 … x … 6 and - 3 … y … 5.

Solve for y. Divide by 3.

Figure P.27 shows the graph of y = -12/32x + 2, or equivalently, the graph of the linear equation 2x + 3y = 6 in the 3-4, 64 by 3- 3, 54 viewing window. Now try Exercise 27.

Parallel and Perpendicular Lines EXPLORATION 1

Investigating Graphs of Linear Equations

1. What do the graphs of y = mx + b and y = mx + c, b Z c, have in com-

mon? How are they different?

Square Viewing Window A square viewing window on a grapher is one in which angles appear to be true. For example, the line y = x will appear to make a 45° angle with the positive x-axis. Furthermore, a distance of 1 on the x- and y-axes will appear to be the same. That is, if Xscl = Yscl, the distance between consecutive tick marks on the x- and y-axes will appear to be the same.

2. Graph y = 2x and y = - 11/22x in a square viewing window (see margin

note). On the calculator we use, the “decimal window” 3-4.7, 4.74 by 3-3.1, 3.14 is square. Estimate the angle between the two lines.

3. Repeat part 2 for y = mx and y = - 11/m2x with m = 1, 3, 4, and 5.

Parallel lines and perpendicular lines were involved in Exploration 1. Using a grapher to decide when lines are parallel or perpendicular is risky. Here is an algebraic test to determine when two lines are parallel or perpendicular.

6965_CH0P_pp001-062.qxd

32

1/14/10

12:44 PM

Page 32

CHAPTER P Prerequisites

Parallel and Perpendicular Lines 1. Two nonvertical lines are parallel if and only if their slopes are equal. 2. Two nonvertical lines are perpendicular if and only if their slopes m 1 and m 2 are opposite reciprocals, that is, if and only if m1 = -

1 . m2

EXAMPLE 5 Finding an Equation of a Parallel Line Find an equation of the line through P11, -22 that is parallel to the line L with equation 3x - 2y = 1. SOLUTION We find the slope of L by writing its equation in slope-intercept form.

3x - 2y = 1 -2y = - 3x + 1 3 1 y = x 2 2

Equation for L Subtract 3x. Divide by - 2.

The slope of L is 3/2. The line whose equation we seek has slope 3/2 and contains the point 1x 1, y12 = 11, -22. Thus, the point-slope form equation for the line we seek is 3 1x - 12 2 3 3 y + 2 = x 2 2 3 7 y = x 2 2 y + 2 =

Distributive property

Now try Exercise 41(a).

EXAMPLE 6 Finding an Equation of a Perpendicular Line Find an equation of the line through P12, -32 that is perpendicular to the line L with equation 4x + y = 3. Support the result with a grapher. SOLUTION We find the slope of L by writing its equation in slope-intercept form.

4x + y = 3 y = - 4x + 3

Equation for L Subtract 4x.

The slope of L is -4. The line whose equation we seek has slope -1/1- 42 = 1/4 and passes through the point 1x 1, y12 = 12, -32. Thus, the point-slope form equation for the line we seek is 1 1x - 22 4 1 2 y + 3 = x 4 4 1 7 y = x 4 2

y - 1 -32 =

[–4.7, 4.7] by [–5.1, 1.1]

FIGURE P.28 The graphs of

y = -4x + 3 and y = 11/42x - 7/2 in this square viewing window appear to intersect at a right angle. (Example 6)

Distributive property

Figure P.28 shows the graphs of the two equations in a square viewing window and suggests that the graphs are perpendicular. Now try Exercise 43(b).

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 33

SECTION P.4

Lines in the Plane

33

Applying Linear Equations in Two Variables Linear equations and their graphs occur frequently in applications. Algebraic solutions to these application problems often require finding an equation of a line and solving a linear equation in one variable. Grapher techniques complement algebraic ones.

EXAMPLE 7 Finding the Depreciation of Real Estate Camelot Apartments purchased a $50,000 building and depreciates it $2000 per year over a 25-year period. (a) Write a linear equation giving the value y of the building in terms of the years x after the purchase. (b) In how many years will the value of the building be $24,500? SOLUTION

(a) We need to determine the value of m and b so that y = mx + b, where 0 … x … 25. We know that y = 50,000 when x = 0, so the line has y-intercept 10, 50,0002 and b = 50,000. One year after purchase 1x = 12, the value of the building is 50,000 - 2,000 = 48,000. So when x = 1, y = 48,000. Using algebra, we find

1

y = mx + b 48,000 = m # 1 + 50,000 -2000 = m

y = 48,000 when x = 1

The value y of the building x years after its purchase is y = -2000x + 50,000.

X=12.75

Y=24500

[0, 23.5] by [0, 60000]

(a)

X 12 12.25 12.5 12.75 13 13.25 13.5

Y1 26000 25500 25000 24500 24000 23500 23000

Y1 = –2000X+50000 (b)

FIGURE P.29 A (a) graph and (b) table of values for y = - 2000x + 50,000. (Example 7)

(b) We need to find the value of x when y = 24,500. y = - 2000x + 50,000 Again, using algebra we find 24,500 = - 2000x + 50,000 -25,500 = - 2000x 12.75 = x

Set y = 24,500. Subtract 50,000.

The depreciated value of the building will be $24,500 exactly 12.75 years, or 12 years 9 months, after purchase by Camelot Apartments. We can support our algebraic work both graphically and numerically. The trace coordinates in Figure P.29a show graphically that 112.75, 24,5002 is a solution of y = - 2000x + 50,000. This means that y = 24,500 when x = 12.75. Figure P.29b is a table of values for y = - 2000x + 50,000 for a few values of x. The fourth line of the table shows numerically that y = 24,500 when x = 12.75. Now try Exercise 45.

Figure P.30 on page 34 shows Americans’ income from 2002 to 2007 in trillions of dollars and a corresponding scatter plot of the data. In Example 8, we model the data in Figure P.30 with a linear equation.

6965_CH0P_pp001-062.qxd

34

1/14/10

12:44 PM

Page 34

CHAPTER P Prerequisites

Year

Amount (trillions of dollars)

2002 2003 2004 2005 2006 2007

8.9 9.2 9.7 10.3 11.0 11.7 [2000, 2010] by [5, 15]

FIGURE P.30 Americans’ Personal Income. Source: U.S. Census Bureau, The World Almanac and Book of Facts 2009. (Example 8)

EXAMPLE 8 Finding a Linear Model for Americans’ Personal Income American’s personal income in trillions of dollars is given in Figure P.30. (a) Write a linear equation for Americans’ income y in terms of the year x using the points 12002, 8.92 and 12003, 9.22. (b) Use the equation in (a) to estimate Americans’ income in 2005. (c) Use the equation in (a) to predict Americans’ income in 2010. (d) Superimpose a graph of the linear equation in (a) on a scatter plot of the data. SOLUTION

(a) Let y = mx + b. The slope of the line through the two points 12002, 8.92 and 12003, 9.22 is 9.2 - 8.9 m = = 0.3. 2003 - 2002 The value of 8.9 trillion dollars in 2002 gives y = 8.9 when x = 2002. y y 8.9 b b

= = = = =

mx + b 0.3x + b 0.3120022 + b 8.9 - 10.32120022 - 591.7

m = 0.3 y = 8.9 when x = 2002

The linear equation we seek is y = 0.3x - 591.7. (b) We need to find the value of y when x = 2005. y = 0.3x - 591.7 y = 0.3120052 - 591.7 y = 9.8

Set x = 2005.

Using the linear model we found in (a) we estimate Americans’ income in 2005 to be 9.8 trillion dollars, a little less than the actual amount 10.3 trillion. (c) We need to find the value of y when x = 2010. y = 0.3x - 591.7 y = 0.3120102 - 591.7 y = 11.3 [2000, 2010] by [5, 15]

FIGURE P.31 Linear model for Americans’ personal income. (Example 8)

Set x = 2010.

Using the linear model we found in (a) we predict Americans’ income in 2010 to be 11.3 trillion dollars. (d) The graph and scatter plot are shown in Figure P.31. Now try Exercise 51.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 35

SECTION P.4

Lines in the Plane

35

Chapter Opener Problem (from page 1) Problem: Assume that the speed of light is approximately 186,000 miles per second. (It took a long time to arrive at this number. See the note below about the speed of light.) (a) If the distance from the Moon to the Earth is approximately 237,000 miles, find the length of time required for light to travel from the Earth to the Moon. (b) If light travels from the Earth to the Sun in 8.32 minutes, approximate the distance from the Earth to the Sun. (c) If it takes 5 hours and 29 seconds for light to travel from the Sun to Pluto, approximate the distance from the Sun to Pluto. Solution: We use the linear equation d = r * t 1distance = rate * time2 to make the calculations with r = 186,000 miles/second. (a) Here d = 237,000 miles, so t =

The Speed of Light Many scientists have tried to measure the speed of light. For example, Galileo Galilei (1564–1642) attempted to measure the speed of light without much success. Visit the following Web site for some interesting information about this topic: http://www.what-is-the-speed-of-light.com/

237,000 miles d = L 1.27 seconds. r 186,000 miles/second

The length of time required for light to travel from the Earth to the Moon is about 1.27 seconds. (b) Here t = 8.32 minutes = 499.2 seconds, so miles d = r * t = 186,000 * 499.2 seconds = 92,851,200 miles. second The distance from the Earth to the Sun is about 93 million miles. (c) Here t = 5 hours and 29 minutes = 329 minutes = 19,740 seconds, so miles d = r * t = 186,000 * 19,740 seconds second = 3,671,640,000 miles. The distance from the Sun to Pluto is about 3.7 * 10 9 miles.

QUICK REVIEW P.4 Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 1–4, solve for x. 1. -75x + 25 = 200 2. 400 - 50x = 150 3. 311 - 2x2 + 412x - 52 = 7 4. 217x + 12 = 511 - 3x2

In Exercises 5–8, solve for y. 1 1 x + y = 2 3 4

5. 2x - 5y = 21

6.

7. 2x + y = 17 + 21x - 2y2

8. x 2 + y = 3x - 2y

In Exercises 9 and 10, simplify the fraction. 9.

9 - 5 - 2 - 1- 82

10.

-4 - 6 -14 - 1 -22

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 36

CHAPTER P Prerequisites

36

SECTION P.4 EXERCISES Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator.

25. The line 2x + 5y = 12

In Exercises 1 and 2, estimate the slope of the line.

26. The line 7x - 12y = 96

y

1.

y

2.

9 8 7 6 5 4 3 2 1 0

In Exercises 27–30, graph the linear equation on a grapher. Choose a viewing window that shows the line intersecting both the x- and y-axes.

9 8 7 6 5 4 3 2 1 1 2

3

4

5

6

x

0

27. 8x + y = 49

28. 2x + y = 35

29. 123x + 7y = 429

30. 2100x + 12y = 3540

In Exercises 31 and 32, the line contains the origin and the point in the upper right corner of the grapher screen. 31. Writing to Learn Which line shown here has the greater slope? Explain.

1

2

3

4

5

6

x

In Exercises 3–6, find the slope of the line through the pair of points. 3. 1 -3, 52 and 14, 92

5. 1 - 2, -52 and 1- 1, 32

4. 1 -2, 12 and 15, -32

6. 15, -32 and 1- 4, 122

In Exercises 7–10, find the value of x or y so that the line through the pair of points has the given slope. Points

[–10, 10] by [–10, 10]

(a)

(b)

32. Writing to Learn Which line shown here has the greater slope? Explain.

Slope

7. 1x, 32 and 15, 92

m = 2

9. (- 3, -5) and 14, y2

m = 3

8. 1- 2, 32 and 14, y2

10. 1-8, - 22 and 1x, 22

m = -3 m = 1/2

In Exercises 11–14, find a point-slope form equation for the line through the point with given slope. Point

Slope

11. 11, 42

m = 2

13. 15, - 42

[–10, 10] by [–15, 15]

m = -2

Point 12. 1- 4, 32

14. 1- 3, 42

Slope m = - 2/3 m = 3

In Exercises 15–20, find a general form equation for the line through the pair of points. 15. 1 -7, - 22 and 11, 62

16. 1 - 3, -82 and 14, -12

19. 1 -1, 22 and 12, 52

20. 14, -12 and 14, 52

17. 11, -32 and 15, -32

18. 1-1, -52 and 1- 4, -22

In Exercises 21–26, find a slope-intercept form equation for the line. 21. The line through 10, 52 with slope m = - 3 22. The line through 11, 22 with slope m = 1/2

23. The line through the points 1 - 4, 52 and 14, 32 24. The line through the points 14, 22 and 1- 3, 12

[–20, 20] by [–35, 35]

[–5, 5] by [–20, 20]

(a)

(b)

In Exercises 33–36, find the value of x and the value of y for which 1x, 142 and 118, y2 are points on the graph. 33. y = 0.5x + 12

34. y = -2x + 18

35. 3x + 4y = 26

36. 3x - 2y = 14

In Exercises 37–40, find the values for Ymin, Ymax, and Yscl that will make the graph of the line appear in the viewing window as shown here. WINDOW Xmin=–10 Xmax=10 Xscl=1 Ymin= Ymax= Yscl= Xres=1

37. y = 3x 39. y =

2 x 3

38. y = 5x 40. y =

5 x 4

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 37

SECTION P.4

(a) Find the slope of this section of the highway.

In Exercises 41–44, (a) find an equation for the line passing through the point and parallel to the given line, and (b) find an equation for the line passing through the point and perpendicular to the given line. Support your work graphically. Point 41. 11, 22

42. 1 - 2, 32 43. 13, 12 44. 16, 12

37

Lines in the Plane

(b) On a highway with a 6% grade what is the horizontal distance required to climb 250 ft? (c) A sign along the highway says 6% grade for the next 7 mi. Estimate how many feet of vertical change there are along those next 7 mi. (There are 5280 ft in 1 mile.)

Line y = 3x - 2 y = - 2x + 4

49. Writing to Learn Building Specifications Asphalt shingles do not meet code specifications on a roof that has less than a 4-12 pitch. A 4-12 pitch means there are 4 ft of vertical change in 12 ft of horizontal change. A certain roof has slope m = 3/8. Could asphalt shingles be used on that roof? Explain.

2x + 3y = 12 3x - 5y = 15

45. Real Estate Appreciation Bob Michaels purchased a house 8 years ago for $42,000. This year it was appraised at $67,500. (a) A linear equation V = mt + b, 0 … t … 15, represents the value V of the house for 15 years after it was purchased. Determine m and b.

50. Revisiting Example 8 Use the linear equation found in Example 8 to estimate Americans’ income in 2004, 2006, 2007 displayed in Figure P.30.

(b) Graph the equation and trace to estimate in how many years after purchase this house will be worth $72,500.

51. Americans’ Spending Americans’ personal consumption expenditures for several years from 1990 to 2007 in trillions of dollars are shown in the table. (Source: U.S. Bureau of Economic Analysis as reported in The World Almanac and Book of Facts 2009.)

(c) Write and solve an equation algebraically to determine how many years after purchase this house will be worth $74,000. (d) Determine how many years after purchase this house will be worth $80,250. 46. Investment Planning Mary Ellen plans to invest $18,000, putting part of the money x into a savings that pays 5% annually and the rest into an account that pays 8% annually.

1990

1995

2000

2005

2006

2007

y

3.8

5.0

6.7

8.7

9.2

9.7

(a) Write a linear equation for Americans’ spending y in terms of the year x, using the points (1990, 3.8) and (1995, 5.0).

(a) What are the possible values of x in this situation? (b) If Mary Ellen invests x dollars at 5%, write an equation that describes the total interest I received from both accounts at the end of one year.

(b) Use the equation in (a) to estimate Americans’ expenditures in 2006. (c) Use the equation in (a) to predict Americans’ expenditures in 2010.

(c) Graph and trace to estimate how much Mary Ellen invested at 5% if she earned $1020 in total interest at the end of the first year.

(d) Superimpose a graph of the linear equation in (a) on a scatter plot of the data.

(d) Use your grapher to generate a table of values for I to find out how much Mary Ellen should invest at 5% to earn $1185 in total interest in one year.

52. U.S. Imports from Mexico The total y in billions of dollars of U.S. imports from Mexico for each year x from 2000 to 2007 is given in the table. (Source: U.S. Census Bureau as reported in The World Almanac and Book of Facts 2009.)

47. Navigation A commercial jet airplane climbs at takeoff with slope m = 3/8. How far in the horizontal direction will the airplane fly to reach an altitude of 12,000 ft above the takeoff point? 48. Grade of a Highway Interstate 70 west of Denver, Colorado, has a section posted as a 6% grade. This means that for a horizontal change of 100 ft there is a 6-ft vertical change.

x

x

2000

2001

2002

2003

2004

2005

2006

2007

y

135

131.3

134.6

138.1

155.9

170.1

198.2

210.7

(a) Use the pairs 12001, 131.32 and 12005, 170.12 to write a linear equation for x and y. (b) Superimpose the graph of the linear equation in (a) on a scatter plot of the data.

6% grade 6% GRADE

(c) Use the equation in (a) to predict the total U.S. imports from Mexico in 2010. 53. The midyear world population in millions for some of the years from 1980 to 2008 is shown in Table P.7.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 38

CHAPTER P Prerequisites

38

In Exercises 57 and 58, determine a and b so that figure ABCD is a parallelogram.

Table P.7 World Population Year

Population (millions)

1980 1990 1995 2000 2007 2008

4453 5282 5691 6085 6628 6707

57.

B(2, 5)

A(0, 0)

Source: U.S. Census Bureau as reported in The Statistical Abstracts of the United States, 2009.

(b) Use the 1980 and 2008 data to write a linear equation for the population y in terms of the year x. Superimpose the graph of the linear equation on the scatter plot of the data. (c) Use the graph in (b) to predict the midyear world population in 2010. 54. U.S. Exports to Canada The total in billions of dollars of U.S. exports to Canada from 2000 to 2009 is given in Table P.8.

Table P.8 U.S. Exports to Canada Year

U.S. Exports (billions of dollars)

2000 2001 2002 2003 2004 2005 2006 2007

178.9 163.4 160.9 169.9 189.9 211.9 230.6 248.9

In Exercises 55 and 56, determine a so that the line segments AB and CD are parallel. y

D(5, a)

B(3, 4) B(1, 2) C(3, 0)

x

A(0, 0)

A(0, 0)

D(5, 0)

x

(a) Is it possible for two lines with positive slopes to be perpendicular? Explain. (b) Is it possible for two lines with negative slopes to be perpendicular? Explain. 60. Group Activity Parallel and Perpendicular Lines (a) Assume that c Z d and a and b are not both zero. Show that ax + by = c and ax + by = d are parallel lines. Explain why the restrictions on a, b, c, and d are necessary. (b) Assume that a and b are not both zero. Show that ax + by = c and bx - ay = d are perpendicular lines. Explain why the restrictions on a and b are necessary.

In Exercises 63–66, you may use a graphing calculator to solve these problems.

(c) Use the equation in (b) to predict the U.S. exports to Canada in 2010.

A(0, 0)

x

62. True or False The graph of any equation of the form ax + by = c, where a and b are not both zero, is always a line. Justify your answer.

(b) Use the 2000 and 2007 data to write a linear equation for the U.S. exports to Canada y in terms of the year x. Superimpose the graph of the linear equation on the scatter plot in (a).

D(a, 8)

D(4, 0)

C(8, 4)

B(a, b)

61. True or False The slope of a vertical line is zero. Justify your answer.

(a) Let x = 0 represent 2000, x = 1 represent 1991, and so forth. Draw a scatter plot of the data.

56.

C(a, b)

Standardized Test Questions

Source: U.S. Census Bureau, The World Almanac and Book of Facts 2009.

y

y

59. Writing to Learn Perpendicular Lines

(a) Let x = 0 represent 1980, x = 1 represent 1981, and so forth. Draw a scatter plot of the data.

55.

58.

y

C(3, 0)

x

63. Multiple Choice Which of the following is an equation of the line through the point 1- 2, 32 with slope 4? (A) y - 3 = 41x + 22

(B) y + 3 = 41x - 22

(C) x - 3 = 41y + 22

(D) x + 3 = 41y - 22

(E) y + 2 = 41x - 32 64. Multiple Choice Which of the following is an equation of the line with slope 3 and y-intercept - 2? (A) y = 3x + 2

(B) y = 3x - 2

(C) y = -2x + 3

(D) x = 3y - 2

(E) x = 3y + 2 65. Multiple Choice Which of the following lines is perpendicular to the line y = -2x + 5? 1 5

(A) y = 2x + 1

(B) y = -2x -

1 1 (C) y = - x + 2 3

1 (D) y = - x + 3 2

(E) y =

1 x - 3 2

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 39

SECTION P.4

66. Multiple Choice Which of the following is the slope of the line through the two points 1 - 2, 12 and 11, -42? (A) -

3 5

(B)

3 5

(C) -

5 3

(D)

5 3

39

Lines in the Plane

Extending the Ideas 69. Connecting Algebra and Geometry Show that if the midpoints of consecutive sides of any quadrilateral (see figure) are connected, the result is a parallelogram. y

(b, c)

y

(E) - 3

Explorations

(d, e) (3, 4)

y x 67. Exploring the Graph of ! " c, a # 0, b # 0 a b

x

(a, 0)

x

Let c = 1. (a) Draw the graph for a = 3, b = - 2. (b) Draw the graph for a = - 2, b = - 3. (c) Draw the graph for a = 5, b = 3. (d) Use your graphs in (a), (b), (c) to conjecture what a and b represent when c = 1. Prove your conjecture. (e) Repeat (a)–(d) for c = 2. (f ) If c = - 1, what do a and b represent? 68. Investigating Graphs of Linear Equations (a) Graph y = mx for m = - 3, -2, - 1, 1, 2, 3 in the window 3 - 8, 84 by 3 - 5, 54. What do these graphs have in common? How are they different? (b) If m 7 0, what do the graphs of y = mx and y = -mx have in common? How are they different?

(c) Graph y = 0.3x + b for b = - 3, -2, - 1, 0, 1, 2, 3 in 3 - 8, 84 by 3 - 5, 54. What do these graphs have in common? How are they different?

Art for Exercise 69

Art for Exercise 70

70. Connecting Algebra and Geometry Consider the semicircle of radius 5 centered at 10, 02 as shown in the figure. Find an equation of the line tangent to the semicircle at the point 13, 42. (Hint: A line tangent to a circle is perpendicular to the radius at the point of tangency.) 71. Connecting Algebra and Geometry Show that in any triangle (see figure), the line segment joining the midpoints of two sides is parallel to the third side and is half as long. y (b, c) A O (0, 0)

B (a, 0)

x

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 40

CHAPTER P Prerequisites

40

P.5 Solving Equations Graphically, Numerically, and Algebraically What you’ll learn about • Solving Equations Graphically • Solving Quadratic Equations • Approximating Solutions of Equations Graphically • Approximating Solutions of Equations Numerically with Tables • Solving Equations by Finding Intersections

Solving Equations Graphically The graph of the equation y = 2x - 5 1in x and y2 can be used to solve the equation 2x - 5 = 0 (in x). Using the techniques of Section P.3, we can show algebraically that x = 5/2 is a solution of 2x - 5 = 0. Therefore, the ordered pair 15/2, 02 is a solution of y = 2x - 5. Figure P.32 suggests that the x-intercept of the graph of the line y = 2x - 5 is the point 15/2, 02 as it should be.

One way to solve an equation graphically is to find all its x-intercepts. There are many graphical techniques that can be used to find x-intercepts.

... and why

1

These basic techniques are involved in using a graphing utility to solve equations in this textbook.

X=2.5

Y=0 [–4.7, 4.7] by [–10, 5]

FIGURE P.32 Using the TRACE feature of a grapher, we see that 12.5, 02 is an x-intercept of the graph of y = 2x - 5 and, therefore, x = 2.5 is a solution of the equation 2x - 5 = 0.

EXAMPLE 1 Solving by Finding x-Intercepts Solve the equation 2x 2 - 3x - 2 = 0 graphically. SOLUTION

Solve Graphically

Objective Students will be able to solve equations involving quadratic, absolute value, and fractional expressions by finding x-intercepts or intersections on graphs, by using algebraic techniques, or by using numerical techniques.

Find the x-intercepts of the graph of y = 2x 2 - 3x - 2 (Figure P.33). We use TRACE to see that 1- 0.5, 02 and 12, 02 are x-intercepts of this graph. Thus, the solutions of this equation are x = - 0.5 and x = 2. Answers obtained graphically are really approximations, although in general they are very good approximations. Solve Algebraically In this case, we can use factoring to find exact values.

1 We can conclude that

2x 2 - 3x - 2 = 0 12x + 121x - 22 = 0

Factor.

2x + 1 = 0 or x - 2 = 0, x = - 1/2 or x = 2.

X=–.5

Y=0 [–4.7, 4.7] by [–5, 5]

FIGURE P.33 It appears that 1 -0.5, 02 and 12, 02 are x-intercepts of the graph of y = 2x 2 - 3x - 2. (Example 1)

So, x = -1/2 and x = 2 are the exact solutions of the original equation. Now try Exercise 1. The algebraic solution procedure used in Example 1 is a special case of the following important property.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 41

SECTION P.5 Solving Equations Graphically, Numerically, and Algebraically

41

Zero Factor Property Let a and b be real numbers. If ab = 0, then a = 0 or b = 0.

Solving Quadratic Equations Linear equations (ax + b = 0) and quadratic equations are two members of the family of polynomial equations, which will be studied in more detail in Chapter 2.

DEFINITION Quadratic Equation in x A quadratic equation in x is one that can be written in the form ax 2 + bx + c = 0, where a, b, and c are real numbers with a Z 0.

We review some of the basic algebraic techniques for solving quadratic equations. One algebraic technique that we have already used in Example 1 is factoring. Quadratic equations of the form 1ax + b22 = c are fairly easy to solve as illustrated in Example 2. Square Root Principle 2

If t = K 7 0, then t = 2K or t = - 2K.

EXAMPLE 2 Solving by Extracting Square Roots Solve 12x - 122 = 9 algebraically. SOLUTION

12x - 122 = 9 2x - 1 = !3 2x = 4 or 2x = -2 x = 2 or

Extract square roots.

x = -1

Now try Exercise 9.

The technique of Example 2 is more general than you might think because every quadratic equation can be written in the form 1x + b22 = c. The procedure we need to accomplish this is completing the square. Completing the Square

To solve x 2 + bx = c by completing the square, add 1b/222 to both sides of the equation and factor the left side of the new equation. b 2 b 2 x 2 + bx + a b = c + a b 2 2 b2 b 2 ax + b = c + 2 4

To solve a quadratic equation by completing the square, we simply divide both sides of the equation by the coefficient of x 2 and then complete the square as illustrated in Example 3.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 42

CHAPTER P Prerequisites

42

EXAMPLE 3 Solving by Completing the Square Solve 4x 2 - 20x + 17 = 0 by completing the square. SOLUTION

4x 2 - 20x + 17 = 0 17 x 2 - 5x + = 0 4 x 2 - 5x = -

Divide by 4.

17 4

Subtract a

Completing the square on the equation above we obtain

5 2 17 5 2 x 2 - 5x + a - b = + a- b 2 4 2 5 2 ax - b = 2 2 5 x - = ! 22 2 5 x = ! 22 2 5 5 x = + 22 L 3.91 or x = - 22 L 1.09 2 2

17 4

b.

Add a -

5 2 b . 2

Factor and simplify. Extract square roots.

Now try Exercise 13.

The procedure of Example 3 can be applied to the general quadratic equation ax 2 + bx + c = 0 to produce the following formula for its solutions (see Exercise 68).

Quadratic Formula The solutions of the quadratic equation ax 2 + bx + c = 0, where a Z 0, are given by the quadratic formula x =

-b ! 2b 2 - 4ac . 2a

EXAMPLE 4 Solving Using the Quadratic Formula Solve the equation 3x 2 - 6x = 5. SOLUTION First we subtract 5 from both sides of the equation to put it in the form ax 2 + bx + c = 0: 3x 2 - 6x - 5 = 0. We can see that a = 3, b = -6, and c = - 5.

x = x =

- b ! 2b 2 - 4ac 2a - 1- 62 ! 21-622 - 41321-52 2132

Quadratic formula a = 3, b = -6, c = -5

6 ! 296 Simplify. 6 6 + 296 6 - 296 x = L 2.63 or x = L - 0.63 6 6

x =

[–5, 5] by [–10, 10]

FIGURE P.34 The graph of 2

y = 3x - 6x - 5. (Example 4)

The graph of y = 3x 2 - 6x - 5 in Figure P.34 supports that the x-intercepts are approximately - 0.63 and 2.63. Now try Exercise 19.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 43

SECTION P.5 Solving Equations Graphically, Numerically, and Algebraically

43

Solving Quadratic Equations Algebraically There are four basic ways to solve quadratic equations algebraically. 1. Factoring (see Example 1) 2. Extracting Square Roots (see Example 2) 3. Completing the Square (see Example 3)

Zero X=1.324718 Y=0

4. Using the Quadratic Formula (see Example 4)

[–4.7, 4.7] by [–3.1, 3.1]

(a)

1.324718 X X3–X–1

1.324718 1.823355E–7

Approximating Solutions of Equations Graphically A solution of the equation x 3 - x - 1 = 0 is a value of x that makes the value of y = x 3 - x - 1 equal to zero. Example 5 illustrates a built-in procedure on graphing calculators to find such values of x.

EXAMPLE 5 Solving Graphically (b)

FIGURE P.35 The graph of y = x 3 -

x - 1. (a) shows that 11.324718, 0) is an approximation to the x-intercept of the graph. (b) supports this conclusion. (Example 5)

Solve the equation x 3 - x - 1 = 0 graphically. SOLUTION Figure P.35a suggests that x = 1.324718 is the solution we seek. Figure P.35b provides numerical support that x = 1.324718 is a close approximation to the solution because, when x = 1.324718, x 3 - x - 1 L 1.82 * 10 - 7, which is nearly zero. Now try Exercise 31.

When solving equations graphically, we usually get approximate solutions and not exact solutions. We will use the following agreement about accuracy in this book.

Agreement About Approximate Solutions For applications, round to a value that is reasonable for the context of the problem. For all others round to two decimal places unless directed otherwise.

With this accuracy agreement, we would report the solution found in Example 5 as 1.32.

Approximating Solutions of Equations Numerically with Tables The table feature on graphing calculators provides a numerical zoom-in procedure that we can use to find accurate solutions of equations. We illustrate this procedure in Example 6 using the same equation of Example 5.

EXAMPLE 6 Solving Using Tables Solve the equation x 3 - x - 1 = 0 using grapher tables. SOLUTION From Figure P.35a, we know that the solution we seek is between

x = 1 and x = 2. Figure P.36a sets the starting point of the table 1TblStart = 12 at x = 1 and increments the numbers in the table 1 ¢Tbl = 0.12 by 0.1. Figure P.36b shows that the zero of x 3 - x - 1 is between x = 1.3 and x = 1.4.

6965_CH0P_pp001-062.qxd

44

1/14/10

12:44 PM

Page 44

CHAPTER P Prerequisites

TABLE SETUP TblStart=1 ∆Tbl=.1 Indpnt: Auto Ask Depend: Auto Ask

X 1 1.1 1.2 1.3 1.4 1.5 1.6

Y1 –1 –.769 –.472 –.103 .344 .875 1.496

Y1 = X3–X–1 (b)

(a)

FIGURE P.36 (a) gives the setup that produces the table in (b). (Example 6) The next two steps in this process are shown in Figure P.37.

X

X

Y1

1.3 1.31 1.32 1.33 1.34 1.35 1.36

–.103 –.0619 –.02 .02264 .0661 .11038 .15546

1.32 1.321 1.322 1.323 1.324 1.325 1.326

Y1 = X3–X–1

Y1 –.02 –.0158 –.0116 –.0073 –.0031 .0012 .00547

Y1 = X3–X–1 (a)

(b)

FIGURE P.37 In (a) TblStart = 1.3 and ¢Tbl = 0.01, and in (b) TblStart = 1.32 and ¢ Tbl = 0.001. (Example 6)

From Figure P.37a, we can read that the zero is between x = 1.32 and x = 1.33; from Figure P.37b, we can read that the zero is between x = 1.324 and x = 1.325. Because all such numbers round to 1.32, we can report the zero as 1.32 with our accuracy agreement. Now try Exercise 37.

EXPLORATION 1

Finding Real Zeros of Equations

2 Consider the equation 4x - 12x + 7 = 0.

1. Use a graph to show that this equation has two real solutions, one between 0

and 1 and the other between 2 and 3. 2. Use the numerical zoom-in procedure illustrated in Example 6 to find each

zero accurate to two decimal places. 3. Use the built-in zero finder (see Example 5) to find the two solutions. Then

round them to two decimal places. 4. If you are familiar with the graphical zoom-in process, use it to find each solu-

tion accurate to two decimal places. 5. Compare the numbers obtained in parts 2, 3, and 4. 6. Support the results obtained in parts 2, 3, and 4 numerically. 7. Use the numerical zoom-in procedure illustrated in Example 6 to find each

zero accurate to six decimal places. Compare with the answer found in part 3 with the zero finder.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 45

SECTION P.5 Solving Equations Graphically, Numerically, and Algebraically

45

Solving Equations by Finding Intersections Sometimes we can rewrite an equation and solve it graphically by finding the points of intersection of two graphs. A point 1a, b2 is a point of intersection of two graphs if it lies on both graphs. We illustrate this procedure with the absolute value equation in Example 7.

EXAMPLE 7 Solving by Finding Intersections Solve the equation ƒ 2x - 1 ƒ = 6. SOLUTION Figure P.38 suggests that the V-shaped graph of y = ƒ 2x - 1 ƒ intersects the graph of the horizontal line y = 6 twice. We can use TRACE or the intersection feature of our grapher to see that the two points of intersection have coordinates 1 -2.5, 62 and 13.5, 62. This means that the original equation has two solutions: -2.5 and 3.5.

We can use algebra to find the exact solutions. The only two real numbers with absolute value 6 are 6 itself and -6. So, if ƒ 2x - 1 ƒ = 6, then 2x - 1 = 6 or 2x - 1 = - 6 5 7 = 3.5 or x = - = -2.5 x = 2 2

Now try Exercise 39.

Intersection X=–2.5 Y=6 [–4.7, 4.7] by [–5, 10]

FIGURE P.38 The graphs of y = ƒ 2x - 1 ƒ and

y = 6 intersect at 1 -2.5, 62 and 13.5, 62. (Example 7)

QUICK REVIEW P.5 In Exercises 1–4, expand the product. 2

1. 13x - 42

3. 12x + 1213x - 52

2

2. 12x + 32

4. 13y - 1215y + 42

In Exercises 9 and 10, combine the fractions and reduce the resulting fraction to lowest terms. 9.

In Exercises 5–8, factor completely. 5. 25x 2 - 20x + 4

7. 3x 3 + x 2 - 15x - 5

6. 15x 3 - 22x 2 + 8x 8. y 4 - 13y 2 + 36

10.

2 x 2x + 1 x + 3 x + 1 x 2 - 5x + 6

-

3x + 11 x2 - x - 6

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 46

CHAPTER P Prerequisites

46

SECTION P.5 EXERCISES In Exercises 1–6, solve the equation graphically by finding x-intercepts. Confirm by using factoring to solve the equation. 1. x 2 - x - 20 = 0 2

2. 2x 2 + 5x - 3 = 0

3. 4x - 8x + 3 = 0

4. x 2 - 8x = -15

5. x13x - 72 = 6

6. x13x + 112 = 20

In Exercises 7–12, solve the equation by extracting square roots. 7. 4x 2 = 25

8. 21x - 522 = 17

2

9. 31x + 42 = 8 11. 2y 2 - 8 = 6 - 2y 2

10. 41u + 122 = 18 12. 12x + 322 = 169

In Exercises 13–18, solve the equation by completing the square. 13. x 2 + 6x = 7 5 15. x 2 - 7x + = 0 4

14. x 2 + 5x - 9 = 0 16. 4 - 6x = x 2

17. 2x 2 - 7x + 9 = 1x - 321x + 12 + 3x

18. 3x 2 - 6x - 7 = x 2 + 3x - x1x + 12 + 3 In Exercises 19–24, solve the equation using the quadratic formula. 19. x 2 + 8x - 2 = 0 21. 3x + 4 = x

2

23. x(x + 5) = 12

35.

X .4 .41 .42 .43 .44 .45 .46

36.

Y1

X

–.04 –.0119 .0164 .0449 .0736 .1025 .1316

–1.735 –1.734 –1.733 –1.732 –1.731 –1.73 –1.729

Y1 –.0177 –.0117 –.0057 3E–4 .0063 .01228 .01826

Y1 = X3–3X

Y1 = X2+2X–1

In Exercises 37 and 38, use tables to find the indicated number of solutions of the equation accurate to two decimal places. 37. Two solutions of x 2 - x - 1 = 0 38. One solution of -x 3 + x + 1 = 0 In Exercises 39–44, solve the equation graphically by finding intersections. Confirm your answer algebraically. 39. ƒ t - 8 ƒ = 2

40. ƒ x + 1 ƒ = 4

41. ƒ 2x + 5 ƒ = 7 43. ƒ 2x - 3 ƒ = x

42. ƒ 3 - 5x ƒ = 4 2

44. ƒ x + 1 ƒ = 2x - 3

45. Interpreting Graphs The graphs in the two viewing windows shown here can be used to solve the equation 32x + 4 = x 2 - 1 graphically.

20. 2x 2 - 3x + 1 = 0 22. x 2 - 5 = 23x

24. x 2 - 2x + 6 = 2x 2 - 6x - 26 In Exercises 25–28, estimate any x- and y-intercepts that are shown in the graph. 25.

26.

[–5, 5] by [–10, 10]

[–5, 5] by [–10, 10]

(a)

(b)

(a) The viewing window in (a) illustrates the intersection method for solving. Identify the two equations that are graphed.

(b) The viewing window in (b) illustrates the x-intercept [–5, 5] by [–5, 5]

[–3, 6] by [–3, 8]

method for solving. Identify the equation that is graphed.

(c) Writing to Learn How are the intersection points in (a) related to the x-intercepts in (b)?

27.

28.

46. Writing to Learn Revisiting Example 6 Explain why all real numbers x that satisfy 1.324 6 x 6 1.325 round to 1.32. In Exercises 47–56, use a method of your choice to solve the equation.

[–5, 5] by [–5, 5]

[–3, 3] by [–3, 3]

In Exercises 29–34, solve the equation graphically by finding x-intercepts. 29. x 2 + x - 1 = 0

30. 4x 2 + 20x + 23 = 0

31. x 3 + x 2 + 2x - 3 = 0

32. x 3 - 4x + 2 = 0

33. x 2 + 4 = 4x

34. x 2 + 2x = -2

In Exercises 35 and 36, the table permits you to estimate a zero of an expression. State the expression and give the zero as accurately as can be read from the table.

47. x 2 + x - 2 = 0

48. x 2 - 3x = 12 - 31x - 22

49. ƒ 2x - 1 ƒ = 5

50. x + 2 - 22x + 3 = 0

51. x 3 + 4x 2 - 3x - 2 = 0 52. x 3 - 4x + 2 = 0 53. ƒ x 2 + 4x - 1 ƒ = 7 2

55. ƒ 0.5x + 3 ƒ = x - 4

54. ƒ x + 5 ƒ = ƒ x - 3 ƒ 56. 2x + 7 = - x 2 + 5

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 47

SECTION P.5 Solving Equations Graphically, Numerically, and Algebraically

57. Group Activity Discriminant of a Quadratic The radicand b 2 - 4ac in the quadratic formula is called the discriminant of the quadratic polynomial ax 2 + bx + c because it can be used to describe the nature of its zeros.

(a) Writing to Learn If b 2 - 4ac 7 0, what can you say about the zeros of the quadratic polynomial ax 2 + bx + c? Explain your answer.

(b) Writing to Learn If b 2 - 4ac = 0, what can you say about the zeros of the quadratic polynomial ax 2 + bx + c? Explain your answer.

47

Standardized Test Questions 62. True or False If 2 is an x-intercept of the graph of y = ax 2 + bx + c, then 2 is a solution of the equation ax 2 + bx + c = 0. Justify your answer. 63. True or False If 2x 2 = 18, then x must be equal to 3. Justify your answer. In Exercises 64–67, you may use a graphing calculator to solve these problems.

(c) Writing to Learn If b - 4ac 6 0, what can you

64. Multiple Choice Which of the following are the solutions of the equation x1x - 32 = 0?

58. Group Activity Discriminant of a Quadratic Use the information learned in Exercise 57 to create a quadratic polynomial with the following numbers of real zeros. Support your answers graphically.

65. Multiple Choice Which of the following replacements for ? make x 2 - 5x + ? a perfect square?

2

say about the zeros of the quadratic polynomial ax 2 + bx + c? Explain your answer.

(a) Two real zeros (b) Exactly one real zero (c) No real zeros

(A) Only x = 3 (C) x = 0 and x = -3 (E) There are no solutions.

(A) -

59. Size of a Soccer Field Several of the World Cup ’94 soccer matches were played in Stanford University’s stadium in Menlo Park, California. The field is 30 yd longer than it is wide, and the area of the field is 8800 yd2. What are the dimensions of this soccer field? 60. Height of a Ladder John’s paint crew knows from experience that its 18-ft ladder is particularly stable when the distance from the ground to the top of the ladder is 5 ft more than the distance from the building to the base of the ladder as shown in the figure. In this position, how far up the building does the ladder reach?

5 2

(B) Only x = - 3 (D) x = 0 and x = 3

5 2

2

2 5

2

(B) a - b

(C) 1- 522

(D) a - b

(E) -6

66. Multiple Choice Which of the following are the solutions of the equation 2x 2 - 3x - 1 = 0?

(A)

3 ! 217 4

(C)

3 ! 217 2

(E)

3!1 4

(B)

3 ! 217 4

(D)

- 3 ! 217 4

67. Multiple Choice Which of the following are the solutions of the equation ƒ x - 1 ƒ = -3?

(A) Only x = 4 (C) Only x = 2 (E) There are no solutions.

x"5

(B) Only x = - 2 (D) x = 4 and x = - 2

18 ft

Explorations x

61. Finding the Dimensions of a Norman Window A Norman window has the shape of a square with a semicircle mounted on it. Find the width of the window if the total area of the square and the semicircle is to be 200 ft 2.

x

x

68. Deriving the Quadratic Formula Follow these steps to use completing the square to solve ax 2 + bx + c = 0, a Z 0.

(a) Subtract c from both sides of the original equation and divide both sides of the resulting equation by a to obtain

x2 +

b c x = - . a a

6965_CH0P_pp001-062.qxd

48

1/14/10

12:44 PM

Page 48

CHAPTER P Prerequisites

(b) Add the square of one-half of the coefficient of x in (a) to both sides and simplify to obtain

ax +

2

tions. (There are many such values.) 2

b b - 4ac b = . 2a 4a 2

(d) Find a value of c for which this equation has no solutions.

-b ! 2b 2 - 4ac . 2a

70. Sums and Products of Solutions of ax2 ! bx ! c " 0, a # 0 Suppose that b 2 - 4ac 7 0.

(c) Extract square roots in (b) and solve for x to obtain the quadratic formula

x =

(c) Find a value of c for which this equation has two solu-

Extending the Ideas 69. Finding Number of Solutions Consider the equation ƒ x 2 - 4 ƒ = c.

(a) Find a value of c for which this equation has four solutions. (There are many such values.)

(b) Find a value of c for which this equation has three solutions. (There is only one such value.)

(There are many such values.)

(e) Writing to Learn Are there any other possible numbers of solutions of this equation? Explain.

(a) Show that the sum of the two solutions of this equation is -1b/a2.

(b) Show that the product of the two solutions of this equation is c/a. 71. Exercise 70 Continued The equation 2x 2 + bx + c = 0 has two solutions x 1 and x 2. If x 1 + x 2 = 5 and x 1 # x 2 = 3, find the two solutions.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 49

SECTION P.6 Complex Numbers

49

P.6 Complex Numbers What you’ll learn about • Complex Numbers • Operations with Complex Numbers • Complex Conjugates and Division • Complex Solutions of Quadratic Equations

... and why The zeros of polynomials are complex numbers.

Complex Numbers Figure P.39 shows that the function f(x) = x 2 + 1 has no real zeros, so x 2 + 1 = 0 has no real-number solutions. To remedy this situation, mathematicians in the 17th century extended the definition of 2a to include negative real numbers a. First the number i = 2 -1 is defined as a solution of the equation i 2 + 1 = 0 and is the imaginary unit. Then for any negative real number 2a = 2 ƒ a ƒ # i.

The extended system of numbers, called the complex numbers, consists of all real numbers and sums of real numbers and real number multiples of i. The following are all examples of complex numbers: -6, 5i,

25,

- 7i,

5 2 i + , 2 3

-2 + 3i, 5-3i,

1 4 + i. 3 5

DEFINITION Complex Number A complex number is any number that can be written in the form a + bi, where a and b are real numbers. The real number a is the real part, the real number b is the imaginary part, and a + bi is the standard form.

[–5, 5] by [–3, 10]

FIGURE P.39 The graph of

f (x) = x 2 + 1 has no x-intercepts.

A real number a is the complex number a + 0i, so all real numbers are also complex numbers. If a = 0 and b Z 0, then a + bi becomes bi, and is an imaginary number. For instance, 5i and -7i are imaginary numbers. Two complex numbers are equal if and only if their real and imaginary parts are equal. For example,

Historical Note René Descartes (1596–1650) coined the term imaginary in a time when negative solutions to equations were considered false. Carl Friedrich Gauss (1777–1855) gave us the term complex number and the symbol i for 2-1. Today practical applications of complex numbers abound.

x + yi = 2 + 5i if and only if x = 2 and y = 5.

Operations with Complex Numbers Adding complex numbers is done by adding their real and imaginary parts separately. Subtracting complex numbers is also done using the same parts.

DEFINITION Addition and Subtraction of Complex Numbers If a + bi and c + di are two complex numbers, then Sum: Difference:

1a + bi2 + 1c + di2 = 1a + c2 + 1b + d 2i, 1a + bi2 - 1c + di2 = 1a - c2 + 1b - d 2i.

EXAMPLE 1 Adding and Subtracting Complex Numbers (a) 17 - 3i2 + 14 + 5i2 = 17 + 42 + 1-3 + 52i = 11 + 2i (b) 12 - i2 - 18 + 3i2 = 12 - 82 + 1-1 - 32i = - 6 - 4i

Now try Exercise 3.

6965_CH0P_pp001-062.qxd

50

1/14/10

12:44 PM

Page 50

CHAPTER P Prerequisites

The additive identity for the complex numbers is 0 = 0 + 0i. The additive inverse of a + bi is - 1a + bi2 = - a - bi because 1a + bi2 + 1- a - bi2 = 0 + 0i = 0.

Many of the properties of real numbers also hold for complex numbers. These include: • Commutative properties of addition and multiplication, • Associative properties of addition and multiplication, and • Distributive properties of multiplication over addition and subtraction. Using these properties and the fact that i 2 = - 1, complex numbers can be multiplied by treating them as algebraic expressions.

EXAMPLE 2 Multiplying Complex Numbers 12 + 3i2 # 15 - i2 = = = =

215 - i2 + 3i15 - i2 10 - 2i + 15i - 3i 2 10 + 13i - 31-12 13 + 13i

Now try Exercise 9.

We can generalize Example 2 as follows:

(7–3i)+(4+5i) (2–i)–(8+3i) (2+3i)*(5–i) N

1a + bi21c + di2 = ac + adi + bci + bdi 2 = 1ac - bd2 + 1ad + bc2i

11+2i –6–4i

Many graphers can perform basic calculations on complex numbers. Figure P.40 shows how the operations of Examples 1 and 2 look on some graphers.

13+13i

We compute positive integer powers of complex numbers by treating them as algebraic expressions.

FIGURE P.40 Complex number operations on a grapher. (Examples 1 and 2)

EXAMPLE 3 Raising a Complex Number to a Power If z =

1 23 + i, find z 2 and z 3. 2 2

SOLUTION

1 23 1 23 + ib a + ib 2 2 2 2 1 23 23 3 + i + i + i2 4 4 4 4 1 223 3 + i + (-1) 4 4 4 1 23 - + i 2 2 1 23 1 23 a- + ib a + ib 2 2 2 2 1 23 23 3 - i + i + i2 4 4 4 4 1 3 - + 0i + 1-12 4 4 -1

z2 = a =

= =

(1/2+i (3)/2)2 –.5+.8660254038i (1/2+i (3)/2)3 –1

z3 = z2 # z = = =

FIGURE P.41 The square and cube of a complex number. (Example 3)

=

Figure P.41 supports these results numerically.

Now try Exercise 27.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 51

SECTION P.6 Complex Numbers

51

Example 3 demonstrates that 1/2 + 123/22i is a cube root of -1 and a solution of x 3 + 1 = 0. In Section 2.5, complex zeros of polynomial functions will be explored in depth.

Complex Conjugates and Division The product of the complex numbers a + bi and a - bi is a positive real number: 1a + bi2 # 1a - bi2 = a 2 - 1bi22 = a 2 + b 2.

We introduce the following definition to describe this special relationship.

DEFINITION Complex Conjugate The complex conjugate of the complex number z = a + bi is z = a + bi = a - bi. The multiplicative identity for the complex numbers is 1 = 1 + 0i. The multiplicative inverse, or reciprocal, of z = a + bi is z -1 =

1 1 1 # a - bi a b = = = 2 - 2 i. 2 z a + bi a + bi a - bi a + b a + b2

In general, a quotient of two complex numbers, written in fraction form, can be simplified as we just simplified 1/z—by multiplying the numerator and denominator of the fraction by the complex conjugate of the denominator.

EXAMPLE 4 Dividing Complex Numbers Write the complex number in standard form. 5 + i 2 - 3i SOLUTION Multiply the numerator and denominator by the complex conjugate of the denominator. (a)

2 3 - i

(b)

(a)

2 2 #3 + i = 3 - i 3 - i 3 + i

(b)

5 + i 5 + i # 2 + 3i = 2 - 3i 2 - 3i 2 + 3i

=

6 + 2i 32 + 12

=

10 + 15i + 2i + 3i 2 22 + 32

=

6 2 + i 10 10

=

7 + 17i 13

=

3 1 + i 5 5

=

17 7 + i 13 13 Now try Exercise 33.

Complex Solutions of Quadratic Equations Recall that the solutions of the quadratic equation ax 2 + bx + c = 0, where a, b, and c are real numbers and a Z 0, are given by the quadratic formula x =

-b ! 2b 2 - 4ac . 2a

The radicand b 2 - 4ac is the discriminant, and tells us whether the solutions are real numbers. In particular, if b 2 - 4ac 6 0, the solutions involve the square root of a

6965_CH0P_pp001-062.qxd

52

1/21/10

7:17 AM

Page 52

CHAPTER P Prerequisites

negative number and so lead to complex-number solutions. In all, there are three cases, which we now summarize:

Discriminant of a Quadratic Equation For a quadratic equation ax 2 + bx + c = 0, where a, b, and c are real numbers and a Z 0, • If b 2 - 4ac 7 0, there are two distinct real solutions. • If b 2 - 4ac = 0, there is one repeated real solution. • If b 2 - 4ac 6 0, there is a complex conjugate pair of solutions.

EXAMPLE 5 Solving a Quadratic Equation Solve x 2 + x + 1 = 0. SOLUTION

Solve Algebraically Using the quadratic formula with a = b = c = 1, we obtain x =

-112 ! 21122 - 4112112 2112

=

-112 ! 2-3 2

= -

1 23 ! i. 2 2

So the solutions are -1/2 + 123/22i and -1/2 - 123/22i, a complex conjugate pair. Confirm Numerically Substituting - 1/2 + 123/22i into the original equation, we obtain a-

1 23 2 1 23 + ib + a - + ib + 1 2 2 2 2 1 23 1 23 ib + a - + ib + 1 = 0. = a- 2 2 2 2

By a similar computation we can confirm the second solution.

Now try Exercise 41.

QUICK REVIEW P.6 In Exercises 1–4, add or subtract, and simplify. 1. 12x + 32 + 1- x + 62

3. 12a + 4d 2 - 1a + 2d2

2. 13y - x2 + 12x - y2 4. 16z - 12 - 1z + 32

In Exercises 5–10, multiply and simplify. 5. 1x - 321x + 22

6. 12x - 121x + 32

7. 1x - 2221x + 222

8. 1x + 2 2321x - 2 232

9. 3x - 11 + 22243x - 11 - 2224

10. 3x - 12 + 23243x - 12 - 2324

SECTION P.6 EXERCISES Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator.

3. 17 - 3i2 + 16 - i2

In Exercises 1–8, write the sum or difference in the standard form a + bi without using a calculator.

6. 125 - 3i2 + 1- 2 + 2-92

1. 12 - 3i2 + 16 + 5i2

2. 12 - 3i2 + 13 - 4i2

5. 12 - i2 + 13 - 2 -32 7. 1i 2 + 32 - 17 + i 32

8. 127 + i 22 - 16 - 2-812

4. (2 + i) - (9i - 3)

6965_CH0P_pp001-062.qxd

2/2/10

11:20 AM

Page 53

SECTION P.6 Complex Numbers

In Exercises 9–16, write the product in standard form without using a calculator. 9. 12 + 3i212 - i2

11. 11 - 4i213 - 2i2 13. 17i - 3212 + 6i2

15. ( -3 - 4i2(1 + 2i2

10. 12 - i211 + 3i2

12. 15i - 3212i + 12

14. 1 2- 4 + i216 - 5i2

16. 12- 2 + 2i216 + 5i2

In Exercises 17–20, write the expression in the form bi, where b is a real number. 17. 2- 16 19. 2- 3

18. 2 -25 20. 2-5

In Exercises 21–24, find the real numbers x and y that make the equation true. 21. 2 + 3i = x + yi

22. 3 + yi = x - 7i

24. 1x + 6i2 = 13 - i2 + 14 - 2yi2

In Exercises 25–28, write the complex number in standard form. 25. 13 + 2i22

22 22 4 + ib 27. a 2 2

26. 11 - i23

23 1 3 + ib 28. a 2 2

In Exercises 29–32, find the product of the complex number and its conjugate. 29. 2 - 3i

30. 5 - 6i

31. - 3 + 4i

32. -1 - 22i

In Exercises 33–40, write the expression in standard form without using a calculator. 1 33. 2 + i

i 34. 2 - i

2 + i 36. 3i 12 - i211 + 2i2 38. 5 + 2i 11 - 22i211 + i2 40. 11 + 22i2 In Exercises 41–44, solve the equation. 2 + i 35. 2 - i 12 + i221- i2 37. 1 + i 11 - i212 - i2 39. 1 - 2i 2

41. x + 2x + 5 = 0 2

43. 4x - 6x + 5 = x + 1

In Exercises 47–50, solve the problem without using a calculator. 47. Multiple Choice Which of the following is the standard form for the product 12 + 3i212 - 3i2?

(A) -5 + 12i (B) 4 - 9i (C) 13 - 3i (D) -5 (E) 13 + 0i

48. Multiple Choice Which of the following is the standard 1 form for the quotient ? i

(A) 1

(B) - 1

2

42. 3x + x + 2 = 0

44. x 2 + x + 11 = 5x - 8

Standardized Test Questions 45. True or False There are no complex numbers z satisfying z = - z. Justify your answer. 46. True or False For the complex number i, i + i 2 + i 3 + i 4 = 0. Justify your answer.

(C) i

(D) - 1/i

(E) 0 - i

49. Multiple Choice Assume that 2 - 3i is a solution of ax 2 + bx + c = 0, where a, b, c are real numbers. Which of the following is also a solution of the equation?

(A) 2 + 3i (B) - 2 - 3i (C) - 2 + 3i (D) 3 + 2i (E)

23. 15 - 2i2 - 7 = x - 13 + yi2

53

1 2 - 3i

50. Multiple Choice Which of the following is the standard form for the power 11 - i23?

(A) -4i (B) -2 + 2i (C) -2 - 2i (D) 2 + 2i (E) 2 - 2i

Explorations 51. Group Activity The Powers of i

(a) Simplify the complex numbers i, i 2, Á , i 8 by evaluating each one.

(b) Simplify the complex numbers i -1, i -2, Á , i -8 by evaluating each one.

(c) Evaluate i 0. (d) Writing to Learn Discuss your results from (a)–(c) with the members of your group, and write a summary statement about the integer powers of i. 52. Writing to Learn Describe the nature of the graph of f (x) = ax 2 + bx + c when a, b, and c are real numbers and the equation ax 2 + bx + c = 0 has nonreal complex solutions.

Extending the Ideas 53. Prove that the difference between a complex number and its conjugate is a complex number whose real part is 0. 54. Prove that the product of a complex number and its complex conjugate is a complex number whose imaginary part is zero. 55. Prove that the complex conjugate of a product of two complex numbers is the product of their complex conjugates. 56. Prove that the complex conjugate of a sum of two complex numbers is the sum of their complex conjugates. 57. Writing to Learn Explain why -i is a solution of x 2 - ix + 2 = 0 but i is not.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 54

CHAPTER P Prerequisites

54

What you’ll learn about • Solving Absolute Value Inequalities • Solving Quadratic Inequalities

P.7 Solving Inequalities Algebraically and Graphically Solving Absolute Value Inequalities The methods for solving inequalities parallel the methods for solving equations. Here are two basic rules we apply to solve absolute value inequalities.

• Approximating Solutions to Inequalities

Solving Absolute Value Inequalities

• Projectile Motion

Let u be an algebraic expression in x and let a be a real number with a Ú 0.

... and why

1. If ƒ u ƒ 6 a, then u is in the interval 1- a, a2. That is,

These techniques are involved in using a graphing utility to solve inequalities in this textbook.

ƒuƒ 6 a

-a 6 u 6 a.

if and only if

2. If ƒ u ƒ 7 a, then u is in the interval 1- q , - a2 or 1a, q 2, that is,

ƒuƒ 7 a

u 6 - a or u 7 a.

if and only if

The inequalities 6 and 7 can be replaced with … and Ú , respectively. See Figure P.42.

y = !u! y=a

a (–a, a)

(a, a)

–a !u!

>a

a !u!

a

FIGURE P.42 The solution of ƒ u ƒ 6 a is represented by the portion of the number line

where the graph of y = ƒ u ƒ is below the graph of y = a. The solution of ƒ u ƒ 7 a is represented by the portion of the number line where the graph of y = ƒ u ƒ is above the graph of y = a.

EXAMPLE 1 Solving an Absolute Value Inequality Solve ƒ x - 4 ƒ 6 8. SOLUTION

ƒx - 4ƒ 6 8 -8 6 x - 4 6 8 -4 6 x 6 12

Original inequality Equivalent double inequality. Add 4.

As an interval the solution is 1- 4, 122. Figure P.43 shows that points on the graph of y = ƒ x - 4 ƒ are below the points on the graph of y = 8 for values of x between - 4 and 12. Now try Exercise 3.

EXAMPLE 2 Solving Another Absolute Value Inequality –4

12 [–7, 15] by [–5, 10]

FIGURE P.43 The graphs of y = ƒ x - 4 ƒ and y = 8. (Example 1)

Solve ƒ 3x - 2 ƒ Ú 5. SOLUTION The solution of this absolute value inequality consists of the solutions of both of these inequalities.

3x - 2 … - 5 or 3x - 2 Ú 5 3x … - 3 or 3x Ú 7 x … - 1 or

x Ú

7 3

Add 2. Divide by 3.

6965_CH0P_pp001-062.qxd

2/3/10

2:59 PM

Page 55

SECTION P.7 Solving Inequalities Algebraically and Graphically

–1

7 3

[–4, 4] by [–4, 10]

FIGURE P.44 The graphs of

y = ƒ 3x - 2 ƒ and y = 5. (Example 2)

Union of Two Sets The union of two sets A and B, denoted by A ´ B, is the set of all objects that belong to A or B or both.

[–10, 10] by [–15, 15]

FIGURE P.45 The graph of

y = x 2 - x - 12 appears to cross the x-axis at x = -3 and x = 4. (Example 3)

55

The solution consists of all numbers that are in either one of the two intervals 1- q , -14 or 37/3, q ), which may be written as 1- q , - 14 h 37/3, q 2. The notation “ h ” is read as “union.” Figure P.44 shows that points on the graph of y = ƒ 3x - 2 ƒ are above or on the points on the graph of y = 5 for values of x to the left of and including -1 and to the right of and including 7/3. Now try Exercise 7.

Solving Quadratic Inequalities To solve a quadratic inequality such as x 2 - x - 12 7 0 we begin by solving the corresponding quadratic equation x 2 - x - 12 = 0. Then we determine the values of x for which the graph of y = x 2 - x - 12 lies above the x-axis.

EXAMPLE 3 Solving a Quadratic Inequality Solve x 2 - x - 12 7 0. SOLUTION First we solve the corresponding equation x 2 - x - 12 = 0.

x 2 - x - 12 1x - 421x + 32 x - 4 = 0 or x + 3 x = 4 or x

= = = =

0 0 0 -3

Factor.

ab = 0 Q a = 0 or b = 0

Solve for x.

The solutions of the corresponding quadratic equation are -3 and 4, and they are not solutions of the original inequality because 0 7 0 is false. Figure P.45 shows that the points on the graph of y = x 2 - x - 12 are above the x-axis for values of x to the left of - 3 and to the right of 4. The solution of the original inequality is 1 - q , -32 ´ 14, q 2. Now try Exercise 11. In Example 4, the quadratic inequality involves the symbol … . In this case, the solutions of the corresponding quadratic equation are also solutions of the inequality.

EXAMPLE 4 Solving Another Quadratic Inequality Solve 2x 2 + 3x … 20. SOLUTION First we subtract 20 from both sides of the inequality to obtain

2x 2 + 3x - 20 … 0. Next, we solve the corresponding quadratic equation 2x 2 + 3x - 20 = 0. 2x 2 + 3x - 20 = 0 1x + 4212x - 52 = 0 x + 4 = 0 or 2x - 5 = 0 x = - 4 or

[–10, 10] by [–25, 25]

FIGURE P.46 The graph of

y = 2x 2 + 3x - 20 appears to be below the x-axis for -4 6 x 6 2.5. (Example 4)

x =

5 2

Factor.

ab = 0 Q a = 0 or b = 0

Solve for x.

The solutions of the corresponding quadratic equation are -4 and 5/2 = 2.5. You can check that they are also solutions of the inequality. Figure P.46 shows that the points on the graph of y = 2x 2 + 3x - 20 are below the x-axis for values of x between -4 and 2.5. The solution of the original inequality is 3-4, 2.54. We use square brackets because the numbers - 4 and 2.5 are also solutions of the inequality. Now try Exercise 9. In Examples 3 and 4 the corresponding quadratic equation factored. If this doesn’t happen we will need to approximate the zeros of the quadratic equation if it has any. Then we use our accuracy agreement from Section P.5 and write the endpoints of any intervals accurate to two decimal places as illustrated in Example 5.

6965_CH0P_pp001-062.qxd

56

1/14/10

12:44 PM

Page 56

CHAPTER P Prerequisites

EXAMPLE 5 Solving a Quadratic Inequality Graphically Solve x 2 - 4x + 1 Ú 0 graphically. SOLUTION We can use the graph of y = x 2 - 4x + 1 in Figure P.47 to deter-

mine that the solutions of the equation x 2 - 4x + 1 = 0 are about 0.27 and 3.73. Thus, the solution of the original inequality is 1 - q , 0.274 h 33.73, q 2. We use square brackets because the zeros of the quadratic equation are solutions of the inequality even though we only have approximations to their values. Now try Exercise 21.

Zero X=.26794919 Y=0 [–3, 7] by [–4, 6]

Zero X=3.7320508 Y=1E–12 [–3, 7] by [–4, 6]

FIGURE P.47 This figure suggests that y = x 2 - 4x + 1 is zero for x L 0.27 and x L 3.73. (Example 5)

EXAMPLE 6 Showing That a Quadratic Inequality Has No Solution

[–5, 5] by [–2, 5] 2

FIGURE P.48 The values of y = x + 2x + 2 are never negative. (Example 6)

Solve x 2 + 2x + 2 6 0. SOLUTION Figure P.48 shows that the graph of y = x 2 + 2x + 2 lies above the

x-axis for all values for x. Thus, the inequality x 2 + 2x + 2 6 0 has no solution. Now try Exercise 25. Figure P.48 also shows that the solution of the inequality x 2 + 2x + 2 7 0 is the set of all real numbers or, in interval notation, 1 - q , q 2. A quadratic inequality can also have exactly one solution (see Exercise 31).

Approximating Solutions to Inequalities To solve the inequality in Example 7 we approximate the zeros of the corresponding graph. Then we determine the values of x for which the corresponding graph is above or on the x-axis.

EXAMPLE 7 Solving a Cubic Inequality Solve x 3 + 2x 2 - 1 Ú 0 graphically. SOLUTION We can use the graph of y = x 3 + 2x 2 - 1 in Figure P.49 to show

that the solutions of the corresponding equation x 3 + 2x 2 - 1 = 0 are approximately -1.62, - 1, and 0.62. The points on the graph of y = x 3 + 2x 2 - 1 are above the x-axis for values of x between -1.62 and -1, and for values of x to the right of 0.62. The solution of the inequality is 3-1.62, - 14 h 30.62, q 2. We use square brackets because the zeros of y = x 3 + 2x 2 - 1 are also solutions of the inequality. Now try Exercise 27.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 57

SECTION P.7 Solving Inequalities Algebraically and Graphically

57

Zero X=–1.618034 Y=0 [–3, 3] by [–2, 2]

FIGURE P.49 The graph of y = x 3 + 2x 2 - 1 appears to be above the x-axis between the two negative x-intercepts and to the right of the positive x-intercept. (Example 7)

Projectile Motion The movement of an object that is propelled vertically, but then subject only to the force of gravity, is an example of projectile motion.

Projectile Motion Suppose an object is launched vertically from a point s0 feet above the ground with an initial velocity of v0 feet per second. The vertical position s (in feet) of the object t seconds after it is launched is s = -16t 2 + v0t + s0.

EXAMPLE 8 Finding Height of a Projectile A projectile is launched straight up from ground level with an initial velocity of 288 ft/sec. (a) When will the projectile’s height above ground be 1152 ft? (b) When will the projectile’s height above ground be at least 1152 ft? SOLUTION Here s0 = 0 and v0 = 288. So, the projectile’s height is

s = - 16t 2 + 288t.

(a) We need to determine when s = 1152. s = - 16t 2 + 288t 1152 = - 16t 2 + 288t 16t 2 - 288t + 1152 = 0 t 2 - 18t + 72 = 0 [0, 20] by [0, 1500]

FIGURE P.50 The graphs of

s = -16t 2 + 288t and s = 1152. We know from Example 8a that the two graphs intersect at 16, 11522 and 112, 11522.

(t - 6)(t - 12) = 0 t = 6 or t = 12

Substitute s = 1152. Add 16t 2 - 288t. Divide by 16. Factor. Solve for t.

The projectile is 1152 ft above ground twice; the first time at t = 6 sec on the way up, and the second time at t = 12 sec on the way down (Figure P.50). (b) The projectile will be at least 1152 ft above ground when s Ú 1152. We can see from Figure P.50 together with the algebraic work in (a) that the solution is [6, 12]. This means that the projectile is at least 1152 ft above ground for times between t = 6 sec and t = 12 sec, including 6 and 12 sec. In Exercise 32 we ask you to use algebra to solve the inequality s = -16t 2 + 288t Ú 1152. Now try Exercise 33.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 58

CHAPTER P Prerequisites

58

QUICK REVIEW P.7 Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator.

7.

In Exercises 1–3, solve for x. 1. - 7 6 2x - 3 6 7

In Exercises 7 and 8, reduce the fraction to lowest terms.

2. 5x - 2 Ú 7x + 4

3. ƒ x + 2 ƒ = 3 In Exercises 4–6, factor the expression completely.

z 2 - 25

8.

x x + 1 + x - 1 3x - 4

10.

4. 4x 2 - 9 5. x 3 - 4x

x 2 + 2x - 35

z - 5z x 2 - 10x + 25 In Exercises 9 and 10, add the fractions and simplify. 9.

2

2x - 1 2

x - x - 2

+

x - 3 2

x - 3x + 2

6. 9x 2 - 16y 2

SECTION P.7 EXERCISES In Exercises 1–8, solve the inequality algebraically. Write the solution in interval notation and draw its number line graph. 1. ƒ x + 4 ƒ Ú 5

2. ƒ 2x - 1 ƒ 7 3.6

3. ƒ x - 3 ƒ 6 2

4. ƒ x + 3 ƒ … 5

5. ƒ 4 - 3x ƒ - 2 6 4

6. ƒ 3 - 2x ƒ + 2 7 5

7. `

x + 2 ` Ú3 3

8. `

x - 5 ` …6 4

In Exercises 9–16, solve the inequality. Use algebra to solve the corresponding equation. 9. 2x 2 + 17x + 21 … 0 2

11. 2x + 7x 7 15 2

10. 6x 2 - 13x + 6 Ú 0 2

12. 4x + 2 6 9x

13. 2 - 5x - 3x 6 0

14. 21 + 4x - x 2 7 0

15. x 3 - x Ú 0

16. x 3 - x 2 - 30x … 0

In Exercises 17–26, solve the inequality graphically. 17. x 2 - 4x 6 1

18. 12x 2 - 25x + 12 Ú 0

19. 6x 2 - 5x - 4 7 0

20. 4x 2 - 1 … 0

21. 9x 2 + 12x - 1 Ú 0

22. 4x 2 - 12x + 7 6 0

23. 4x 2 + 1 7 4x

24. x 2 + 9 … 6x

2

25. x - 8x + 16 6 0

26. 9x 2 + 12x + 4 Ú 0

In Exercises 27–30, solve the cubic inequality graphically. 27. 3x 3 - 12x + 2 Ú 0

28. 8x - 2x 3 - 1 6 0

29. 2x 3 + 2x 7 5

30. 4 … 2x 3 + 8x

31. Group Activity Give an example of a quadratic inequality with the indicated solution.

(a) All real numbers (c) Exactly one solution

(e) 1 - q , - 12 h 14, q 2

(b) No solution (d) 3 - 2, 54

(f) 1 - q , 04 h 34, q 2

32. Revisiting Example 8 Solve the inequality - 16t 2 + 288t Ú 1152 algebraically and compare your answer with the result obtained in Example 10.

33. Projectile Motion A projectile is launched straight up from ground level with an initial velocity of 256 ft/sec.

(a) When will the projectile’s height above ground be 768 ft? (b) When will the projectile’s height above ground be at least 768 ft?

(c) When will the projectile’s height above ground be less than or equal to 768 ft? 34. Projectile Motion A projectile is launched straight up from ground level with an initial velocity of 272 ft/sec.

(a) When will the projectile’s height above ground be 960 ft? (b) When will the projectile’s height above ground be more than 960 ft?

(c) When will the projectile’s height above ground be less than or equal to 960 ft? 35. Writing to Learn Explain the role of equation solving in the process of solving an inequality. Give an example. 36. Travel Planning Barb wants to drive to a city 105 mi from her home in no more than 2 h. What is the lowest average speed she must maintain on the drive? 37. Connecting Algebra and Geometry Consider the collection of all rectangles that have length 2 in. less than twice their width.

(a) Find the possible widths (in inches) of these rectangles if their perimeters are less than 200 in.

(b) Find the possible widths (in inches) of these rectangles if their areas are less than or equal to 1200 in.2.

38. Boyle’s Law For a certain gas, P = 400/V, where P is pressure and V is volume. If 20 … V … 40, what is the corresponding range for P? 39. Cash-Flow Planning A company has current assets (cash, property, inventory, and accounts receivable) of $200,000 and current liabilities (taxes, loans, and accounts payable) of $50,000. How much can it borrow if it wants its ratio of assets to liabilities to be no less than 2? Assume the amount borrowed is added to both current assets and current liabilities.

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 59

SECTION P.7 Solving Inequalities Algebraically and Graphically

Standardized Test Questions 40. True or False The absolute value inequality ƒ x - a ƒ 6 b, where a and b are real numbers, always has at least one solution. Justify your answer. 41. True or False Every real number is a solution of the absolute value inequality ƒ x - a ƒ Ú 0, where a is a real number. Justify your answer. In Exercises 42–45, solve these problems without using a calculator. 42. Multiple Choice Which of the following is the solution to ƒ x - 2 ƒ 6 3? (A) x = - 1 or x = 5 (C) 3 - 1, 54

(E) 1 -1, 52

(B) 3 - 1, 52

(D) 1 - q , -12 ´ 15, q 2

43. Multiple Choice Which of the following is the solution to x 2 - 2x + 2 Ú 0? (A) 30, 24

(C) 1 - q , 04 ´ 32, q 2

(B) 1 - q , 02 ´ 12, q 2

(D) All real numbers

45. Multiple Choice Which of the following is the solution to x 2 … 1? (A) 1 - q , 14

(B) 1- 1, 12

(C) 31, q 2

(D) 3-1, 14

(E) There is no solution.

Explorations 46. Constructing a Box with No Top An open box is formed by cutting squares from the corners of a regular piece of cardboard (see figure) and folding up the flaps.

(a) What size corner squares should be cut to yield a box with a volume of 125 in.3?

(b) What size corner squares should be cut to yield a box with a volume more than 125 in.3? 15 in.

(c) What size corner squares should be cut to yield a box with a volume of at most 125 in.3?

(A) 1 - q , 02 ´ 11, q 2 (C) 11, q 2

(E) There is no solution.

(B) 1 - q , 04 ´ 31, q )2

(D) 10, q 2

x

Extending the Ideas In Exercises 47 and 48, use a combination of algebraic and graphical techniques to solve the inequalities. 47. ƒ 2x 2 + 7x - 15 ƒ 6 10

48. ƒ 2x 2 + 3x - 20 ƒ Ú 10

CHAPTER P Key Ideas Properties, Theorems, and Formulas Trichotomy Property 4 Properties of Algebra 6 Properties of Equality 21 Properties of Inequalities 23 Distance Formulas 13, 14 Midpoint Formula (Coordinate Plane) Quadratic Formula 42

Equations of a Line 30 Equations of a Circle 15

Procedures

15

x

12 in.

(E) There is no solution.

44. Multiple Choice Which of the following is the solution to x 2 7 x?

59

Completing the Square 41 Solving Quadratic Equations Algebraically 43 Agreement About Approximate Solutions 43

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 60

CHAPTER P Prerequisites

60

CHAPTER P Review Exercises Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 1 and 2, find the endpoints and state whether the interval is bounded or unbounded. 2. 12, q 2

1. 30, 54

3. Distributive Property Use the distributive property to write the expanded form of 21x 2 - x2. 4. Distributive Property Use the distributive property to write the factored form of 2x 3 + 4x 2.

In Exercises 5 and 6, simplify the expression. Assume that denominators are not zero. 2 3

5.

1uv 2

13. -5 and 14

15. Right triangle: 1 - 2, 12, 13, 112, 17, 92

16. Equilateral triangle: 10, 12, 14, 12, 12, 1 - 2 232

In Exercises 17 and 18, find the standard form equation for the circle. 17. Center 10, 02, radius 2

18. Center 15, -32, radius 4

In Exercises 19 and 20, find the center and radius of the circle.

7. The mean distance from Pluto to the Sun is about 3,680,000,000 miles. 8. The diameter of a red blood corpuscle is about 0.000007 meter.

21. (a) Find the length of the sides of the triangle in the figure.

(b) Writing to Learn Show

(–3, 2) x (–1, –2)

9. Our solar system is about 5 * 10 9 years old. 10. The mass of an electron is about 9.1094 * 10 - 28 g (gram). 11. The data in Table P.9 give the Fiscal 2009 final budget for some Department of Education programs. Using scientific notation and no calculator, write the amount in dollars for the programs.

Fiscal 2009 Budget

Title 1 district grants Title 1 school improvement grants IDEA (Individuals with Disabilities Education Act) state grants Teacher Incentive Fund Head Start

(5, 6)

that the triangle is a right triangle.

In Exercises 9 and 10, write the number in decimal form.

Program

y

20. x 2 + y 2 = 1

In Exercises 7 and 8, write the number in scientific notation.

Table P.9

14. 1 - 4, 32 and 15, -12

In Exercises 15 and 16, show that the figure determined by the points is the indicated type.

19. (x + 5)2 + ( y + 4)2 = 9

6. 13x 2y 32 - 2

v 2u 3

In Exercises 13 and 14, find (a) the distance between the points and (b) the midpoint of the line segment determined by the points.

Amount $14.5 billion $545.6 million $11.5 billion $97 million $7.1 billion

Source: U.S. Departments of Education, Health and Human Services as reported in Education Week, May 13, 2009.

22. Distance and Absolute Value Use absolute value notation to write the statement that the distance between z and - 3 is less than or equal to 1. 23. Finding a Line Segment with Given Midpoint Let 13, 52 be the midpoint of the line segment with endpoints 1-1, 12 and 1a, b2. Determine a and b.

24. Finding Slope Find the slope of the line through the points 1-1, -22 and 14, - 52.

25. Finding Point-Slope Form Equation Find an equation in point-slope form for the line through the point 12, -12 with slope m = - 2/3.

26. Find an equation of the line through the points 1 -5, 42 and 12, -52 in the general form Ax + By + C = 0.

In Exercises 27–32, find an equation in slope-intercept form for the line. 27. The line through 13, -22 with slope m = 4/5

(a) Title 1 district grants

28. The line through the points 1-1, - 42 and 13, 2)

(b) Title 1 school improvement grants

30. The line 3x - 4y = 7

(c) IDEA state grants (d) Teacher Incentive Fund (e) Head Start 12. Decimal Form Find the decimal form for - 5/11. State whether it repeats or terminates.

29. The line through 1 -2, 42 with slope m = 0

31. The line through 12, -32 and parallel to the line 2x + 5y = 3

32. The line through 12, -32 and perpendicular to the line 2x + 5y = 3

6965_CH0P_pp001-062.qxd

1/14/10

12:44 PM

Page 61

CHAPTER P Review Exercises

33. SAT Math Scores The SAT scores are measured on an 800-point scale. The data in Table P.10 show the average SAT math score for several years.

Table P.10

Average SAT Math Scores

Year

SAT Math Score

2000 2001 2002 2003 2004 2005 2006 2007 2008

514 514 516 519 518 520 518 515 515

Source: The World Almanac and Book of Facts, The New York Times, June, 2009.

(a) Let x = 0 represent 2000, x = 1 represent 2001, and so forth. Draw a scatter plot of the data. (b) Use the 2001 and 2006 data to write a linear equation for the average SAT math score y in terms of the year x. Superimpose the graph of the linear equation on the scatter plot in (a). (c) Use the equation in (b) to estimate the average SAT math score in 2007. Compare with the actual value of 515. (d) Use the equation in (b) to predict the average SAT math score in 2010. 34. Consider the point 1 - 6, 32 and Line L: 4x - 3y = 5. Write an equation (a) for the line passing through this point and parallel to L, and (b) for the line passing through this point and perpendicular to L. Support your work graphically. In Exercises 35 and 36, assume that each graph contains the origin and the upper right-hand corner of the viewing window. 35. Find the slope of the line in the figure.

[–10, 10] by [–25, 25]

36. Writing to Learn Which line has the greater slope? Explain.

[–6, 6] by [–4, 4]

[–15, 15] by [–12, 12]

(a)

(b)

61

In Exercises 37–52, solve the equation algebraically without using a calculator. x - 2 x + 5 1 37. 3x - 4 = 6x + 5 38. + = 3 2 3 39. 215 - 2y2 - 311 - y2 = y + 1 40. 313x - 122 = 21

41. x 2 - 4x - 3 = 0

42. 16x 2 - 24x + 7 = 0

43. 6x 2 + 7x = 3

44. 2x 2 + 8x = 0

45. x12x + 52 = 41x + 72

46. ƒ 4x + 1 ƒ = 3

47. 4x 2 - 20x + 25 = 0

48. -9x 2 + 12x - 4 = 0

49. x 2 = 3x

50. 4x 2 - 4x + 2 = 0

51. x 2 - 6x + 13 = 0

52. x 2 - 2x + 4 = 0 53. Completing the Square Use completing the square to solve the equation 2x 2 - 3x - 1 = 0. 54. Quadratic Formula Use the quadratic formula to solve the equation 3x 2 + 4x - 1 = 0. In Exercises 55–58, solve the equation graphically. 55. 3x 3 - 19x 2 - 14x = 0

56. x 3 + 2x 2 - 4x - 8 = 0

57. x 3 - 2x 2 - 2 = 0

58. ƒ 2x - 1 ƒ = 4 - x 2

In Exercises 59 and 60, solve the inequality and draw a number line graph of the solution. 59. -2 6 x + 4 … 7

60. 5x + 1 Ú 2x - 4

In Exercises 61–72, solve the inequality. 61.

3x - 5 … -1 4

62. ƒ 2x - 5 ƒ 6 7

63. ƒ 3x + 4 ƒ Ú 2

64. 4x 2 + 3x 7 10

65. 2x 2 - 2x - 1 7 0

66. 9x 2 - 12x - 1 … 0

67. x 3 - 9x … 3

68. 4x 3 - 9x + 2 7 0

6965_CH0P_pp001-062.qxd

62

1/14/10

12:44 PM

Page 62

CHAPTER P Prerequisites

69. `

x + 7 ` 72 5

71. 4x 2 + 12x + 9 Ú 0

70. 2x 2 + 3x - 35 6 0 72. x 2 - 6x + 9 6 0

In Exercises 73–80, perform the indicated operation, and write the result in the standard form a + bi without using a calculator. 73. 13 - 2i2 + 1- 2 + 5i2

74. 15 - 7i2 - 13 - 2i2

79. 2-16

80.

75. 11 + 2i213 - 2i2

77. 11 + 2i2211 - 2i22

76. 11 + i23 78. i 29

2 + 3i 1 - 5i

81. Projectile Motion A projectile is launched straight up from ground level with an initial velocity of 320 ft/sec.

(a) When will the projectile’s height above ground be 1538 ft?

(b) When will the projectile’s height above ground be at most 1538 ft?

(c) When will the projectile’s height above ground be greater than or equal to 1538 ft? 82. Navigation A commercial jet airplane climbs at takeoff with slope m = 4/9. How far in the horizontal direction will the airplane fly to reach an altitude of 20,000 ft above the takeoff point? 83. Connecting Algebra and Geometry Consider the collection of all rectangles that have length 1 cm more than three times their width w.

(a) Find the possible widths (in cm) of these rectangles if their perimeters are less than or equal to 150 cm.

(b) Find the possible widths (in cm) of these rectangles if their areas are greater than 1500 cm2.

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 63

CHAPTER 1

Functions and Graphs 1.1 Modeling and Equation

Solving

1.2 Functions and Their

Properties

1.3 Twelve Basic Functions 1.4 Building Functions

from Functions

1.5 Parametric Relations

and Inverses

1.6 Graphical Transformations 1.7 Modeling with Functions

One of the central principles of economics is that the value of money is not constant; it is a function of time. Since fortunes are made and lost by people attempting to predict the future value of money, much attention is paid to quantitative measures like the Consumer Price Index, a basic measure of inflation in various sectors of the economy. See page 146 for a look at how the Consumer Price Index for housing has behaved over time.

63

6965_CH01_pp063-156.qxd

64

1/14/10

1:00 PM

Page 64

CHAPTER 1 Functions and Graphs

Chapter 1 Overview In this chapter we begin the study of functions that will continue throughout the book. Your previous courses have introduced you to some basic functions. These functions can be visualized using a graphing calculator, and their properties can be described using the notation and terminology that will be introduced in this chapter. A familiarity with this terminology will serve you well in later chapters when we explore properties of functions in greater depth.

1.1 Modeling and Equation Solving What you’ll learn about • Numerical Models • Algebraic Models • Graphical Models • The Zero Factor Property • Problem Solving • Grapher Failure and Hidden Behavior • A Word About Proof

... and why Numerical, algebraic, and graphical models provide different methods to visualize, analyze, and understand data.

Numerical Models Scientists and engineers have always used mathematics to model the real world and thereby to unravel its mysteries. A mathematical model is a mathematical structure that approximates phenomena for the purpose of studying or predicting their behavior. Thanks to advances in computer technology, the process of devising mathematical models is now a rich field of study itself, mathematical modeling. We will be concerned primarily with three types of mathematical models in this book: numerical models, algebraic models, and graphical models. Each type of model gives insight into real-world problems, but the best insights are often gained by switching from one kind of model to another. Developing the ability to do that will be one of the goals of this course. Perhaps the most basic kind of mathematical model is the numerical model, in which numbers (or data) are analyzed to gain insights into phenomena. A numerical model can be as simple as the major league baseball standings or as complicated as the network of interrelated numbers that measure the global economy.

EXAMPLE 1 Tracking the Minimum Wage Table 1.1 The Minimum Hourly Wage Minimum Purchasing Hourly Wage Power in Year (MHW) 1996 Dollars 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

0.75 1.00 1.25 1.60 2.10 3.10 3.35 3.80 4.25 5.15 5.15

Source: www.infoplease.com

4.39 5.30 6.23 6.47 6.12 5.90 4.88 4.56 4.38 4.69 4.15

The numbers in Table 1.1 show the growth of the minimum hourly wage (MHW) from 1955 to 2005. The table also shows the MHW adjusted to the purchasing power of 1996 dollars (using the CPI-U, the Consumer Price Index for all Urban Consumers). Answer the following questions using only the data in the table. (a) In what five-year period did the actual MHW increase the most? (b) In what year did a worker earning the MHW enjoy the greatest purchasing power? (c) A worker on minimum wage in 1980 was earning nearly twice as much as a worker on minimum wage in 1970, and yet there was great pressure to raise the minimum wage again. Why? SOLUTION

(a) In the period 1975 to 1980 it increased by $1.00. Notice that the minimum wage never goes down, so we can tell that there were no other increases of this magnitude even though the table does not give data from every year. (b) In 1970. (c) Although the MHW increased from $1.60 to $3.10 in that period, the purchasing power actually dropped by $0.57 (in 1996 dollars). This is one way inflation can affect the economy. Now try Exercise 11.

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 65

SECTION 1.1

Modeling and Equation Solving

65

The numbers in Table 1.1 provide a numerical model for one aspect of the U.S. economy by using another numerical model, the urban Consumer Price Index (CPI-U), to adjust the data. Working with large numerical models is standard operating procedure in business and industry, where computers are relied upon to provide fast and accurate data processing.

EXAMPLE 2 Analyzing Prison Populations Table 1.2 shows the growth in the number of prisoners incarcerated in state and federal prisons at year’s end for selected years between 1980 and 2005. Is the proportion of female prisoners increasing over the years?

Table 1.2 U.S. Prison Population (thousands) Year

Total

Male

Female

1980 1985 1990 1995 2000 2005

329 502 774 1125 1391 1526

316 479 730 1057 1298 1418

13 23 44 68 93 108

Source: U.S. Justice Department.

Table 1.3 Female Percentage of U.S. Prison Population Year

Female

1980 1985 1990 1995 2000 2005

3.9 4.6 5.7 6 6.7 7.1

SOLUTION The number of female prisoners over the years is certainly increasing, but so is the total number of prisoners, so it is difficult to discern from the data whether the proportion of female prisoners is increasing. What we need is another column of numbers showing the ratio of female prisoners to total prisoners.

We could compute all the ratios separately, but it is easier to do this kind of repetitive calculation with a single command on a computer spreadsheet. You can also do this on a graphing calculator by manipulating lists (see Exercise 19). Table 1.3 shows the percentage of the total population each year that consists of female prisoners. With these data to extend our numerical model, it is clear that the proportion of female prisoners is increasing. Now try Exercise 19.

Algebraic Models An algebraic model uses formulas to relate variable quantities associated with the phenomena being studied. The added power of an algebraic model over a numerical model is that it can be used to generate numerical values of unknown quantities by relating them to known quantities.

Source: U.S. Justice Department.

EXAMPLE 3 Comparing Pizzas A pizzeria sells a rectangular 18– by 24– pizza for the same price as its large round pizza (24– diameter). If both pizzas are of the same thickness, which option gives the most pizza for the money? SOLUTION We need to compare the areas of the pizzas. Fortunately, geometry

has provided algebraic models that allow us to compute the areas from the given information. For the rectangular pizza: Area = l * w = 18 * 24 = 432 square inches. For the circular pizza: Area = pr 2 = pa

24 2 b = 144p L 452.4 square inches. 2

The round pizza is larger and therefore gives more for the money. Now try Exercise 21. The algebraic models in Example 3 come from geometry, but you have probably encountered algebraic models from many other sources in your algebra and science courses.

6965_CH01_pp063-156.qxd

66

1/14/10

1:00 PM

Page 66

CHAPTER 1 Functions and Graphs

Exploration Extensions Suppose that after the sale, the merchandise prices are increased by 25%. If m represents the marked price before the sale, find an algebraic model for the post-sale price, including tax.

EXPLORATION 1

Designing an Algebraic Model

A department store is having a sale in which everything is discounted 25% off the marked price. The discount is taken at the sales counter, and then a state sales tax of 6.5% and a local sales tax of 0.5% are added on. 1. The discount price d is related to the marked price m by the formula d = km,

where k is a certain constant. What is k?

2. The actual sale price s is related to the discount price d by the formula

s = d + td, where t is a constant related to the total sales tax. What is t? 3. Using the answers from steps 1 and 2 you can find a constant p that relates s

directly to m by the formula s = pm. What is p? 4. If you only have $30, can you afford to buy a shirt marked $36.99? 5. If you have a credit card but are determined to spend no more than $100, what

is the maximum total value of your marked purchases before you present them at the sales counter?

The ability to generate numbers from formulas makes an algebraic model far more useful as a predictor of behavior than a numerical model. Indeed, one optimistic goal of scientists and mathematicians when modeling phenomena is to fit an algebraic model to numerical data and then (even more optimistically) to analyze why it works. Not all models can be used to make accurate predictions. For example, nobody has ever devised a successful formula for predicting the ups and downs of the stock market as a function of time, although that does not stop investors from trying. If numerical data do behave reasonably enough to suggest that an algebraic model might be found, it is often helpful to look at a picture first. That brings us to graphical models.

Graphical Models A graphical model is a visible representation of a numerical model or an algebraic model that gives insight into the relationships between variable quantities. Learning to interpret and use graphs is a major goal of this book.

EXAMPLE 4 Visualizing Galileo’s Gravity Experiments Galileo Galilei (1564–1642) spent a good deal of time rolling balls down inclined planes, carefully recording the distance they traveled as a function of elapsed time. His experiments are commonly repeated in physics classes today, so it is easy to reproduce a typical table of Galilean data.

Elapsed time (seconds)

0

1

2

3

4

5

6

7

8

Distance traveled (inches)

0

0.75

3

6.75

12

18.75

27

36.75

48

What graphical model fits the data? Can you find an algebraic model that fits? (continued)

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 67

SECTION 1.1

67

Modeling and Equation Solving

SOLUTION A scatter plot of the data is shown in Figure 1.1.

Galileo’s experience with quadratic functions suggested to him that this figure was a parabola with its vertex at the origin; he therefore modeled the effect of gravity as a quadratic function: d = kt 2.

[–1, 18] by [–8, 56]

FIGURE 1.1 A scatter plot of the data from a Galileo gravity experiment. (Example 4)

Because the ordered pair 11, 0.752 must satisfy the equation, it follows that k = 0.75, yielding the equation d = 0.75t 2.

You can verify numerically that this algebraic model correctly predicts the rest of the data points. We will have much more to say about parabolas in Chapter 2. Now try Exercise 23.

This insight led Galileo to discover several basic laws of motion that would eventually be named after Isaac Newton. While Galileo had found the algebraic model to describe the path of the ball, it would take Newton’s calculus to explain why it worked.

EXAMPLE 5 Fitting a Curve to Data We showed in Example 2 that the percentage of females in the U.S. prison population has been steadily growing over the years. Model this growth graphically and use the graphical model to suggest an algebraic model. SOLUTION Let t be the number of years after 1980, and let F be the percentage of

females in the prison population from year 0 to year 25. From the data in Table 1.3 we get the corresponding data in Table 1.4: [–2, 28] by [3, 8]

FIGURE 1.2 A scatter plot of the data in Table 1.4. (Example 5)

Table 1.4 Percentage (F) of Females in the Prison Population t years after 1980 t F

0 3.9

5 4.6

10 5.7

15 6.0

20 6.7

25 7.1

Source: U.S. Justice Department.

[–2, 28] by [3, 8]

FIGURE 1.3 The line with equation y = 0.128x + 3.9 is a good model for the data in Table 1.4. (Example 5)

A scatter plot of the data is shown in Figure 1.2. This pattern looks linear. If we use a line as our graphical model, we can find an algebraic model by finding the equation of the line. We will describe in Chapter 2 how a statistician would find the best line to fit the data, but we can get a pretty good fit for now by finding the line through the points 10, 3.92 and 125, 7.12. The slope is 17.1 - 3.92/125 - 02 = 0.128 and the y-intercept is 3.9. Therefore, the line has equation y = 0.128x + 3.9. You can see from Figure 1.3 that this line does a very nice job of modeling the data. Now try Exercises 13 and 15.

6965_CH01_pp063-156.qxd

68

1/14/10

1:00 PM

Page 68

CHAPTER 1 Functions and Graphs

Exploration Extensions What are the advantages of a linear model over a quadratic model for these data?

EXPLORATION 2

Interpreting the Model

The parabola in Example 4 arose from a law of physics that governs falling objects, which should inspire more confidence than the linear model in Example 5. We can repeat Galileo’s experiment many times with differently sloped ramps, with different units of measurement, and even on different planets, and a quadratic model will fit it every time. The purpose of this Exploration is to think more deeply about the linear model in the prison example. 1. The linear model we found will not continue to predict the percentage of

female prisoners in the United States indefinitely. Why must it eventually fail? 2. Do you think that our linear model will give an accurate estimate of the per-

centage of female prisoners in the United States in 2009? Why or why not? 3. The linear model is such a good fit that it actually calls our attention to the un-

usual jump in the percentage of female prisoners in 1990. Statisticians would look for some unusual “confounding” factor in 1990 that might explain the jump. What sort of factors do you think might explain it? 4. Does Table 1.1 suggest a possible factor that might influence female crime

statistics?

Prerequisite Chapter In the Prerequisite chapter we defined solution of an equation, solving an equation, x-intercept, and graph of an equation in x and y.

There are other ways of graphing numerical data that are particularly useful for statistical studies. We will treat some of them in Chapter 9. The scatter plot will be our choice of data graph for the time being, as it provides the closest connection to graphs of functions in the Cartesian plane.

The Zero Factor Property The main reason for studying algebra through the ages has been to solve equations. We develop algebraic models for phenomena so that we can solve problems, and the solutions to the problems usually come down to finding solutions of algebraic equations. If we are fortunate enough to be solving an equation in a single variable, we might proceed as in the following example.

EXAMPLE 6 Solving an Equation Algebraically Find all real numbers x for which 6x 3 = 11x 2 + 10x. SOLUTION We begin by changing the form of the equation to

6x 3 - 11x 2 - 10x = 0.

We can then solve this equation algebraically by factoring:

x = 0 x = 0

6x 3 - 11x 2 - 10x x16x 2 - 11x - 102 x12x - 5213x + 22 or 2x - 5 = 0 or 5 x = or or 2

= 0 = 0 = 0 3x + 2 = 0 2 3 Now try Exercise 31.

x = -

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 69

SECTION 1.1

Modeling and Equation Solving

69

In Example 6, we used the important Zero Factor Property of real numbers.

The Zero Factor Property A product of real numbers is zero if and only if at least one of the factors in the product is zero. It is this property that algebra students use to solve equations in which an expression is set equal to zero. Modern problem solvers are fortunate to have an alternative way to find such solutions. If we graph the expression, then the x-intercepts of the graph of the expression will be the values for which the expression equals 0.

EXAMPLE 7 Solving an Equation: Comparing Methods Solve the equation x 2 = 10 - 4x. SOLUTION

Solve Algebraically The given equation is equivalent to x 2 + 4x - 10 = 0. This quadratic equation has irrational solutions that can be found by the quadratic formula. x =

-4 + 216 + 40 L 1.7416574 2

and x =

Zero X=–5.741657 Y=0 [–8, 6] by [–20, 20]

FIGURE 1.4 The graph of y = x 2 + 4x - 10. (Example 7)

Solving Equations with Technology Example 7 shows one method of solving an equation with technology. Some graphers could also solve the equation in Example 7 by finding the intersection of the graphs of y = x 2 and y = 10 - 4x. Some graphers have built-in equation solvers. Each method has its advantages and disadvantages, but we recommend the “finding the x-intercepts” technique for now because it most closely parallels the classical algebraic techniques for finding roots of equations, and makes the connection between the algebraic and graphical models easier to follow and appreciate.

-4 - 216 + 40 L -5.7416574 2

While the decimal answers are certainly accurate enough for all practical purposes, it is important to note that only the expressions found by the quadratic formula give the exact real number answers. The tidiness of exact answers is a worthy mathematical goal. Realistically, however, exact answers are often impossible to obtain, even with the most sophisticated mathematical tools. Solve Graphically We first find an equivalent equation with 0 on the right-hand side: x 2 + 4x - 10 = 0. We next graph the equation y = x 2 + 4x - 10, as shown in Figure 1.4. We then use the grapher to locate the x-intercepts of the graph: x L 1.7416574 and x L -5.741657. Now try Exercise 35. We used the graphing utility of the calculator to solve graphically in Example 7. Most calculators also have solvers that would enable us to solve numerically for the same decimal approximations without considering the graph. Some calculators have computer algebra systems that will solve numerically to produce exact answers in certain cases. In this book we will distinguish between these two technological methods and the traditional pencil-and-paper methods used to solve algebraically. Every method of solving an equation usually comes down to finding where an expression equals zero. If we use ƒ1x2 to denote an algebraic expression in the variable x, the connections are as follows:

6965_CH01_pp063-156.qxd

70

1/14/10

1:00 PM

Page 70

CHAPTER 1 Functions and Graphs

Fundamental Connection If a is a real number that solves the equation ƒ1x2 = 0, then these three statements are equivalent: 1. The number a is a root (or solution) of the equation ƒ1x2 = 0. 2. The number a is a zero of y = ƒ1x2. 3. The number a is an x-intercept of the graph of y = ƒ1x2. (Sometimes the point 1a, 02 is referred to as an x-intercept.)

Problem Solving George Pólya (1887–1985) is sometimes called the father of modern problem solving, not only because he was good at it (as he certainly was) but also because he published the most famous analysis of the problem-solving process: How to Solve It: A New Aspect of Mathematical Method. His “four steps” are well known to most mathematicians: Pólya’s Four Problem-Solving Steps

1. Understand the problem. 2. Devise a plan. 3. Carry out the plan. 4. Look back.

The problem-solving process that we recommend you use throughout this course will be the following version of Pólya’s four steps. A Problem-Solving Process

Step 1—Understand the problem. • Read the problem as stated, several times if necessary. • Be sure you understand the meaning of each term used. • Restate the problem in your own words. Discuss the problem with others if you can. • Identify clearly the information that you need to solve the problem. • Find the information you need from the given data. Step 2—Develop a mathematical model of the problem. • Draw a picture to visualize the problem situation. It usually helps. • Introduce a variable to represent the quantity you seek. (In some cases there may be more than one.) • Use the statement of the problem to find an equation or inequality that relates the variables you seek to quantities that you know. Step 3—Solve the mathematical model and support or confirm the solution. • Solve algebraically using traditional algebraic methods and support graphically or support numerically using a graphing utility. • Solve graphically or numerically using a graphing utility and confirm algebraically using traditional algebraic methods. • Solve graphically or numerically because there is no other way possible.

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 71

SECTION 1.1

Modeling and Equation Solving

71

Step 4—Interpret the solution in the problem setting. • Translate your mathematical result into the problem setting and decide whether the result makes sense.

EXAMPLE 8 Applying the Problem-Solving Process The engineers at an auto manufacturer pay students $0.08 per mile plus $25 per day to road test their new vehicles. (a) How much did the auto manufacturer pay Sally to drive 440 miles in one day? (b) John earned $93 test-driving a new car in one day. How far did he drive? SOLUTION

Model A picture of a car or of Sally or John would not be helpful, so we go directly to designing the model. Both John and Sally earned $25 for one day, plus $0.08 per mile. Multiply dollars/mile by miles to get dollars. So if p represents the pay for driving x miles in one day, our algebraic model is p = 25 + 0.08x. Solve Algebraically (a) To get Sally’s pay we let x = 440 and solve for p:

X=440

Y=60.2

p = 25 + 0.0814402 = 60.20 (b) To get John’s mileage we let p = 93 and solve for x:

[0, 940] by [0, 150]

(a)

Support Graphically

X=850

Y=93 [0, 940] by [0, 150]

(b) FIGURE 1.5 Graphical support for the algebraic solutions in Example 8.

93 = 25 + 0.08x 68 = 0.08x 68 x = 0.08 x = 850

Figure 1.5a shows that the point 1440, 60.202 is on the graph of y = 25 + 0.08x, supporting our answer to (a). Figure 1.5b shows that the point 1850, 932 is on the graph of y = 25 + 0.08x, supporting our answer to (b). (We could also have supported our answer numerically by simply substituting in for each x and confirming the value of p.) Interpret Sally earned $60.20 for driving 440 miles in one day. John drove 850 miles in one day to earn $93.00. Now try Exercise 47. It is not really necessary to show written support as part of an algebraic solution, but it is good practice to support answers wherever possible simply to reduce the chance for error. We will often show written support of our solutions in this book in order to highlight the connections among the algebraic, graphical, and numerical models.

Grapher Failure and Hidden Behavior While the graphs produced by computers and graphing calculators are wonderful tools for understanding algebraic models and their behavior, it is important to keep in mind that machines have limitations. Occasionally they can produce graphical models that

6965_CH01_pp063-156.qxd

72

1/14/10

1:00 PM

Page 72

CHAPTER 1 Functions and Graphs

misrepresent the phenomena we wish to study, a problem we call grapher failure. Sometimes the viewing window will be too large, obscuring details of the graph which we call hidden behavior. We will give an example of each just to illustrate what can happen, but rest assured that these difficulties rarely occur with graphical models that arise from real-world problems.

EXAMPLE 9 Seeing Grapher Failure Technology Note One way to get the table in Figure 1.6b is to use the “Ask” feature of your graphing calculator and enter each x-value separately.

Look at the graph of y = 3 -

1 2

2x - 1 calculator. Are there any x-intercepts?

in the ZDecimal window on a graphing

SOLUTION The graph is shown in Figure 1.6a.

X .8 .9 1 1.1 1.2 1.3 1.4

Y1 ERROR ERROR ERROR .81782 1.4924 1.7961 1.9794

Y1 = 3–1/ (X2–1) [–4.7, 4.7] by [–3.1, 3.1]

(a) (b) FIGURE 1.6 (a) A graph with no apparent intercepts. (b) The function y = 3 - 1/ 2x 2 - 1 is undefined when ƒ x ƒ … 1. The graph seems to have no x-intercepts, yet we can find some by solving the equation 0 = 3 - 1/2x 2 - 1 algebraically: 0 = 3 - 1/2x 2 - 1 1/2x 2 - 1 = 3 2x 2 - 1 = 1/3

x 2 - 1 = 1/9 x 2 = 10/9 x = ! 210/9 L !1.054

There should be x-intercepts at about !1.054. What went wrong? The answer is a simple form of grapher failure. As the table shows, the function is undefined for the sampled x-values until x = 1.1, at which point the graph “turns on,” beginning with the pixel at 11.1, 0.817822 and continuing from there to the right. Similarly, the graph coming from the left “turns off ” at x = - 1, before it gets to the x-axis. The x-intercepts might well appear in other windows, but for this particular function in this particular window, the behavior we expect to see is not there. Now try Exercise 49.

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 73

SECTION 1.1

Modeling and Equation Solving

73

EXAMPLE 10 Not Seeing Hidden Behavior Solve graphically: x 3 - 1.1x 2 - 65.4x + 229.5 = 0. SOLUTION Figure 1.7a shows the graph in the standard 3- 10, 104 by 3-10, 104

window, an inadequate choice because too much of the graph is off the screen. Our horizontal dimensions look fine, so we adjust our vertical dimensions to 3-500, 5004, yielding the graph in Figure 1.7b.

[–10, 10] by [–500, 500]

[–10, 10] by [–10, 10]

(a)

(b) 3

2

FIGURE 1.7 The graph of y = x - 1.1x - 65.4x + 229.5 in two viewing windows. (Example 10) We use the grapher to locate an x-intercept near -9 (which we find to be -9) and then an x-intercept near 5 (which we find to be 5). The graph leads us to believe that we have finished. However, if we zoom in closer to observe the behavior near x = 5, the graph tells a new story (Figure 1.8). In this graph we see that there are actually two x-intercepts near 5 (which we find to be 5 and 5.1). There are therefore three roots (or zeros) of the equation x 3 - 1.1x 2 65.4x + 229.5 = 0: x = - 9, x = 5, and x = 5.1. Now try Exercise 51. [4.95, 5.15] by [–0.1, 0.1]

FIGURE 1.8 A closer look at the graph of y = x 3 - 1.1x 2 - 65.4x + 229.5. (Example 10)

You might wonder if there could be still more hidden x-intercepts in Example 10! We will learn in Chapter 2 how the Fundamental Theorem of Algebra guarantees that there are not.

A Word About Proof While Example 10 is still fresh in our minds, let us point out a subtle, but very important, consideration about our solution.

Teacher Note Sometimes it is impossible to show all of the details of a graph in a single window. For example, in Example 10 the graph in Figure 1.8 reveals minute details of the graph, but it hides the overall shape of the graph.

We solved graphically to find two solutions, then eventually three solutions, to the given equation. Although we did not show the steps, it is easy to confirm numerically that the three numbers found are actually solutions by substituting them into the equation. But the problem asked us to find all solutions. While we could explore that equation graphically in a hundred more viewing windows and never find another solution, our failure to find them would not prove that they are not out there somewhere. That is why the Fundamental Theorem of Algebra is so important. It tells us that there can be at most three real solutions to any cubic equation, so we know for a fact that there are no more. Exploration is encouraged throughout this book because it is how mathematical progress is made. Mathematicians are never satisfied, however, until they have proved their results. We will show you proofs in later chapters and we will ask you to produce proofs occasionally in the exercises. That will be a time for you to set the technology aside, get out a pencil, and show in a logical sequence of algebraic steps that something is undeniably and universally true. This process is called deductive reasoning.

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 74

CHAPTER 1 Functions and Graphs

74

EXAMPLE 11 Proving a Peculiar Number Fact Prove that 6 is a factor of n 3 - n for every positive integer n. SOLUTION You can explore this expression for various values of n on your calculator. Table 1.5 shows it for the first 12 values of n.

Table 1.5 The First 12 Values of n3 ! n n

1

2

3

4

5

6

7

8

9

10

11

12

n - n

0

6

24

60

120

210

336

504

720

990

1320

1716

3

All of these numbers are divisible by 6, but that does not prove that they will continue to be divisible by 6 for all values of n. In fact, a table with a billion values, all divisible by 6, would not constitute a proof. Here is a proof: Let n be any positive integer. • We can factor n 3 - n as the product of three numbers: 1n - 121n21n + 12. • The factorization shows that n 3 - n is always the product of three consecutive integers. • Every set of three consecutive integers must contain a multiple of 3. • Since 3 divides a factor of n 3 - n, it follows that 3 is a factor of n 3 - n itself. • Every set of three consecutive integers must contain a multiple of 2. • Since 2 divides a factor of n 3 - n, it follows that 2 is a factor of n 3 - n itself. • Since both 2 and 3 are factors of n 3 - n, we know that 6 is a factor of n 3 - n. Now try Exercise 53.

End of proof!

QUICK REVIEW 1.1

(For help, go to Section A.2.)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. Factor the following expressions completely over the real numbers. 1. x 2 - 16

5. 16h4 - 81 6. x 2 + 2xh + h2 7. x 2 + 3x - 4 8. x 2 - 3x + 4

2. x 2 + 10x + 25

9. 2x 2 - 11x + 5

3. 81y 2 - 4

10. x 4 + x 2 - 20

4. 3x 3 - 15x 2 + 18x

SECTION 1.1 EXERCISES In Exercises 1–10, match the numerical model to the corresponding graphical model 1a–j2 and algebraic model 1k–t2.

3.

x y

2 4

4 10

6 16

8 22

10 28

12 34

1.

x y

3 6

5 10

7 14

9 18

12 24

15 30

4.

x y

5 90

10 80

15 70

20 60

25 50

30 40

2.

x y

0 2

1 3

2 6

3 11

4 18

5 27

5.

x y

1 39

2 36

3 31

4 24

5 15

6 4

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 75

SECTION 1.1

6.

Modeling and Equation Solving

x y

1 5

2 7

3 9

4 11

5 13

6 15

(k) y = x 2 + x

(l) y = 40 - x 2

x y

5 1

7 2

9 3

11 4

13 5

15 6

(m) y = 1x + 121x - 12 (o) y = 100 - 2x

(n) y = 2x - 3

(p) y = 3x - 2

(q) y = 2x

(r) y = x 2 + 2

8.

x y

4 20

8 72

12 156

14 210

18 342

24 600

(s) y = 2x + 3

(t) y =

9.

x y

3 8

4 15

5 24

6 35

7 48

8 63

10.

x y

4 1

7 2

12 3

19 4

28 5

39 6

7.

[–2, 14] by [–4, 36]

[–1, 6] by [–2, 20]

(a)

(b)

75

x - 3 2

Exercises 11–18 refer to the data in Table 1.6 below, showing the percentage of the female and male populations in the United States employed in the civilian work force in selected years from 1954 to 2004.

Table 1.6 Employment Statistics Year

Female

Male

1954 1959 1964 1969 1974 1979 1984 1989 1994 1999 2004

32.3 35.1 36.9 41.1 42.8 47.7 50.1 54.9 56.2 58.5 57.4

83.5 82.3 80.9 81.1 77.9 76.5 73.2 74.5 72.6 74.0 71.9

Source: www.bls.gov

[–4, 40] by [–1, 7]

[–3, 18] by [–2, 32]

(c)

(d)

11. (a) According to the numerical model, what has been the trend in females joining the work force since 1954? (b) In what 5-year interval did the percentage of women who were employed change the most? 12. (a) According to the numerical model, what has been the trend in males joining the work force since 1954?

[–1, 7] by [–4, 40]

[–1, 7] by [–4, 40]

(e)

(f)

(b) In what 5-year interval did the percentage of men who were employed change the most? 13. Model the data graphically with two scatter plots on the same graph, one showing the percentage of women employed as a function of time and the other showing the same for men. Measure time in years since 1954.

[–1, 16] by [–1, 9]

[–5, 30] by [–5, 100]

(g)

(h)

14. Are the male percentages falling faster than the female percentages are rising, or vice versa? 15. Model the data algebraically with linear equations of the form y = mx + b. Write one equation for the women’s data and another equation for the men’s data. Use the 1954 and 1999 ordered pairs to compute the slopes. 16. If the percentages continue to follow the linear models you found in Exercise 15, what will the employment percentages for women and men be in the year 2009?

[–3, 9] by [–2, 60]

[–5, 40] by [–10, 650]

(i)

( j)

17. If the percentages continue to follow the linear models you found in Exercise 15, when will the percentages of women and men in the civilian work force be the same? What percentage will that be?

6965_CH01_pp063-156.qxd

76

1/14/10

1:00 PM

Page 76

CHAPTER 1 Functions and Graphs

18. Writing to Learn Explain why the percentages cannot continue indefinitely to follow the linear models that you wrote in Exercise 15. 19. Doing Arithmetic with Lists Enter the data from the “Total” column of Table 1.2 of Example 2 into list L1 in your calculator. Enter the data from the “Female” column into list L2. Check a few computations to see that the procedures in (a) and (b) cause the calculator to divide each element of L2 by the corresponding entry in L1, multiply it by 100, and store the resulting list of percentages in L3. (a) On the home screen, enter the command: 100 * L2/L1 : L3. (b) Go to the top of list L3 and enter L3 = 1001L2/L12.

20. Comparing Cakes A bakery sells a 9– by 13– cake for the same price as an 8– diameter round cake. If the round cake is twice the height of the rectangular cake, which option gives the most cake for the money? 21. Stepping Stones A garden shop sells 12– by 12– square stepping stones for the same price as 13– round stones. If all of the stepping stones are the same thickness, which option gives the most rock for the money? 22. Free Fall of a Smoke Bomb At the Oshkosh, WI, air show, Jake Trouper drops a smoke bomb to signal the official beginning of the show. Ignoring air resistance, an object in free fall will fall d feet in t seconds, where d and t are related by the algebraic model d = 16t 2. (a) How long will it take the bomb to fall 180 feet?

23. Physics Equipment A physics student obtains the following data involving a ball rolling down an inclined plane, where t is the elapsed time in seconds and y is the distance traveled in inches. 0 0

1 1.2

2 4.8

3 10.8

4 19.2

5 30

Find an algebraic model that fits the data. 24. U.S. Air Travel The number of revenue passengers enplaned in the United States over the 14-year period from 1994 to 2007 is shown in Table 1.7.

Table 1.7 U.S. Air Travel Year

Passengers (millions)

Year

Passengers (millions)

1994 1995 1996 1997 1998 1999 2000

528.8 547.8 581.2 594.7 612.9 636.0 666.2

2001 2002 2003 2004 2005 2006 2007

622.1 614.1 646.5 702.9 738.3 744.2 769.2

Source: www.airlines.org

(b) From 1994 to 2000 the data seem to follow a linear model. Use the 1994 and 2000 points to find an equation of the line and superimpose the line on the scatter plot. (c) According to the linear model, in what year did the number of passengers seem destined to reach 900 million? (d) What happened to disrupt the linear model? Exercises 25–28 refer to the graph below, which shows the minimum salaries in major league baseball over a recent 18-year period and the average salaries in major league baseball over the same period. Salaries are measured in dollars and time is measured after the starting year (year 0). y 1,400,000 1,260,000 1,120,000 980,000 840,000 700,000 560,000 420,000 280,000 140,000 –1

1

2

3

4 5 6

7

8

9 10 11 12 13 14 15 16 17 18

x

Source: Major League Baseball Players Association.

(b) If the smoke bomb is in free fall for 12.5 seconds after it is dropped, how high was the airplane when the smoke bomb was dropped?

t y

(a) Graph a scatter plot of the data. Let x be the number of years since 1994.

25. Which line is which, and how do you know? 26. After Peter Ueberroth’s resignation as baseball commissioner in 1988 and his successor’s untimely death in 1989, the team owners broke free of previous restrictions and began an era of competitive spending on player salaries. Identify where the 1990 salaries appear in the graph and explain how you can spot them. 27. The owners attempted to halt the uncontrolled spending by proposing a salary cap, which prompted a players’ strike in 1994. The strike caused the 1995 season to be shortened and left many fans angry. Identify where the 1995 salaries appear in the graph and explain how you can spot them. 28. Writing to Learn Analyze the general patterns in the graphical model and give your thoughts about what the longterm implications might be for (a) the players; (b) the team owners; (c) the baseball fans. In Exercises 29–38, solve the equation algebraically and confirm graphically. 29. v 2 - 5 = 8 - 2v 2 30. 1x + 1122 = 121

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 77

SECTION 1.1

31. 2x 2 - 5x + 2 = 1x - 321x - 22 + 3x 32. x 2 - 7x -

3 = 0 4

33. x12x - 52 = 12 34. x12x - 12 = 10

(b) Write an equation that can be solved to find the x-intercepts of the graph of y3. (c) Writing to Learn How does the graphical model reflect the fact that the answers to (a) and (b) are equivalent algebraically? (d) Confirm numerically that the x-intercepts of y3 give the same values when substituted into the expressions for y1 and y2.

35. x1x + 72 = 14

49. Exploring Grapher Failure Let y = 1x 20021/200.

36. x 2 - 3x + 4 = 2x 2 - 7x - 8

(a) Explain algebraically why y = x for all x Ú 0.

37. x + 1 - 2 2x + 4 = 0 38. 2x + x = 1

In Exercises 39–46, solve the equation graphically by converting it to an equivalent equation with 0 on the right-hand side and then finding the x-intercepts. 39. 2x - 5 = 2x + 4

41. ƒ 2x - 5 ƒ = 4 - ƒ x - 3 ƒ 3

40. ƒ 3x - 2 ƒ = 22x + 8 42. 2x + 6 = 6 - 225 - x 44. x + 1 = x 3 - 2x - 5

43. 2x - 3 = x - 5 45. 1x + 12-1 = x -1 + x

(b) Graph the equation y = 1x 20021/200 in the window [0, 1] by 30, 14. (c) Is the graph different from the graph of y = x? (d) Can you explain why the grapher failed?

50. Connecting Algebra and Geometry Explain how the algebraic equation 1x + b22 = x 2 + 2bx + b 2 models the areas of the regions in the geometric figure shown below on the left:

46. x 2 = ƒ x ƒ

x

b

x

47. Swan Auto Rental charges $32 per day plus $0.18 per mile for an automobile rental. (a) Elaine rented a car for one day and she drove 83 miles. How much did she pay? (b) Ramon paid $69.80 to rent a car for one day. How far did he drive? 48. Connecting Graphs and Equations The curves on the graph below are the graphs of the three curves given by y1 = 4x + 5 y2 = x 3 + 2x 2 - x + 3 y3 = - x 3 - 2x 2 + 5x + 2.

x

b 2

b

(Ex. 50)

(Ex. 52)

(a) y = 10x 3 + 7.5x 2 - 54.85x + 37.95 (b) y = x 3 + x 2 - 4.99x + 3.03 52. Connecting Algebra and Geometry The geometric figure shown on the right above is a large square with a small square missing.

15 10

(a) Find the area of the figure.

5 –2 –1

x

b 2

51. Exploring Hidden Behavior Solving graphically, find all real solutions to the following equations. Watch out for hidden behavior.

y

–5 –4

77

Modeling and Equation Solving

(b) What area must be added to complete the large square? 1

3 4 5

x

–5 –10

(a) Write an equation that can be solved to find the points of intersection of the graphs of y1 and y2.

(c) Explain how the algebraic formula for completing the square models the completing of the square in (b). 53. Proving a Theorem Prove that if n is a positive integer, then n 2 + 2n is either odd or a multiple of 4. Compare your proof with those of your classmates.

6965_CH01_pp063-156.qxd

78

1/14/10

1:00 PM

Page 78

CHAPTER 1 Functions and Graphs

54. Writing to Learn The graph below shows the distance from home against time for a jogger. Using information from the graph, write a paragraph describing the jogger’s workout.

Distance

y

x

Explorations 61. Analyzing the Market Both Ahmad and LaToya watch the stock market throughout the year for stocks that make significant jumps from one month to another. When they spot one, each buys 100 shares. Ahmad’s rule is to sell the stock if it fails to perform well for three months in a row. LaToya’s rule is to sell in December if the stock has failed to perform well since its purchase. The graph below shows the monthly performance in dollars (Jan–Dec) of a stock that both Ahmad and LaToya have been watching.

Time

Standardized Test Questions 55. True or False A product of real numbers is zero if and only if every factor in the product is zero. Justify your answer. 56. True or False An algebraic model can always be used to make accurate predictions.

(B) y = x 2 + 5

(C) y = 12 - 3x

(D) y = 4x + 3

130 120 110 100

Ja

(A) y = 2x + 3

Stock Index

n Fe . b M . ar . Ap r. M ay Ju ne Ju ly Au g Se . pt O . ct N . ov De . c.

In Exercises 57–60, you may use a graphing calculator to decide which algebraic model corresponds to the given graphical or numerical model.

140

(E) y = 28 - x

57. Multiple Choice (a) Both Ahmad and LaToya bought the stock early in the year. In which month? (b) At approximately what price did they buy the stock? (c) When did Ahmad sell the stock? (d) How much did Ahmad lose on the stock?

[0, 6] by [–9, 15]

(e) Writing to Learn Explain why LaToya’s strategy was better than Ahmad’s for this particular stock in this particular year.

58. Multiple Choice

(f) Sketch a 12-month graph of a stock’s performance that would favor Ahmad’s strategy over LaToya’s. 62. Group Activity Creating Hidden Behavior You can create your own graphs with hidden behavior. Working in groups of two or three, try this exploration.

[0, 9] by [0, 6]

59. Multiple Choice x 1 2 y 6 9 60. Multiple Choice x 0 2 y 3 7

3 14

4 11

4 21

6 15

5 30

8 19

6 41

10 23

(a) Graph the equation y = 1x + 221x 2 - 4x + 42 in the window 3- 4, 44 by 3-10, 104.

(b) Confirm algebraically that this function has zeros only at x = -2 and x = 2. (c) Graph the equation y = 1x + 221x 2 - 4x + 4.012 in the window 3-4, 44 by 3-10, 104

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 79

SECTION 1.1

(d) Confirm algebraically that this function has only one zero, at x = - 2. (Use the discriminant.) (e) Graph the equation 1x + 221x 2 - 4x + 3.992 in the window 3 -4, 44 by 3 - 10, 104. (f) Confirm algebraically that this function has three zeros. (Use the discriminant.)

Extending the Ideas 63. The Proliferation of Cell Phones Table 1.8 shows the number of cellular phone subscribers in the United States and their average monthly bill in the years from 1998 to 2007.

Table 1.8 Cellular Phone Subscribers Year

Subscribers (millions)

Average Local Monthly Bill ($)

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

69.2 86.0 109.5 128.4 140.8 158.7 182.1 207.9 233.0 255.4

39.43 41.24 45.27 47.37 48.40 49.91 50.64 49.98 50.56 49.79

Source: Cellular Telecommunication & Internet Association.

Modeling and Equation Solving

79

(a) Graph the scatter plots of the number of subscribers and the average local monthly bill as functions of time, letting time t = the number of years after 1990. (b) One of the scatter plots clearly suggests a linear model in the form y = mx + b. Use the points at t = 8 and t = 16 to find a linear model. (c) Superimpose the graph of the linear model onto the scatter plot. Does the fit appear to be good? (d) Why does a linear model seem inappropriate for the other scatter plot? Can you think of a function that might fit the data better? (e) In 1995 there were 33.8 million subscribers with an average local monthly bill of $51.00. Add these points to the scatter plots. (f) Writing to Learn The 1995 points do not seem to fit the models used to represent the 1998–2004 data. Give a possible explanation for this. 64. Group Activity (Continuation of Exercise 63) Discuss the economic forces suggested by the two models in Exercise 63 and speculate about the future by analyzing the graphs.

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 80

CHAPTER 1 Functions and Graphs

80

1.2 Functions and Their Properties What you’ll learn about • Function Definition and Notation • Domain and Range • Continuity • Increasing and Decreasing Functions • Boundedness

In this section we will introduce the terminology that is used to describe functions throughout this book. Feel free to skim over parts with which you are already familiar, but take the time to become comfortable with concepts that might be new to you (like continuity and symmetry). Even if it takes several days to cover this section, it will be precalculus time well spent.

• Local and Absolute Extrema

Function Definition and Notation

• Symmetry

Mathematics and its applications abound with examples of formulas by which quantitative variables are related to each other. The language and notation of functions is ideal for that purpose. A function is actually a simple concept; if it were not, history would have replaced it with a simpler one by now. Here is the definition.

• Asymptotes • End Behavior

... and why Functions and graphs form the basis for understanding the mathematics and applications you will see both in your workplace and in coursework in college.

x

DEFINITION Function, Domain, and Range A function from a set D to a set R is a rule that assigns to every element in D a unique element in R. The set D of all input values is the domain of the function, and the set R of all output values is the range of the function.

There are many ways to look at functions. One of the most intuitively helpful is the “machine” concept (Figure 1.9), in which values of the domain 1x2 are fed into the machine 1the function ƒ2 to produce range values 1 y2. To indicate that y comes from the function acting on x, we use Euler’s elegant function notation y = ƒ1x2 1 which we read as “ y equals ƒ of x” or “the value of ƒ at x”2. Here x is the independent variable and y is the dependent variable.

f f (x)

FIGURE 1.9 A “machine” diagram for a function.

A function can also be viewed as a mapping of the elements of the domain onto the elements of the range. Figure 1.10a shows a function that maps elements from the domain X onto elements of the range Y. Figure 1.10b shows another such mapping, but this one is not a function, since the rule does not assign the element x 1 to a unique element of Y.

A Bit of History The word function in its mathematical sense is generally attributed to Gottfried Leibniz (1646–1716), one of the pioneers in the methods of calculus. His attention to clarity of notation is one of his greatest contributions to scientific progress, which is why we still use his notation in calculus courses today. Ironically, it was not Leibniz but Leonhard Euler (1707–1783) who introduced the familiar notation ƒ1x2.

X

Y

X

y2

x2 x1

y2

x2 x1

y1

y4

x3

x3

y3

x4

Y

x4

y1

y3

Range

Domain A function

Not a function

(a)

(b)

FIGURE 1.10 The diagram in (a) depicts a mapping from X to Y that is a function. The diagram in (b) depicts a mapping from X to Y that is not a function.

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 81

SECTION 1.2

Functions and Their Properties

81

This uniqueness of the range value is very important to us as we study function behavior. Knowing that ƒ122 = 8 tells us something about ƒ, and that understanding would be contradicted if we were to discover later that ƒ122 = 4. That is why you will never see a function defined by an ambiguous formula like ƒ1x2 = 3x ! 2.

EXAMPLE 1 Defining a Function Does the formula y = x 2 define y as a function of x? SOLUTION Yes, y is a function of x. In fact, we can write the formula in function

notation: ƒ1x2 = x 2. When a number x is substituted into the function, the square of x will be the output, and there is no ambiguity about what the square of x is. Now try Exercise 3.

Another useful way to look at functions is graphically. The graph of the function y = ƒ1x2 is the set of all points 1x, ƒ1x22, x in the domain of ƒ. We match domain values along the x-axis with their range values along the y-axis to get the ordered pairs that yield the graph of y = ƒ1x2.

EXAMPLE 2 Seeing a Function Graphically Of the three graphs shown in Figure 1.11, which is not the graph of a function? How can you tell? SOLUTION The graph in (c) is not the graph of a function. There are three points

on the graph with x-coordinate 0, so the graph does not assign a unique value to 0. (Indeed, we can see that there are plenty of numbers between -2 and 2 to which the graph assigns multiple values.) The other two graphs do not have a comparable problem because no vertical line intersects either of the other graphs in more than one point. Graphs that pass this vertical line test are the graphs of functions. Now try Exercise 5.

[–4.7, 4.7] by [–3.3, 3.3]

(a)

[–4.7, 4.7] by [–3.3, 3.3] (b)

[–4.7, 4.7] by [–3.3, 3.3] (c)

FIGURE 1.11 One of these is not the graph of a function. (Example 2)

Vertical Line Test A graph 1set of points 1x, y22 in the xy-plane defines y as a function of x if and only if no vertical line intersects the graph in more than one point.

Domain and Range We will usually define functions algebraically, giving the rule explicitly in terms of the domain variable. The rule, however, does not tell the complete story without some consideration of what the domain actually is.

6965_CH01_pp063-156.qxd

82

1/14/10

1:00 PM

Page 82

CHAPTER 1 Functions and Graphs

What About Data? When moving from a numerical model to an algebraic model we will often use a function to approximate data pairs that by themselves violate our definition. In Figure 1.12 we can see that several pairs of data points fail the vertical line test, and yet the linear function approximates the data quite well.

For example, we can define the volume of a sphere as a function of its radius by the formula V1r2 =

4 3 pr 1Note that this is “V of r ”—not “V # r ”2. 3

This formula is defined for all real numbers, but the volume function is not defined for negative r-values. So, if our intention were to study the volume function, we would restrict the domain to be all r Ú 0.

Agreement

[–1, 10] by [–1, 11]

FIGURE 1.12 The data points fail the vertical line test but are nicely approximated by a linear function.

Unless we are dealing with a model (like volume) that necessitates a restricted domain, we will assume that the domain of a function defined by an algebraic expression is the same as the domain of the algebraic expression, the implied domain. For models, we will use a domain that fits the situation, the relevant domain.

EXAMPLE 3 Finding the Domain of a Function Find the domain of each of these functions: (a) ƒ1x2 = 2x + 3 2x (b) g1x2 = x - 5 (c) A1s2 = 123/42s 2, where A1s2 is the area of an equilateral triangle with sides of length s. SOLUTION

Solve Algebraically

Note The symbol “´ ” is read “union.” It means that the elements of the two sets are combined to form one set.

(a) The expression under a radical may not be negative. We set x + 3 Ú 0 and solve to find x Ú -3. The domain of ƒ is the interval 3- 3, q 2. (b) The expression under a radical may not be negative; therefore x Ú 0. Also, the denominator of a fraction may not be zero; therefore x Z 5. The domain of g is the interval 30, q 2 with the number 5 removed, which we can write as the union of two intervals: 30, 52 ´ 15, q 2. (c) The algebraic expression has domain all real numbers, but the behavior being modeled restricts s from being negative. The domain of A is the interval 30, q 2. Support Graphically

We can support our answers in (a) and (b) graphically, as the calculator should not plot points where the function is undefined. (a) Notice that the graph of y = 2x + 3 (Figure 1.13a) shows points only for x Ú -3, as expected. (b) The graph of y = 2x/1x - 52 (Figure 1.13b) shows points only for x Ú 0, as expected. Some calculators might show an unexpected line through the x-axis at x = 5. This line, another form of grapher failure, should not be there. Ignoring it, we see that 5, as expected, is not in the domain. (c) The graph of y = 123/42s 2 (Figure 1.13c) shows the unrestricted domain of the algebraic expression: all real numbers. The calculator has no way of knowing that s is the length of a side of a triangle. Now try Exercise 11.

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 83

SECTION 1.2

[–10, 10] by [–4, 4] (a)

Functions and Their Properties

[–10, 10] by [–4, 4] (b)

83

[–10, 10] by [–4, 4] (c)

FIGURE 1.13 Graphical support of the algebraic solutions in Example 3. The vertical line in (b) should be ignored because it results from grapher failure. The points in (c) with negative x-coordinates should be ignored because the calculator does not know that x is a length (but we do). Finding the range of a function algebraically is often much harder than finding the domain, although graphically the things we look for are similar: To find the domain we look for all x-coordinates that correspond to points on the graph, and to find the range we look for all y-coordinates that correspond to points on the graph. A good approach is to use graphical and algebraic approaches simultaneously, as we show in Example 4.

EXAMPLE 4 Finding the Range of a Function Find the range of the function ƒ1x2 =

2 . x

SOLUTION

Solve Graphically The graph of y =

2 is shown in Figure 1.14. x

Function Notation A grapher typically does not use function notation. So the function ƒ1x2 = x 2 + 1 is entered as y1 = x 2 + 1. On some graphers you can evaluate ƒ at x = 3 by entering y1132 on the home screen. On the other hand, on other graphers y1132 means y1 * 3.

[–5, 5] by [–3, 3]

FIGURE 1.14 The graph of y = 2/x. Is y = 0 in the range? It appears that x = 0 is not in the domain (as expected, because a denominator cannot be zero). It also appears that the range consists of all real numbers except 0. Confirm Algebraically We confirm that 0 is not in the range by trying to solve 2/x = 0: 2 = 0 x

2 = 0#x 2 = 0 (continued)

6965_CH01_pp063-156.qxd

84

1/14/10

1:00 PM

Page 84

CHAPTER 1 Functions and Graphs

Since the equation 2 = 0 is never true, 2/x = 0 has no solutions, and so y = 0 is not in the range. But how do we know that all other real numbers are in the range? We let k be any other real number and try to solve 2/x = k: 2 = k x

2 = k#x 2 x = k

As you can see, there was no problem finding an x this time, so 0 is the only number not in the range of ƒ. We write the range 1- q , 02 ´ 10, q 2. Now try Exercise 17. You can see that this is considerably more involved than finding a domain, but we are hampered at this point by not having many tools with which to analyze function behavior. We will revisit the problem of finding ranges in Exercise 86, after having developed the tools that will simplify the analysis.

Continuity One of the most important properties of the majority of functions that model real-world behavior is that they are continuous. Graphically speaking, a function is continuous at a point if the graph does not come apart at that point. We can illustrate the concept with a few graphs (Figure 1.15): y

y

y

y

y

f (a)

x

Continuous at all x

x

a

Removable discontinuity

a

x

Removable discontinuity

a

a

x

x

Jump discontinuity

Infinite discontinuity

FIGURE 1.15 Some points of discontinuity. y

Let’s look at these cases individually. This graph is continuous everywhere. Notice that the graph has no breaks. This means that if we are studying the behavior of the function ƒ for x-values close to any particular real number a, we can be assured that the ƒ1x2-values will be close to ƒ1a2.

x

Continuous at all x

This graph is continuous everywhere except for the “hole” at x = a. If we are studying the behavior of this function ƒ for x-values close to a, we cannot be assured that the ƒ1x2values will be close to ƒ1a2. In this case, ƒ1x2 is smaller than ƒ1a2 for x near a. This is called a removable discontinuity because it can be patched by redefining ƒ1a2 so as to plug the hole.

y f (a)

a

Removable discontinuity

x

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 85

SECTION 1.2

Functions and Their Properties

85

y

This graph also has a removable discontinuity at x = a. If we are studying the behavior of this function ƒ for x-values close to a, we are still not assured that the ƒ1x2-values will be close to ƒ1a2, because in this case ƒ1a2 doesn’t even exist. It is removable because we could define ƒ1a2 in such a way as to plug the hole and make ƒ continuous at a.

x

a

Removable discontinuity y

Here is a discontinuity that is not removable. It is a jump discontinuity because there is more than just a hole at x = a; there is a jump in function values that makes the gap impossible to plug with a single point 1a, ƒ1a22, no matter how we try to redefine ƒ1a2.

a

x

Jump discontinuity

y

This is a function with an infinite discontinuity at x = a. It is definitely not removable. A Limited Use of Limits While the notation of limits is easy to understand, the algebraic definition of a limit can be a little intimidating and is best left to future courses. We will have more to say about limits in Chapter 10. For now, if you understand the statement lim 1x 2 - 12 = 24, you are where you

a

x

Infinite discontinuity

x: 5

need to be.

The simple geometric concept of an unbroken graph at a point is a visual notion that is extremely difficult to communicate accurately in the language of algebra. The key concept from the pictures seems to be that we want the point 1x, ƒ1x22 to slide smoothly onto the point 1a, ƒ1a22 without missing it from either direction. We say that ƒ1x2 approaches ƒ1a2 as a limit as x approaches a, and we write lim ƒ1x2 = ƒ1a2. This “limit x:a notation” reflects graphical behavior so naturally that we will use it throughout this book as an efficient way to describe function behavior, beginning with this definition of continuity. A function ƒ is continuous at x ! a if lim ƒ1x2 = ƒ1a2. A function is x :a discontinuous at x ! a if it is not continuous at x = a.

EXAMPLE 5 Identifying Points of Discontinuity Judging from the graphs, which of the following figures shows functions that are discontinuous at x = 2? Are any of the discontinuities removable? SOLUTION Figure 1.16 shows a function that is undefined at x = 2 and hence not

continuous there. The discontinuity at x = 2 is not removable.

The function graphed in Figure 1.17 is a quadratic polynomial whose graph is a parabola, a graph that has no breaks because its domain includes all real numbers. It is continuous for all x. The function graphed in Figure 1.18 is not defined at x = 2 and so cannot be continuous there. The graph looks like the graph of the line y = x + 2, except that there is a hole where the point 12, 42 should be. This is a removable discontinuity. Now try Exercise 21.

6965_CH01_pp063-156.qxd

86

1/14/10

1:00 PM

Page 86

CHAPTER 1 Functions and Graphs

[–9.4, 9.4] by [–6, 6]

[–9.4, 9.4] by [–6.2, 6.2]

[–5, 5] by [–10, 10]

x + 3 x - 2

FIGURE 1.16 ƒ1x2 =

FIGURE 1.17 g1x2 = 1x + 321x - 22

FIGURE 1.18 h1x2 =

x2 - 4 x - 2

Increasing and Decreasing Functions Another function concept that is easy to understand graphically is the property of being increasing, decreasing, or constant on an interval. We illustrate the concept with a few graphs (Figure 1.19): y y

y 3 2 1

3 2 1 –5 –4 –3

–1 –1 –2 –3

y

1 2 3 4 5

Increasing

x

–5 –4 –3 –2 –1 –1 –2 –3

3 2

3

1 2 3 4 5

x

1 –5 –4 –3 –2 –1 –1 –2 –3

Decreasing

1 2 3 4 5

x

–5 –4 –3 –2 –1 –1 –2 –3

1 2 3 4 5

x

Decreasing on (–", –2] Constant on [–2, 2] Increasing on [2, ")

Constant

FIGURE 1.19 Examples of increasing, decreasing, or constant on an interval.

Once again the idea is easy to communicate graphically, but how can we identify these properties of functions algebraically? Exploration 1 will help to set the stage for the algebraic definition.

EXPLORATION 1

Increasing, Decreasing, and Constant Data

1. Of the three tables of numerical data below, which would be modeled by a

function that is (a) increasing, (b) decreasing, (c) constant? X

Y1

X

Y2

X

Y3

-2

12

-2

3

-2

-5

-1

12

-1

1

-1

-3

0

12

0

0

0

-1

1

12

1

-2

1

1

3

12

3

-6

3

4

7

12

7

-12

7

10

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 87

SECTION 1.2

≤ List on a Calculator Your calculator might be able to help you with the numbers in Exploration 1. Some calculators have a “ ¢List” operation that will calculate the changes as you move down a list. For example, the command “ ¢List (L1) : L3” will store the differences from L1 into L3. Note that ¢List (L1) is always one entry shorter than L1 itself.

Functions and Their Properties

87

2. Make a list of ¢ Y1, the change in Y1 values as you move down the list. As

you move from Y1 = a to Y1 = b, the change is ¢Y1 = b - a. Do the same for the values of Y2 and Y3. X moves ¢ X ¢ Y1 from - 2 to -1 1

X moves ¢ X ¢ Y2 from - 2 to - 1 1

X moves ¢ X ¢ Y3 from -2 to -1 1

- 1 to 0

1

- 1 to 0

1

-1 to 0

1

0 to 1

1

0 to 1

1

0 to 1

1

1 to 3

2

1 to 3

2

1 to 3

2

3 to 7

4

3 to 7

4

3 to 7

4

3. What is true about the quotients ¢Y/¢X for an increasing function? For a

decreasing function? For a constant function? 4. Where else have you seen the quotient ¢Y/¢X? Does this reinforce your answers in part 3?

Your analysis of the quotients ¢Y/¢X in the exploration should help you to understand the following definition.

DEFINITION Increasing, Decreasing, and Constant Function on an Interval A function ƒ is increasing on an interval if, for any two points in the interval, a positive change in x results in a positive change in ƒ1x2. A function ƒ is decreasing on an interval if, for any two points in the interval, a positive change in x results in a negative change in ƒ1x2. A function ƒ is constant on an interval if, for any two points in the interval, a positive change in x results in a zero change in ƒ1x2.

EXAMPLE 6 Analyzing a Function for IncreasingDecreasing Behavior For each function, tell the intervals on which it is increasing and the intervals on which it is decreasing. x2 (a) ƒ1x2 = 1x + 222 (b) g1x2 = 2 x - 1 SOLUTION

Solve Graphically (a) We see from the graph in Figure 1.20 that ƒ is decreasing on 1 - q , - 24 and increasing on 3-2, q 2. (Notice that we include -2 in both intervals. Don’t worry that this sets up some contradiction about what happens at - 2, because we only talk about functions increasing or decreasing on intervals, and -2 is not an interval.) (continued)

6965_CH01_pp063-156.qxd

88

1/14/10

1:00 PM

Page 88

CHAPTER 1 Functions and Graphs

[–5, 5] by [–3, 5]

FIGURE 1.20 The function ƒ1x2 = 1x + 222 decreases on 1- q , - 24 and increases on 3- 2, q 2. (Example 6)

(b) We see from the graph in Figure 1.21 that g is increasing on 1- q , -12, increasing again on 1-1, 04, decreasing on 30, 12, and decreasing again on 11, q 2.

[–4.7, 4.7] by [–3.1, 3.1]

FIGURE 1.21 The function g1x2 = x 2/1x 2 - 12 increases on 1- q , - 12 and 1- 1, 04; the function decreases on 30, 12 and 11, q 2. (Example 6)

Now try Exercise 33.

You may have noticed that we are making some assumptions about the graphs. How do we know that they don’t turn around somewhere off the screen? We will develop some ways to answer that question later in the book, but the most powerful methods will await you when you study calculus.

Boundedness The concept of boundedness is fairly simple to understand both graphically and algebraically. We will move directly to the algebraic definition after motivating the concept with some typical graphs (Figure 1.22).

y

y

y y

x

x

x x

Not bounded above Not bounded below

Not bounded above Bounded below

Bounded above Not bounded below

FIGURE 1.22 Some examples of graphs bounded and not bounded above and below.

Bounded

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 89

SECTION 1.2

Functions and Their Properties

89

DEFINITION Lower Bound, Upper Bound, and Bounded A function ƒ is bounded below if there is some number b that is less than or equal to every number in the range of ƒ. Any such number b is called a lower bound of ƒ. A function ƒ is bounded above if there is some number B that is greater than or equal to every number in the range of ƒ. Any such number B is called an upper bound of ƒ. A function ƒ is bounded if it is bounded both above and below. We can extend the above definition to the idea of bounded on an interval by restricting the domain of consideration in each part of the definition to the interval we wish to consider. For example, the function ƒ1x2 = 1/x is bounded above on the interval 1- q , 02 and bounded below on the interval 10, q 2.

EXAMPLE 7 Checking Boundedness

[–4, 4] by [–5, 5] (a)

Identify each of these functions as bounded below, bounded above, or bounded. x (a) w1x2 = 3x 2 - 4 (b) p1x2 = 1 + x2 SOLUTION

Solve Graphically The two graphs are shown in Figure 1.23. It appears that w is bounded below, and p is bounded. Confirm Graphically We can confirm that w is bounded below by finding a lower bound, as follows:

[–8, 8] by [–1, 1] (b)

FIGURE 1.23 The graphs for Example 7. Which are bounded where?

x2 3x 2 3x 2 - 4 3x 2 - 4

Ú Ú Ú Ú

0 0 0 - 4 -4

x 2 is nonnegative. Multiply by 3. Subtract 4.

Thus, -4 is a lower bound for w1x2 = 3x 2 - 4. We leave the verification that p is bounded as an exercise (Exercise 77). Now try Exercise 37.

Local and Absolute Extrema Many graphs are characterized by peaks and valleys where they change from increasing to decreasing and vice versa. The extreme values of the function (or local extrema) can be characterized as either local maxima or local minima. The distinction can be easily seen graphically. Figure 1.24 shows a graph with three local extrema: local maxima at points P and R and a local minimum at Q.

y R

P x Q

FIGURE 1.24 The graph suggests that ƒ has a local maximum at P, a local minimum at Q, and a local maximum at R.

This is another function concept that is easier to see graphically than to describe algebraically. Notice that a local maximum does not have to be the maximum value of a function; it only needs to be the maximum value of the function on some tiny interval. We have already mentioned that the best method for analyzing increasing and decreasing behavior involves calculus. Not surprisingly, the same is true for local extrema. We will generally be satisfied in this course with approximating local extrema using a graphing calculator, although sometimes an algebraic confirmation will be possible when we learn more about specific functions.

6965_CH01_pp063-156.qxd

90

1/14/10

1:00 PM

Page 90

CHAPTER 1 Functions and Graphs

DEFINITION Local and Absolute Extrema A local maximum of a function ƒ is a value ƒ1c2 that is greater than or equal to all range values of ƒ on some open interval containing c. If ƒ1c2 is greater than or equal to all range values of ƒ, then ƒ1c2 is the maximum 1or absolute maximum2 value of ƒ.

A local minimum of a function ƒ is a value ƒ1c2 that is less than or equal to all range values of ƒ on some open interval containing c. If ƒ1c2 is less than or equal to all range values of ƒ, then ƒ1c2 is the minimum (or absolute minimum) value of ƒ. Local extrema are also called relative extrema. Minimum X=–2.056546

Y=–24.05728

[–5, 5] by [–35, 15] 4

FIGURE 1.25 A graph of y = x 7x 2 + 6x. (Example 8)

Using a Grapher to Find Local Extrema Most modern graphers have built-in “maximum” and “minimum” finders that identify local extrema by looking for sign changes in ¢y. It is not easy to find local extrema by zooming in on them, as the graphs tend to flatten out and hide the very behavior you are looking at. If you use this method, keep narrowing the vertical window to maintain some curve in the graph.

EXAMPLE 8 Identifying Local Extrema Decide whether ƒ1x2 = x 4 - 7x 2 + 6x has any local maxima or local minima. If so, find each local maximum value or minimum value and the value of x at which each occurs. SOLUTION The graph of y = x 4 - 7x 2 + 6x (Figure 1.25) suggests that there

are two local minimum values and one local maximum value. We use the graphing calculator to approximate local minima as -24.06 (which occurs at x L -2.06) and -1.77 (which occurs at x L 1.60). Similarly, we identify the (approximate) local maximum as 1.32 (which occurs at x L 0.46). Now try Exercise 41.

Symmetry In the graphical sense, the word “symmetry” in mathematics carries essentially the same meaning as it does in art: The picture (in this case, the graph) “looks the same” when viewed in more than one way. The interesting thing about mathematical symmetry is that it can be characterized numerically and algebraically as well. We will be looking at three particular types of symmetry, each of which can be spotted easily from a graph, a table of values, or an algebraic formula, once you know what to look for. Since it is the connections among the three models (graphical, numerical, and algebraic) that we need to emphasize in this section, we will illustrate the various symmetries in all three ways, side-by-side.

Symmetry with respect to the y-axis Example: f (x) = x 2 Graphically

Numerically

y

(–x, y)

–x

x

y

(x, y)

x

x

FIGURE 1.26 The graph looks the same to the left of the y-axis as it does to the right of it.

ƒ1x2

-3 -2 -1 1 2

9 4 1 1 4

3

9

Algebraically For all x in the domain of ƒ, ƒ1- x2 = ƒ1x2. Functions with this property (for example, x n, n even) are even functions.

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 91

SECTION 1.2

Functions and Their Properties

91

Symmetry with respect to the x-axis Example: x = y 2 Graphically

Numerically

y

(x, y)

y

x

x

–y

(x, –y)

FIGURE 1.27 The graph looks the same above the x-axis as it does below it.

x

y

9 4 1 1 4 9

-3 -2 -1 1 2 3

Algebraically Graphs with this kind of symmetry are not functions (except the zero function), but we can say that 1x, - y2 is on the graph whenever 1x, y2 is on the graph.

Symmetry with respect to the origin Example: f (x) = x3 Graphically

Numerically

y (x, y)

y

–x (–x, –y)

x

x

–y

x

y

-3 -2 -1 1 2 3

-27 -8 -1 1 8 27

Algebraically For all x in the domain of ƒ, ƒ1-x2 = - ƒ1x2. Functions with this property (for example, x n, n odd) are odd functions.

FIGURE 1.28 The graph looks the same upside-down as it does rightside-up.

EXAMPLE 9 Checking Functions for Symmetry Tell whether each of the following functions is odd, even, or neither. (a) ƒ1x2 = x 2 - 3

(b) g1x2 = x 2 - 2x - 2

(c) h1x2 =

x3 4 - x2

SOLUTION

(a) Solve Graphically The graphical solution is shown in Figure 1.29.

[–5, 5] by [–4, 4]

FIGURE 1.29 This graph appears to be symmetric with respect to the y-axis, so we conjecture that ƒ is an even function. (continued)

6965_CH01_pp063-156.qxd

92

1/14/10

1:00 PM

Page 92

CHAPTER 1 Functions and Graphs

Confirm Algebraically We need to verify that ƒ1- x2 = ƒ1x2 for all x in the domain of ƒ. ƒ1-x2 = 1-x22 - 3 = x 2 - 3 = ƒ1x2 Since this identity is true for all x, the function ƒ is indeed even. (b) Solve Graphically The graphical solution is shown in Figure 1.30. Confirm Algebraically We need to verify that g1- x2 Z g1x2 and g1-x2 Z - g1x2. g1- x2 = 1- x22 - 21-x2 - 2 = x 2 + 2x - 2 g1x2 = x 2 - 2x - 2 -g1x2 = - x 2 + 2x + 2

[–5, 5] by [–4, 4]

FIGURE 1.30 This graph does not appear to be symmetric with respect to either the yaxis or the origin, so we conjecture that g is neither even nor odd.

So g1-x2 Z g1x2 and g1-x2 Z - g1x2. We conclude that g is neither odd nor even. (c) Solve Graphically The graphical solution is shown in Figure 1.31. Confirm Algebraically We need to verify that h1- x2 = - h1x2 for all x in the domain of h. [–4.7, 4.7] by [–10, 10]

h1-x2 =

4 - 1-x22 = - h1x2

FIGURE 1.31 This graph appears to be symmetric with respect to the origin, so we conjecture that h is an odd function.

1-x23

=

-x3 4 - x2

Since this identity is true for all x except !2 (which are not in the domain of h), the function h is odd. Now try Exercise 49.

y

Asymptotes

6 5 4 3 2 1 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6

Consider the graph of the function ƒ1x2 =

1 2 3 4 5

FIGURE 1.32 The graph of

x

ƒ1x2 = 2x 2/14 - x 22 has two vertical asymptotes and one horizontal asymptote.

2x 2 in Figure 1.32. 4 - x2

The graph appears to flatten out to the right and to the left, getting closer and closer to the horizontal line y = - 2. We call this line a horizontal asymptote. Similarly, the graph appears to flatten out as it goes off the top and bottom of the screen, getting closer and closer to the vertical lines x = - 2 and x = 2. We call these lines vertical asymptotes. If we superimpose the asymptotes onto Figure 1.32 as dashed lines, you can see that they form a kind of template that describes the limiting behavior of the graph (Figure 1.33 on the next page). Since asymptotes describe the behavior of the graph at its horizontal or vertical extremities, the definition of an asymptote can best be stated with limit notation. In this definition, note that x : a - means “x approaches a from the left,” while x : a + means “x approaches a from the right.”

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 93

SECTION 1.2

y

–1 –1

93

DEFINITION Horizontal and Vertical Asymptotes The line y = b is a horizontal asymptote of the graph of a function y = ƒ1x2 if ƒ1x2 approaches a limit of b as x approaches + q or - q .

6 5 4 3 2 1 –5 –4 –3

Functions and Their Properties

In limit notation:

1

3 4 5

lim ƒ1x2 = b

x: -q

x

–3 –4 –5 –6

or

lim ƒ1x2 = b

x: +q

The line x = a is a vertical asymptote of the graph of a function y = ƒ1x2 if ƒ1x2 approaches a limit of + q or - q as x approaches a from either direction. In limit notation: lim ƒ1x2 = ! q

x:a -

or

lim ƒ1x2 = ! q

x :a +

FIGURE 1.33 The graph of

ƒ1x2 = 2x 2/14 - x 22 with the asymptotes shown as dashed lines.

EXAMPLE 10 Identifying the Asymptotes of a Graph Identify any horizontal or vertical asymptotes of the graph of y =

x . x - x - 2 2

SOLUTION The quotient x/1x 2 - x - 22 = x/11x + 121x - 222 is undefined at

x = - 1 and x = 2, which makes them likely sites for vertical asymptotes. The graph (Figure 1.34) provides support, showing vertical asymptotes of x = - 1 and x = 2. [–4.7, 4.7] by [–3, 3]

FIGURE 1.34 The graph of

y = x/1x 2 - x - 22 has vertical asymptotes of x = -1 and x = 2 and a horizontal asymptote of y = 0. (Example 10)

For large values of x, the numerator (a large number) is dwarfed by the denominator (a product of two large numbers), suggesting that lim x/11x + 121x - 222 = 0. This x: q would indicate a horizontal asymptote of y = 0. The graph (Figure 1.34) provides support, showing a horizontal asymptote of y = 0 as x : q . Similar logic suggests that lim x/11x + 121x - 222 = - 0 = 0, indicating the same horizontal asymptote x: - q as x : - q . Again, the graph provides support for this. Now try Exercise 57.

End Behavior A horizontal asymptote gives one kind of end behavior for a function because it shows how the function behaves as it goes off toward either “end” of the x-axis. Not all graphs approach lines, but it is helpful to consider what does happen “out there.” We illustrate with a few examples.

EXAMPLE 11 Matching Functions Using End Behavior Match the functions with the graphs in Figure 1.35 by considering end behavior. All graphs are shown in the same viewing window. 3x x + 1 3x 3 (c) y = 2 x + 1 (a) y =

2

3x 2 x + 1 3x 4 (d) y = 2 x + 1 (b) y =

2

(continued)

6965_CH01_pp063-156.qxd

94

1/14/10

1:00 PM

Page 94

CHAPTER 1 Functions and Graphs

Tips on Zooming

SOLUTION When x is very large, the denominator x 2 + 1 in each of these func-

tions is almost the same number as x 2. If we replace x 2 + 1 in each denominator by x 2 and then reduce the fractions, we get the simpler functions

Zooming out is often a good way to investigate end behavior with a graphing calculator. Here are some useful zooming tips: • Start with a “square” window.

(a) y =

• Set Xscl and Yscl to zero to avoid fuzzy axes.

3 1close to y = 0 for large x2 x

(b) y = 3

(c) y = 3x (d) y = 3x 2. So, we look for functions that have end behavior resembling, respectively, the functions (a) y = 0 (b) y = 3 (c) y = 3x (d) y = 3x 2.

• Be sure the zoom factors are both the same. (They will be unless you change them.)

Graph (iv) approaches the line y = 0. Graph (iii) approaches the line y = 3. Graph (ii) approaches the line y = 3x. Graph (i) approaches the parabola y = 3x 2. So, (a) matches (iv), (b) matches (iii), (c) matches (ii), and (d) matches (i). Now try Exercise 65.

[–4.7, 4.7] by [–3.5, 3.5] (i)

[–4.7, 4.7] by [–3.5, 3.5] (iii)

[–4.7, 4.7] by [–3.5, 3.5] (ii)

[–4.7, 4.7] by [–3.5, 3.5] (iv)

FIGURE 1.35 Match the graphs with the functions in Example 11. For more complicated functions we are often content with knowing whether the end behavior is bounded or unbounded in either direction.

QUICK REVIEW 1.2

(For help, go to Sections A.3, P.3, and P.5.)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 1–4, solve the equation or inequality. 1. x 2 - 16 = 0

2. 9 - x 2 = 0

3. x - 10 6 0

4. 5 - x … 0

In Exercises 5–10, find all values of x algebraically for which the algebraic expression is not defined. Support your answer graphically.

5.

x x - 16

6.

7. 2x - 16 9. 10.

8.

x x 2 - 16 2x 2 + 1 x2 - 1

2x + 2

23 - x x 2 - 2x x2 - 4

SECTION 1.2 EXERCISES In Exercises 1–4, determine whether the formula determines y as a function of x. If not, explain why not. 1. y = 2x - 4 3. x = 2y 2

5.

6.

y

y

2. y = x 2 ; 3 4. x = 12 - y

In Exercises 5–8, use the vertical line test to determine whether the curve is the graph of a function.

x

x

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 95

SECTION 1.2

y

7.

y

8.

Functions and Their Properties

95

In Exercises 29–34, graph the function and identify intervals on which the function is increasing, decreasing, or constant. 29. ƒ1x2 = ƒ x + 2 ƒ - 1

x

30. ƒ1x2 = ƒ x + 1 ƒ + ƒ x - 1 ƒ - 3

x

31. g1x2 = ƒ x + 2 ƒ + ƒ x - 1 ƒ - 2 32. h1x2 = 0.51x + 222 - 1

In Exercises 9–16, find the domain of the function algebraically and support your answer graphically. 5 10. h1x2 = x - 3

9. ƒ1x2 = x 2 + 4

x

13. g1x2 =

14. h1x2 =

2

x - 5x 24 - x

15. h1x2 =

24 - x x - 3

2

18. g1x2 = 5 + 24 - x x2

20. g1x2 =

1 - x2

3 + x2 4 - x2

3 x

ƒxƒ x

22. h1x2 =

x3 + x x

24. g1x2 =

x x - 2

39. y = 21 - x 2

40. y = x - x 3

41. ƒ1x2 = 4 - x + x 2

42. g1x2 = x 3 - 4x + 1

43. h1x2 = -x 3 + 2x - 3

44. ƒ1x2 = 1x + 321x - 122

y

26.

(5, 7)

y

(5, 5)

47. ƒ1x2 = 2x 4

48. g1x2 = x 3

49. ƒ1x2 = 2x 2 + 2

50. g1x2 =

53. g1x2 = 2x 3 - 3x

54. h1x2 =

1 + x2 51. ƒ1x2 = -x + 0.03x + 5 52. ƒ1x2 = x 3 + 0.04x 2 + 3

y

27.

x

y (1, 6)

(1, 5) (#1, 3)

(5, 4)

(3, 3) (5, 1)

x

(#1, 1)

(3, 1)

x

x - 1 x

x x - 1

56. q1x2 =

57. g1x2 =

x + 2 3 - x

58. q1x2 = 1.5x

61. g1x2 =

(1, 2)

28.

1 x

55. ƒ1x2 =

(3, 3)

x

3

2

59. ƒ1x2 =

(#1, 4)

(2, 2)

46. g1x2 = x ƒ 2x + 5 ƒ

In Exercises 55–62, use a method of your choice to find all horizontal and vertical asymptotes of the function.

In Exercises 25–28, state whether each labeled point identifies a local minimum, a local maximum, or neither. Identify intervals on which the function is decreasing and increasing. 25.

38. y = 2 - x

In Exercises 47–54, state whether the function is odd, even, or neither. Support graphically and confirm algebraically.

In Exercises 21–24, graph the function and tell whether or not it has a point of discontinuity at x = 0. If there is a discontinuity, tell whether it is removable or nonremovable.

23. ƒ1x2 =

37. y = 2

45. h1x2 = x 2 2x + 4

17. ƒ1x2 = 10 - x 2

21. g1x2 =

36. y = 2 - x 2

In Exercises 41–46, use a grapher to find all local maxima and minima and the values of x where they occur. Give values rounded to two decimal places.

In Exercises 17–20, find the range of the function.

19. ƒ1x2 =

In Exercises 35–40, determine whether the function is bounded above, bounded below, or bounded on its domain. x

16. ƒ1x2 = 2x 4 - 16x 2

1x + 121x 2 + 12

34. ƒ1x2 = x 3 - x 2 - 2x

35. y = 32

3x - 1 5 1 12. ƒ1x2 = + x 1x + 321x - 12 x - 3

11. ƒ1x2 =

33. g1x2 = 3 - 1x - 122

x2 + 2 2

60. p1x2 =

3

62. h1x2 =

x - 1 4x - 4 x - 8

4 2

x + 1 2x - 4 x2 - 4

6965_CH01_pp063-156.qxd

96

1/14/10

1:00 PM

Page 96

CHAPTER 1 Functions and Graphs

In Exercises 63–66, match the function with the corresponding graph by considering end behavior and asymptotes. All graphs are shown in the same viewing window. x2 + 2 64. y = 2x + 1

x + 2 63. y = 2x + 1 65. y =

x + 2

66. y =

2x 2 + 1

(b) Show how you can add a single point to the graph of ƒ and get a graph that does intersect its vertical asymptote. (c) Is the graph in (b) the graph of a function? 70. Writing to Learn Explain why a graph cannot have more than two horizontal asymptotes.

x3 + 2 2x 2 + 1

Standardized Test Questions 71. True or False The graph of function ƒ is defined as the set of all points 1x, ƒ1x22, where x is in the domain of ƒ. Justify your answer. 72. True or False A relation that is symmetric with respect to the x-axis cannot be a function. Justify your answer.

[–4.7, 4.7] by [–3.1, 3.1] (a)

[–4.7, 4.7] by [–3.1, 3.1] (b)

In Exercises 73–76, answer the question without using a calculator. 73. Multiple Choice Which function is continuous? (A) Number of children enrolled in a particular school as a function of time (B) Outdoor temperature as a function of time (C) Cost of U.S. postage as a function of the weight of the letter

[–4.7, 4.7] by [–3.1, 3.1] (c)

[–4.7, 4.7] by [–3.1, 3.1] (d)

67. Can a Graph Cross Its Own Asymptote? The Greek roots of the word “asymptote” mean “not meeting,” since graphs tend to approach, but not meet, their asymptotes. Which of the following functions have graphs that do intersect their horizontal asymptotes? (a) ƒ1x2 = (b) g1x2 = (c) h1x2 =

x x2 - 1 x x2 + 1 x2

x3 + 1 68. Can a Graph Have Two Horizontal Asymptotes? Although most graphs have at most one horizontal asymptote, it is possible for a graph to have more than one. Which of the following functions have graphs with more than one horizontal asymptote? ƒx3 + 1ƒ (a) ƒ1x2 = 8 - x3 ƒx - 1ƒ (b) g1x2 = 2 x - 4 x (c) h1x2 = 2 2x - 4 69. Can a Graph Intersect Its Own Vertical x - ƒxƒ + 1. Asymptote? Graph the function ƒ1x2 = x2 (a) The graph of this function does not intersect its vertical asymptote. Explain why it does not.

(D) Price of a stock as a function of time (E) Number of soft drinks sold at a ballpark as a function of outdoor temperature 74. Multiple Choice Which function is not continuous? (A) Your altitude as a function of time while flying from Reno to Dallas (B) Time of travel from Miami to Pensacola as a function of driving speed (C) Number of balls that can fit completely inside a particular box as a function of the radius of the balls (D) Area of a circle as a function of radius (E) Weight of a particular baby as a function of time after birth 75. Decreasing Function Which function is decreasing? (A) Outdoor temperature as a function of time (B) The Dow Jones Industrial Average as a function of time (C) Air pressure in the Earth’s atmosphere as a function of altitude (D) World population since 1900 as a function of time (E) Water pressure in the ocean as a function of depth 76. Increasing or Decreasing Which function cannot be classified as either increasing or decreasing? (A) Weight of a lead brick as a function of volume (B) Strength of a radio signal as a function of distance from the transmitter (C) Time of travel from Buffalo to Syracuse as a function of driving speed (D) Area of a square as a function of side length (E) Height of a swinging pendulum as a function of time

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 97

SECTION 1.2

Explorations

(a) Graph the function and find the smallest integer k that appears to be an upper bound. (b) Verify that x/11 + x 22 6 k by proving the equivalent inequality kx 2 - x + k 7 0. (Use the quadratic formula to show that the quadratic has no real zeros.) (c) From the graph, find the greatest integer k that appears to be a lower bound. 2

(d) Verify that x/11 + x 2 7 k by proving the equivalent inequality kx 2 - x + k 6 0.

78. Baylor School Grade Point Averages Baylor School uses a sliding scale to convert the percentage grades on its transcripts to grade point averages (GPAs). Table 1.9 shows the GPA equivalents for selected grades.

Table 1.9 Converting Grades 60 65 70 75 80 85 90 95 100

97

(d) ƒ102 = ƒ152 = 2;

77. Bounded Functions As promised in Example 7 of this section, we will give you a chance to prove algebraically that p1x2 = x/11 + x 22 is bounded.

Grade 1x2

Functions and Their Properties

GPA 1y2 0.00 1.00 2.05 2.57 3.00 3.36 3.69 4.00 4.28

Source: Baylor School College Counselor.

(a) Considering GPA 1y2 as a function of percentage grade 1x2, is it increasing, decreasing, constant, or none of these? (b) Make a table showing the change 1¢y2 in GPA as you move down the list. (See Exploration 1.)

(c) Make a table showing the change in ¢y as you move down the list. (This is ¢¢y.) Considering the change 1¢y2 in GPA as a function of percentage grade 1x2, is it increasing, decreasing, constant, or none of these? (d) In general, what can you say about the shape of the graph if y is an increasing function of x and ¢y is a decreasing function of x?

(e) Sketch the graph of a function y of x such that y is a decreasing function of x and ¢y is an increasing function of x. 79. Group Activity Sketch (freehand) a graph of a function ƒ with domain all real numbers that satisfies all of the following conditions: (a) ƒ is continuous for all x; (b) ƒ is increasing on 1- q , 04 and on 33, 54; (c) ƒ is decreasing on 30, 34 and on 35, q 2;

(e) ƒ132 = 0. 80. Group Activity Sketch (freehand) a graph of a function ƒ with domain all real numbers that satisfies all of the following conditions: (a) ƒ is decreasing on 1- q , 02 and decreasing on 10, q 2; (b) ƒ has a nonremovable point of discontinuity at x = 0; (c) ƒ has a horizontal asymptote at y = 1; (d) ƒ102 = 0; (e) ƒ has a vertical asymptote at x = 0. 81. Group Activity Sketch (freehand) a graph of a function ƒ with domain all real numbers that satisfies all of the following conditions: (a) ƒ is continuous for all x; (b) ƒ is an even function; (c) ƒ is increasing on 30, 24 and decreasing on 32, q 2; (d) ƒ122 = 3.

82. Group Activity Get together with your classmates in groups of two or three. Sketch a graph of a function, but do not show it to the other members of your group. Using the language of functions (as in Exercises 79–81), describe your function as completely as you can. Exchange descriptions with the others in your group and see if you can reproduce each other’s graphs.

Extending the Ideas 83. A function that is bounded above has an infinite number of upper bounds, but there is always a least upper bound, i.e., an upper bound that is less than all the others. This least upper bound may or may not be in the range of ƒ. For each of the following functions, find the least upper bound and tell whether or not it is in the range of the function. (a) ƒ1x2 = 2 - 0.8x 2 (b) g1x2 = (c) h1x2 =

3x 2 3 + x2 1 - x

x2 (d) p1x2 = 2 sin 1x2 (e) q1x2 =

4x

x 2 + 2x + 1 84. Writing to Learn A continuous function ƒ has domain all real numbers. If ƒ1-12 = 5 and ƒ112 = - 5, explain why ƒ must have at least one zero in the interval 3- 1, 14. (This generalizes to a property of continuous functions known as the Intermediate Value Theorem.)

6965_CH01_pp063-156.qxd

98

1/14/10

1:00 PM

Page 98

CHAPTER 1 Functions and Graphs

85. Proving a Theorem Prove that the graph of every odd function with domain all real numbers must pass through the origin. 86. Finding the Range Graph the function f 1x2 = in the window 3 - 6, 64 by 3 - 2, 24.

3x 2 - 1 2x 2 + 1

(a) What is the apparent horizontal asymptote of the graph? (b) Based on your graph, determine the apparent range of ƒ. (c) Show algebraically that -1 …

3x 2 - 1

6 1.5 for all x,

2x 2 + 1 thus confirming your conjecture in part (b).

87. Looking Ahead to Calculus A key theorem in calculus, the Extreme Value Theorem, states, if a function ƒ is continuous on a closed interval 3a, b4 then ƒ has both a maximum value and a minimum value on the interval. For each of the following functions, verify that the function is continuous on the given interval and find the maximum and minimum values of the function and the x values at which these extrema occur. (a) ƒ1x2 = x 2 - 3, 3- 2, 44 (b) ƒ1x2 =

1 , 31, 54 x

(c) ƒ1x2 = ƒ x + 1 ƒ + 2, 3-4, 14 (d) ƒ1x2 = 2x 2 + 9, 3-4, 44

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 99

SECTION 1.3

Twelve Basic Functions

99

1.3 Twelve Basic Functions What you’ll learn about • What Graphs Can Tell Us • Twelve Basic Functions • Analyzing Functions Graphically

... and why As you continue to study mathematics, you will find that the twelve basic functions presented here will come up again and again. By knowing their basic properties, you will recognize them when you see them.

What Graphs Can Tell Us The preceding section has given us a vocabulary for talking about functions and their properties. We have an entire book ahead of us to study these functions in depth, but in this section we want to set the scene by just looking at the graphs of twelve “basic” functions that are available on your graphing calculator. You will find that function attributes such as domain, range, continuity, asymptotes, extrema, increasingness, decreasingness, and end behavior are every bit as graphical as they are algebraic. Moreover, the visual cues are often much easier to spot than the algebraic ones. In future chapters you will learn more about the algebraic properties that make these functions behave as they do. Only then will you able to prove what is visually apparent in these graphs.

Twelve Basic Functions The Identity Function y 3 2 1 –5 –4 –3 –2 –1 –1 –2 –3

1 2 3 4 5

x

ƒ1x2 = x Interesting fact: This is the only function that acts on every real number by leaving it alone.

FIGURE 1.36

The Squaring Function y 5 4 3 2 1 –5 –4 –3 –2 –1 –1

1 2 3 4 5

x

ƒ1x2 = x 2 Interesting fact: The graph of this function, called a parabola, has a reflection property that is useful in making flashlights and satellite dishes.

FIGURE 1.37

6965_CH01_pp063-156.qxd

100

1/14/10

1:00 PM

Page 100

CHAPTER 1 Functions and Graphs

The Reciprocal Function

The Cubing Function

y

y 3 2 1

3 2 1 –5 –4 –3 –2 –1 –1 –2 –3

x

1 2 3 4 5

ƒ1x2 = x

–5 –4

ƒ1x2 =

3

Interesting fact: The origin is called a “point of inflection” for this curve because the graph changes curvature at that point.

1 x

Interesting fact: This curve, called a hyperbola, also has a reflection property that is useful in satellite dishes.

FIGURE 1.38

FIGURE 1.39

The Exponential Function

The Square Root Function

y

y 5 4 3 2 1

5 4 3 2 1 –1 –1

x

1 2 3 4 5

–1

x

1 2 3 4 5 6 7 8

–4 –3 –2 –1 –1

1 2 3 4

x

ƒ1x2 = ex

ƒ1x2 = 2x

Interesting fact: Put any positive number into your calculator. Take the square root. Then take the square root again. Then take the square root again, and so on. Eventually you will always get 1.

Interesting fact: The number e is an irrational number (like p) that shows up in a variety of applications. The symbols e and p were both brought into popular use by the great Swiss mathematician Leonhard Euler (1707–1783).

FIGURE 1.40

FIGURE 1.41

The Sine Function y

The Natural Logarithm Function y

4 3 2 1

3 2 1 –2 –1 –1 –2 –3

1 2 3 4 5 6

x

ƒ1x2 = ln x Interesting fact: This function increases very slowly. If the x-axis and y-axis were both scaled with unit lengths of one inch, you would have to travel more than two and a half miles along the curve just to get a foot above the x-axis.

FIGURE 1.42

–6 –5 –4 –3

–1 –1 –2 –3 –4

1 2 3

5 6 7

x

ƒ1x2 = sin x Interesting fact: This function and the sinus cavities in your head derive their names from a common root: the Latin word for “bay.” This is due to a 12th-century mistake made by Robert of Chester, who translated a word incorrectly from an Arabic manuscript.

FIGURE 1.43

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 101

SECTION 1.3

Twelve Basic Functions

101

The Absolute Value Function y

The Cosine Function y 5 4 3 2 1

4 3 2 –7 –6 –5

–3

–1 –1 –2 –3 –4

1

3

5 6 7

x

ƒ1x2 = cos x Interesting fact: The local extrema of the cosine function occur exactly at the zeros of the sine function, and vice versa.

FIGURE 1.44

–5 –4 –3 –2 –1 –1 –2 –3

x

ƒ1x2 = ƒ x ƒ = abs 1x2

Interesting fact: This function has an abrupt change of direction (a “corner”) at the origin, while our other functions are all “smooth” on their domains.

FIGURE 1.45 The Logistic Function

The Greatest Integer Function

y

y 5 4 3 2 1 –5 –4 –3 –2 –1

1 2 3 4 5

1

1 2 3 4 5

x

–5 –4 –3 –2 –1

1 2 3 4 5

x

–1 2

–2 –3 –4 –5

ƒ1x2 =

ƒ1x2 = int 1x2

Interesting fact: This function has a jump discontinuity at every integer value of x. Similar-looking functions are called step functions.

FIGURE 1.46

1 1 + e -x

Interesting fact: There are two horizontal asymptotes, the x-axis and the line y = 1. This function provides a model for many applications in biology and business.

FIGURE 1.47

EXAMPLE 1 Looking for Domains (a) Nine of the functions have domain the set of all real numbers. Which three do not? (b) One of the functions has domain the set of all reals except 0. Which function is it, and why isn’t zero in its domain? (c) Which two functions have no negative numbers in their domains? Of these two, which one is defined at zero? SOLUTION

(a) Imagine dragging a vertical line along the x-axis. If the function has domain the set of all real numbers, then the line will always intersect the graph. The intersection might occur off screen, but the TRACE function on the calculator will show the y-coordinate if there is one. Looking at the graphs in Figures 1.39, 1.40, and 1.42, we conjecture that there are vertical lines that do not intersect (continued)

6965_CH01_pp063-156.qxd

102

1/14/10

1:00 PM

Page 102

CHAPTER 1 Functions and Graphs

the curve. A TRACE at the suspected x-coordinates confirms our conjecture (Figure 1.48). The functions are y = 1/x, y = 2x, and y = ln x. (b) The function y = 1/x, with a vertical asymptote at x = 0, is defined for all real numbers except 0. This is explained algebraically by the fact that division by zero is undefined. (c) The functions y = 2x and y = ln x have no negative numbers in their domains. (We already knew that about the square root function.) While 0 is in the domain of y = 2x, we can see by tracing that it is not in the domain of y = ln x. We will see the algebraic reason for this in Chapter 3. Now try Exercise 13.

[–3.7, 5.7] by [–3.1, 3.1] (a)

1 EXAMPLE 2 Looking for Continuity Only two of twelve functions have points of discontinuity. Are these points in the domain of the function?

X=–2

SOLUTION All of the functions have continuous, unbroken graphs except for

Y=

y = 1/x, and y = int 1x2.

[–3.7, 5.7] by [–3.1, 3.1] (b)

1

X=0

Y=

The graph of y = 1/x clearly has an infinite discontinuity at x = 0 (Figure 1.39). We saw in Example 1 that 0 is not in the domain of the function. Since y = 1/x is continuous for every point in its domain, it is called a continuous function. The graph of y = int 1x2 has a discontinuity at every integer value of x (Figure 1.46). Since this function has discontinuities at points in its domain, it is not a continuous Now try Exercise 15. function.

EXAMPLE 3 Looking for Boundedness Only three of the twelve basic functions are bounded (above and below). Which three?

[–4.7, 4.7] by [–3.1, 3.1] (c)

FIGURE 1.48 (a) A vertical line through

- 2 on the x-axis appears to miss the graph of y = ln x. (b) A TRACE confirms that -2 is not in the domain. (c) A TRACE at x = 0 confirms that 0 is not in the domain of y = 1/x. (Example 1)

SOLUTION A function that is bounded must have a graph that lies entirely be-

tween two horizontal lines. The sine, cosine, and logistic functions have this property (Figure 1.49). It looks like the graph of y = 2x might also have this property, but we know that the end behavior of the square root function is unbounded: lim 2x = q , so it is really only bounded below. You will learn in Chapter 4 why x: q the sine and cosine functions are bounded. Now try Exercise 17.

[–2π , 2π ] by [–4, 4]

[–2π , 2π ] by [–4, 4]

[–4.7, 4.7] by [–0.5, 1.5]

(a)

(b)

(c)

FIGURE 1.49 The graphs of y = sin x, y = cos x, and y = 1/11 + e -x2 lie entirely between two horizontal lines and are therefore graphs of bounded functions. (Example 3)

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 103

SECTION 1.3

Twelve Basic Functions

103

EXAMPLE 4 Looking for Symmetry Three of the twelve basic functions are even. Which are they? SOLUTION Recall that the graph of an even function is symmetric with respect to

the y-axis. Three of the functions exhibit the required symmetry: y = x 2, y = cos x, Now try Exercise 19. and y = ƒ x ƒ (Figure 1.50).

[–2π , 2π ] by [–4, 4]

[–4.7, 4.7] by [–1.1, 5.1] (a)

(b)

[–4.7, 4.7] by [–1.1, 5.1] (c)

FIGURE 1.50 The graphs of y = x 2, y = cos x, and y = ƒ x ƒ are symmetric with respect to the y-axis, indicating that the functions are even. (Example 4)

Analyzing Functions Graphically We could continue to explore the twelve basic functions as in the first four examples, but we also want to make the point that there is no need to restrict ourselves to the basic twelve. We can alter the basic functions slightly and see what happens to their graphs, thereby gaining further visual insights into how functions behave.

EXAMPLE 5 Analyzing a Function Graphically Graph the function y = 1x - 222. Then answer the following questions: (a) (b) (c) (d)

On what interval is the function increasing? On what interval is it decreasing? Is the function odd, even, or neither? Does the function have any extrema? How does the graph relate to the graph of the basic function y = x 2?

SOLUTION The graph is shown in Figure 1.51.

[–4.7, 4.7] by [–1.1, 5.1]

FIGURE 1.51 The graph of y = 1x - 222. (Example 5)

(continued)

6965_CH01_pp063-156.qxd

104

1/14/10

1:00 PM

Page 104

CHAPTER 1 Functions and Graphs

(a) The function is increasing if its graph is headed upward as it moves from left to right. We see that it is increasing on the interval 32, q 2. The function is decreasing if its graph is headed downward as it moves from left to right. We see that it is decreasing on the interval 1 - q , 24. (b) The graph is not symmetric with respect to the y-axis, nor is it symmetric with respect to the origin. The function is neither. (c) Yes, we see that the function has a minimum value of 0 at x = 2. (This is easily confirmed by the algebraic fact that 1x - 222 Ú 0 for all x.2 (d) We see that the graph of y = 1x - 222 is just the graph of y = x 2 moved two Now try Exercise 35. units to the right. EXPLORATION 1

Looking for Asymptotes

1. Two of the basic functions have vertical asymptotes at x = 0. Which two? 2. Form a new function by adding these functions together. Does the new func-

tion have a vertical asymptote at x = 0? 3. Three of the basic functions have horizontal asymptotes at y = 0. Which

three?

4. Form a new function by adding these functions together. Does the new func-

tion have a horizontal asymptote at y = 0? 5. Graph ƒ1x2 = 1/x, g1x2 = 1/12x 2 - x2, and h1x2 = ƒ1x2 + g1x2. Does h1x2

have a vertical asymptote at x = 0?

EXAMPLE 6 Identifying a Piecewise-Defined Function Which of the twelve basic functions has the following piecewise definition over separate intervals of its domain? ƒ1x2 = e

x -x

if x Ú 0 if x 6 0

SOLUTION You may recognize this as the definition of the absolute value

function (Chapter P). Or, you can reason that the graph of this function must look just like the line y = x to the right of the y-axis, but just like the graph of the line y = - x to the left of the y-axis. That is a perfect description of the absolute value graph in Figure 1.45. Either way, we recognize this as a piecewise definition of ƒ1x2 = ƒ x ƒ . Now try Exercise 45. y

EXAMPLE 7 Defining a Function Piecewise 4 3 2 1 –5 –4 –3 –2 –1 –1 –2

Using basic functions from this section, construct a piecewise definition for the function whose graph is shown in Figure 1.52. Is your function continuous? 1 2 3 4 5

x

FIGURE 1.52 A piecewise-defined function. (Example 7)

SOLUTION This appears to be the graph of y = x 2 to the left of x = 0 and the

graph of y = 2x to the right of x = 0. We can therefore define it piecewise as The function is continuous.

ƒ1x2 = e

x2 2x

if x … 0 if x 7 0.

Now try Exercise 47.

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 105

SECTION 1.3

Twelve Basic Functions

105

You can go a long way toward understanding a function’s behavior by looking at its graph. We will continue that theme in the exercises and then revisit it throughout the book. However, you can’t go all the way toward understanding a function by looking at its graph, as Example 8 shows.

EXAMPLE 8 Looking for a Horizontal Asymptote Does the graph of y = ln x (see Figure 1.42) have a horizontal asymptote? [–600, 5000] by [–5, 12]

FIGURE 1.53 The graph of y = ln x still

appears to have a horizontal asymptote, despite the much larger window than in Figure 1.42. (Example 8)

SOLUTION In Figure 1.42 it certainly looks like there is a horizontal asymptote that the graph is approaching from below. If we choose a much larger window (Figure 1.53), it still looks that way. In fact, we could zoom out on this function all day long and it would always look like it is approaching some horizontal asymptote— but it is not. We will show algebraically in Chapter 3 that the end behavior of this function is lim ln x = q , so its graph must eventually rise above the level of x: q any horizontal line. That rules out any horizontal asymptote, even though there is no visual evidence of that fact that we can see by looking at its graph. Now try Exercise 55.

EXAMPLE 9 Analyzing a Function Give a complete analysis of the basic function ƒ1x2 = ƒ x ƒ . SOLUTION

BASIC FUNCTION The Absolute Value Function ƒ1x2 = ƒ x ƒ Domain: All reals Range: 30, q 2 Continuous

Decreasing on 1- q , 04; increasing on 30, q 2

Symmetric with respect to the y-axis (an even function)

[–6, 6] by [–1, 7]

Bounded below

FIGURE 1.54 The graph of ƒ1x2 = ƒ x ƒ .

Local minimum at 10, 02

No horizontal asymptotes No vertical asymptotes End behavior:

lim ƒ x ƒ = q and lim ƒ x ƒ = q q

x: - q

x:

Now try Exercise 67.

QUICK REVIEW 1.3

(For help, go to Sections P.1, P.2, 3.1, and 3.3.)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 1–10, evaluate the expression without using a calculator. 1. ƒ - 59.34 ƒ

2. ƒ 5 - p ƒ

3. ƒ p - 7 ƒ 5. ln 112

6. e0

3 323 7. 12 3 -8 9. 2

4. 21 -322

2

3 1- 1523 8. 2

10. ƒ 1 - p ƒ - p

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 106

CHAPTER 1 Functions and Graphs

106

SECTION 1.3 EXERCISES In Exercises 1–12, each graph is a slight variation on the graph of one of the twelve basic functions described in this section. Match the graph to one of the twelve functions (a)–(l) and then support your answer by checking the graph on your calculator. (All graphs are shown in the window 3 - 4.7, 4.74 by 3 - 3.1, 3.14.)

1.

(a) y = -sin x

(b) y = cos x + 1

(c) y = ex - 2

(d) y = 1x + 223

(e) y = x 3 + 1

(g) y = ƒ x ƒ - 2

(h) y = - 1/x

(f) y = 1x - 122

(j) y = - 2x

(k) y = int1x + 12

(l) y = 2 - 4/11 + e -x2

(i) y = - x

2.

15. The two functions that have at least one point of discontinuity 16. The function that is not a continuous function 17. The six functions that are bounded below 18. The four functions that are bounded above In Exercises 19–28, identify which of the twelve basic functions fit the description given. 19. The four functions that are odd 20. The six functions that are increasing on their entire domains 21. The three functions that are decreasing on the interval 1- q , 02

22. The three functions with infinitely many local extrema 23. The three functions with no zeros 24. The three functions with range 5all real numbers6 25. The four functions that do not have end behavior lim ƒ1x2 = + q x: + q

3.

4.

26. The three functions with end behavior

lim ƒ1x2 = - q

x: -q

27. The four functions whose graphs look the same when turned upside-down and flipped about the y-axis 28. The two functions whose graphs are identical except for a horizontal shift

5.

7.

6.

8.

In Exercises 29–34, use your graphing calculator to produce a graph of the function. Then determine the domain and range of the function by looking at its graph. 29. ƒ1x2 = x 2 - 5

30. g1x2 = ƒ x - 4 ƒ

31. h1x2 = ln 1x + 62

32. k1x2 = 1/x + 3

33. s1x2 = int 1x/22

34. p1x2 = 1x + 322

In Exercises 35–42, graph the function. Then answer the following questions: (a) On what interval, if any, is the function increasing? Decreasing? (b) Is the function odd, even, or neither?

9.

10.

(c) Give the function’s extrema, if any. (d) How does the graph relate to a graph of one of the twelve basic functions? 35. r1x2 = 2x - 10

-x

11.

12.

37. ƒ1x2 = 3/11 + e 2 39. h1x2 = ƒ x ƒ - 10 41. s1x2 = ƒ x - 2|

36. ƒ1x2 = sin 1x2 + 5 38. q1x2 = ex + 2

40. g1x2 = 4 cos 1x2

42. ƒ1x2 = 5 - abs 1x2

43. Find the horizontal asymptotes for the graph shown in Exercise 11. 44. Find the horizontal asymptotes for the graph of ƒ1x2 in Exercise 37. In Exercises 13–18, identify which of Exercises 1–12 display functions that fit the description given. 13. The function whose domain excludes zero 14. The function whose domain consists of all nonnegative real numbers

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 107

SECTION 1.3

In Exercises 45–52, sketch the graph of the piecewise-defined function. 1Try doing it without a calculator.) In each case, give any points of discontinuity. x 45. ƒ1x2 = e 2 x 46. g1x2 = e

47. h1x2 = e

x3 ex

1/x 2x

if x 6 0 if x Ú 0

48. w1x2 = e 49. ƒ1x2 = e

50. g1x2 = e

if x … 0 if x 7 0 if x 6 0 if x Ú 0

if x … 0 if x 7 0

cos x ex

ƒxƒ x2

if x 6 0 if x Ú 0

-3 - x 51. ƒ1x2 = c 1 x2

if x … 0 if 0 6 x 6 1 if x Ú 1

x2 52. ƒ1x2 = c ƒ x ƒ int 1x2

if x 6 - 1 if - 1 … x 6 1 if x Ú 1

53. Writing to Learn The function ƒ1x2 = 2x 2 is one of our twelve basic functions written in another form. (a) Graph the function and identify which basic function it is. (b) Explain algebraically why the two functions are equal. 54. Uncovering Hidden Behavior The function g1x2 = 2x 2 + 0.0001 - 0.01 is not one of our twelve basic functions written in another form. (a) Graph the function and identify which basic function it appears to be. (b) Verify numerically that it is not the basic function that it appears to be. 55. Writing to Learn The function ƒ1x2 = ln 1ex2 is one of our twelve basic functions written in another form. (a) Graph the function and identify which basic function it is.

(b) Explain how the equivalence of the two functions in (a) shows that the natural logarithm function is not bounded above (even though it appears to be bounded above in Figure 1.42). 56. Writing to Learn Let ƒ1x2 be the function that gives the cost, in cents, to mail a first-class package that weighs x ounces. In August of 2009, the cost was $1.22 for a package that weighed up to 1 ounce, plus 17 cents for each additional ounce or portion thereof (up to 13 ounces). (Source: United States Postal Service.) (a) Sketch a graph of ƒ1x2. (b) How is this function similar to the greatest integer function? How is it different?

107

Packages

if x … 0 if x 7 0

ƒxƒ sin x

Twelve Basic Functions

Weight Not Over

Price

1 ounce 2 ounces 3 ounces 4 ounces 5 ounces 6 ounces 7 ounces 8 ounces 9 ounces 10 ounces 11 ounces 12 ounces 13 ounces

$1.22 $1.39 $1.56 $1.73 $1.90 $2.07 $2.24 $2.41 $2.58 $2.75 $2.92 $3.09 $3.26

57. Analyzing a Function Set your calculator to DOT mode and graph the greatest integer function, y = int 1x2, in the window 3-4.7, 4.74 by 3-3.1, 3.14. Then complete the following analysis.

BASIC FUNCTION The Greatest Integer Function ƒ1x2 = int x Domain: Range: Continuity: Increasing/decreasing behavior: Symmetry: Boundedness: Local extrema: Horizontal asymptotes: Vertical asymptotes: End behavior:

Standardized Test Questions 58. True or False The greatest integer function has an inverse function. Justify your answer. 59. True or False The logistic function has two horizontal asymptotes. Justify your answer. In Exercises 60–63, you may use a graphing calculator to answer the question. 60. Multiple Choice Which function has range {all real numbers}? (A) ƒ1x2 = 4 + ln x (B) ƒ1x2 = 3 - 1/x (C) ƒ1x2 = 5/11 + e -x2 (D) ƒ1x2 = int 1x - 22 (E) ƒ1x2 = 4 cos x

6965_CH01_pp063-156.qxd

108

1/14/10

1:00 PM

Page 108

CHAPTER 1 Functions and Graphs

61. Multiple Choice Which function is bounded both above and below? (A) ƒ1x2 = x 2 - 4 (B) ƒ1x2 = 1x - 323 (C) ƒ1x2 = 3ex

(D) ƒ1x2 = 3 + 1/11 + e -x2 (E) ƒ1x2 = 4 - ƒ x ƒ

62. Multiple Choice Which of the following is the same as the restricted-domain function ƒ1x2 = int 1x2, 0 … x 6 2? 0 (A) ƒ1x2 = c 1 2

if 0 … x 6 1 if x = 1 if 1 6 x 6 2

0 (B) ƒ1x2 = c 1 2

if x = 0 if 0 6 x … 1 if 1 6 x 6 2

0 1

if 0 … x 6 1 if 1 … x 6 2

1 (D) ƒ1x2 = e 2

if 0 … x 6 1 if 1 … x 6 2

(C) ƒ1x2 = e

x (E) ƒ1x2 = e 1 + x

if 0 … x 6 1 if 1 … x 6 2

63. Multiple Choice Increasing Functions Which function is increasing on the interval 1 - q , q 2?

66. Group Activity Assign to each student in the class the name of one of the twelve basic functions, but secretly so that no student knows the “name” of another. (The same function name could be given to several students, but all the functions should be used at least once.) Let each student make a onesentence self-introduction to the class that reveals something personal “about who I am that really identifies me.” The rest of the students then write down their guess as to the function’s identity. Hints should be subtle and cleverly anthropomorphic. (For example, the absolute value function saying “I have a very sharp smile” is subtle and clever, while “I am absolutely valuable” is not very subtle at all.) 67. Pepperoni Pizzas For a statistics project, a student counted the number of pepperoni slices on pizzas of various sizes at a local pizzeria, compiling the following table:

Table 1.10 Type of Pizza Personal Medium Large Extra large

Radius

Pepperoni Count

4– 6– 7– 8–

12 27 37 48

(C) ƒ1x2 = 2x 2

(a) Explain why the pepperoni count 1P2 ought to be proportional to the square of the radius 1r2.

(D) ƒ1x2 = sin x

(c) Does the algebraic model fit the rest of the data well?

(A) ƒ1x2 = 23 + x (B) ƒ1x2 = int 1x2

(E) ƒ1x2 = 3/11 + e -x2

Explorations

64. Which Is Bigger? For positive values of x, we wish to compare the values of the basic functions x 2, x, and 2x. (a) How would you order them from least to greatest?

(b) Graph the three functions in the viewing window 30, 304 by 30, 204. Does the graph confirm your response in (a)? (c) Now graph the three functions in the viewing window 30, 24 by 30, 1.54.

(d) Write a careful response to the question in (a) that accounts for all positive values of x.

(b) Assuming that P = k # r 2, use the data pair 14, 122 to find the value of k. (d) Some pizza places have charts showing their kitchen staff how much of each topping should be put on each size of pizza. Do you think this pizzeria uses such a chart? Explain.

Extending the Ideas 68. Inverse Functions Two functions are said to be inverses of each other if the graph of one can be obtained from the graph of the other by reflecting it across the line y = x. For example, the functions with the graphs shown below are inverses of each other:

65. Odds and Evens There are four odd functions and three even functions in the gallery of twelve basic functions. After multiplying these functions together pairwise in different combinations and exploring the graphs of the products, make a conjecture about the symmetry of: (a) a product of two odd functions; (b) a product of two even functions; (c) a product of an odd function and an even function.

[–4.7, 4.7] by [–3.1, 3.1] (a)

[–4.7, 4.7] by [–3.1, 3.1] (b)

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 109

SECTION 1.3

(a) Two of the twelve basic functions in this section are inverses of each other. Which are they? (b) Two of the twelve basic functions in this section are their own inverses. Which are they? (c) If you restrict the domain of one of the twelve basic functions to 30, q 2, it becomes the inverse of another one. Which are they?

69. Identifying a Function by Its Properties

(a) Seven of the twelve basic functions have the property that ƒ102 = 0. Which five do not?

Twelve Basic Functions

109

(b) Only one of the twelve basic functions has the property that ƒ1x + y2 = ƒ1x2 + ƒ1y2 for all x and y in its domain. Which one is it? (c) One of the twelve basic functions has the property that ƒ1x + y2 = ƒ1x2ƒ1y2 for all x and y in its domain. Which one is it? (d) One of the twelve basic functions has the property that ƒ1xy2 = ƒ1x2 + ƒ1y2 for all x and y in its domain. Which one is it? (e) Four of the twelve basic functions have the property that ƒ1x2 + ƒ1- x2 = 0 for all x in their domains. Which four are they?

6965_CH01_pp063-156.qxd

110

1/14/10

1:00 PM

Page 110

CHAPTER 1 Functions and Graphs

1.4 Building Functions from Functions What you’ll learn about • Combining Functions Algebraically • Composition of Functions • Relations and Implicitly Defined Functions

Combining Functions Algebraically Knowing how a function is “put together” is an important first step when applying the tools of calculus. Functions have their own algebra based on the same operations we apply to real numbers (addition, subtraction, multiplication, and division). One way to build new functions is to apply these operations, using the following definitions.

... and why Most of the functions that you will encounter in calculus and in real life can be created by combining or modifying other functions.

DEFINITION Sum, Difference, Product, and Quotient of Functions Let ƒ and g be two functions with intersecting domains. Then for all values of x in the intersection, the algebraic combinations of ƒ and g are defined by the following rules: Sum: Difference: Product: Quotient:

1ƒ + g21x2 1ƒ - g21x2 1ƒg21x2 ƒ a b 1x2 g

= ƒ1x2 + g1x2 = ƒ1x2 - g1x2 = ƒ1x2g1x2 ƒ1x2 = , provided g1x2 Z 0 g1x2

In each case, the domain of the new function consists of all numbers that belong to both the domain of ƒ and the domain of g. As noted, the zeros of the denominator are excluded from the domain of the quotient.

Euler’s function notation works so well in the above definitions that it almost obscures what is really going on. The “ + ” in the expression “1ƒ + g21x2” stands for a brand new operation called function addition. It builds a new function, ƒ + g, from the given functions ƒ and g. Like any function, ƒ + g is defined by what it does: It takes a domain value x and returns a range value ƒ1x2 + g1x2. Note that the “ + ” sign in “ƒ1x2 + g1x2” does stand for the familiar operation of real number addition. So, with the same symbol taking on different roles on either side of the equal sign, there is more to the above definitions than first meets the eye. Fortunately, the definitions are easy to apply.

EXAMPLE 1 Defining New Functions Algebraically Let ƒ1x2 = x 2 and g1x2 = 2x + 1. Find formulas for the functions ƒ + g, ƒ - g, ƒg, ƒ/g, and gg. Give the domain of each. (continued)

6965_CH01_pp063-156.qxd

1/14/10

1:00 PM

Page 111

SECTION 1.4

Building Functions from Functions

111

SOLUTION We first determine that ƒ has domain all real numbers and that g has domain 3-1, q 2. These domains overlap, the intersection being the interval 3-1, q 2. So:

1ƒ + g21x2 = ƒ1x2 + g1x2 = x 2 + 2x + 1 with domain 3-1, q 2. 2 1ƒ - g21x2 = ƒ1x2 -g1x2 = x - 2x + 1 with domain 3-1, q 2. 2 with domain 1ƒg21x2 = ƒ1x2g1x2 = x 2x + 1 3-1, q 2. 2 ƒ1x2 ƒ x with domain a b 1x2 = = g g1x2 2x + 1 1 -1, q 2. 1gg21x2 = g1x2g1x2 = 1 2x + 122

with domain 3-1, q 2.

Note that we could express 1gg21x2 more simply as x + 1. That would be fine, but the simplification would not change the fact that the domain of gg is (by definition) the interval 3-1, q 2. Under other circumstances the function h1x2 = x + 1 would have domain all real numbers, but under these circumstances it cannot; it is a product of two functions with restricted domains. Now try Exercise 3.

Composition of Functions It is not hard to see that the function sin 1x 22 is built from the basic functions sin x and x 2, but the functions are not put together by addition, subtraction, multiplication, or division. Instead, the two functions are combined by simply applying them in order—first the squaring function, then the sine function. This operation for combining functions, which has no counterpart in the algebra of real numbers, is called function composition.

DEFINITION Composition of Functions Let ƒ and g be two functions such that the domain of ƒ intersects the range of g. The composition ƒ of g, denoted ƒ ! g, is defined by the rule 1ƒ ! g21x) = ƒ1g1x22.

The domain of ƒ ! g consists of all x-values in the domain of g that map to g1x2-values in the domain of ƒ. (See Figure 1.55.) The composition g of ƒ, denoted g ! ƒ, is defined similarly. In most cases g ! ƒ and ƒ ! g are different functions. (In the language of algebra, “function composition is not commutative.”) f!g

g(x) f(g(x)) x

x must be in the domain of g

f

g

and

g(x) must be in the domain of f

FIGURE 1.55 In the composition ƒ ! g, the function g is applied first and then ƒ. This is the reverse of the order in which we read the symbols.

6965_CH01_pp063-156.qxd

112

1/14/10

1:00 PM

Page 112

CHAPTER 1 Functions and Graphs

EXAMPLE 2 Composing Functions Let ƒ1x2 = ex and g1x2 = 2x. Find 1ƒ ! g21x2 and 1g ! ƒ21x2 and verify numerically that the functions ƒ ! g and g ! ƒ are not the same. SOLUTION

1ƒ ! g21x2 = ƒ1g1x22 = ƒ11x2 = e 1x 1g ! ƒ21x2 = g1ƒ1x22 = g1ex2 = 2e x

One verification that these functions are not the same is that they have different domains: ƒ ! g is defined only for x Ú 0, while g ! ƒ is defined for all real numbers. We could also consider their graphs (Figure 1.56), which agree only at x = 0 and x = 4.

[–2, 6] by [–1, 15]

FIGURE 1.56 The graphs of y = e 1x and y = 2ex are not the same. (Example 2)

Finally, the graphs suggest a numerical verification: Find a single value of x for which ƒ1g1x22 and g1ƒ1x22 give different values. For example, ƒ1g1122 = e and g1ƒ1122 = 2e. The graph helps us to make a judicious choice of x. You do not want to check the functions at x = 0 and x = 4 and conclude that they are the same! Now try Exercise 15. EXPLORATION 1

Composition Calisthenics

One of the ƒ functions in column B can be composed with one of the g functions in column C to yield each of the basic ƒ ! g functions in column A. Can you match the columns successfully without a graphing calculator? If you are having trouble, try it with a graphing calculator. A

B

C

ƒ ! g

ƒ

g

x

x - 3

x 0.6

x2

2x - 3

x2

ƒxƒ

2x

1x - 221x + 22

ln x

ƒ 2x + 4 ƒ

x 2

sin x

1- 2x 2

cos x

2 sin x cos x

x3

x5

2

ln 1e3 x2 x + 3 2 x sin a b 2

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 113

SECTION 1.4

Building Functions from Functions

113

EXAMPLE 3 Finding the Domain of a Composition Let ƒ1x2 = x 2 - 1 and let g1x2 = 2x. Find the domains of the composite functions (a) g ! ƒ

(b) ƒ ! g.

SOLUTION

(a) We compose the functions in the order specified: 1 g ! ƒ21x2 = g1ƒ1x22

= 2x 2 - 1

Caution We might choose to express 1ƒ ! g2 more simply as x - 1. However, you must remember that the composition is restricted to the domain of g1x2 = 2x, or 30, q 4. The domain of x - 1 is all real numbers. It is a good idea to work out the domain of a composition before you simplify the expression for ƒ(g(x22. One way to simplify and maintain the restriction on the domain in Example 3 is to write 1ƒ ! g21x2 = x - 1, x Ú 0.

For x to be in the domain of g ! ƒ, we must first find ƒ1x2 = x 2 - 1, which we can do for all real x. Then we must take the square root of the result, which we can only do for nonnegative values of x 2 - 1. Therefore, the domain of g ! ƒ consists of all real numbers for which x 2 - 1 Ú 0, namely the union 1- q , -14 ´ 31, q 2.

(b) Again, we compose the functions in the order specified: 1 ƒ ! g21x2 = ƒ1g1x22

= 12x22 - 1

For x to be in the domain of ƒ ! g, we must first be able to find g1x2 = 2x, which we can only do for nonnegative values of x. Then we must be able to square the result and subtract 1, which we can do for all real numbers. Therefore, the domain of ƒ ! g consists of the interval 30, q 2.

Support Graphically

We can graph the composition functions to see if the grapher respects the domain restrictions. The screen to the left of each graph shows the setup in the “Y = ” editor. Figure 1.57b shows the graph of y = 1g ! ƒ21x2, while Figure 1.57d shows the graph of y = 1ƒ ! g21x2. The graphs support our algebraic work quite nicely. Now try Exercise 17. Plot1

Plot2

\Y1= (X) \Y2=X2–1 \Y3=Y1(Y2(X)) \Y4= \Y5= \Y6= \Y7=

(a)

Plot1

Plot3

Plot2

\Y1= (X) \Y2=X2–1 \Y3=Y2(Y1(X)) \Y4= \Y5= \Y6= \Y7= [–4.7, 4.7] by [–3.1, 3.1] (b)

Plot3

(c)

[–4.7, 4.7] by [–3.1, 3.1] (d)

FIGURE 1.57 The functions Y1 and Y2 are composed to get the graphs of y = 1g ! ƒ)1x2 and y = 1ƒ ! g21x2, respectively. The graphs support our conclusions about the domains of the two composite functions. (Example 3) In Examples 2 and 3 two functions were composed to form new functions. There are times in calculus when we need to reverse the process. That is, we may begin with a function h and decompose it by finding functions whose composition is h.

EXAMPLE 4 Decomposing Functions For each function h, find functions ƒ and g such that h1x2 = ƒ1g1x22. (a) h1x2 = 1x + 122 - 31x + 12 + 4 (b) h1x2 = 2x 3 + 1

(continued)

6965_CH01_pp063-156.qxd

114

1/14/10

1:01 PM

Page 114

CHAPTER 1 Functions and Graphs

SOLUTION

(a) We can see that h is quadratic in x + 1. Let ƒ1x2 = x 2 - 3x + 4 and let g1x2 = x + 1. Then h1x2 = ƒ1g1x22 = ƒ1x + 12 = 1x + 122 - 31x + 12 + 4.

(b) We can see that h is the square root of the function x 3 + 1. Let ƒ1x2 = 2x and let g1x2 = x 3 + 1. Then h1x2 = ƒ1g1x22 = ƒ1x 3 + 12 = 2x 3 + 1.

Now try Exercise 25.

There is often more than one way to decompose a function. For example, an alternate way to decompose h1x2 = 2x 3 + 1 in Example 4b is to let ƒ1x2 = 2x + 1 and let g1x2 = x 3. Then h1x2 = ƒ1g1x22 = ƒ1x 32 = 2x 3 + 1.

EXAMPLE 5 Modeling with Function Composition In the medical procedure known as angioplasty, doctors insert a catheter into a heart vein (through a large peripheral vein) and inflate a small, spherical balloon on the tip of the catheter. Suppose the balloon is inflated at a constant rate of 44 cubic millimeters per second (Figure 1.58). (a) Find the volume after t seconds. (b) When the volume is V, what is the radius r? (c) Write an equation that gives the radius r as a function of the time. What is the radius after 5 seconds?

FIGURE 1.58 (Example 5)

SOLUTION

(a) After t seconds, the volume will be 44t. (b) Solve Algebraically 4 3 pr = v 3 3v 4p 3v r = 3 A 4p

r3 =

3 # 44t 33t . After 5 seconds, the (c) Substituting 44t for V gives r = 3 or r = 3 B p B 4p 33 # 5 Now try Exercise 31. L 3.74 mm. radius will be r = 3 B p

y 3 1 –5 –4 –3

–1 –1

1

3 4 5

x

–3

FIGURE 1.59 A circle of radius 2 centered at the origin. This set of ordered pairs 1x, y2 defines a relation that is not a function, because the graph fails the vertical line test.

Relations and Implicitly Defined Functions There are many useful curves in mathematics that fail the vertical line test and therefore are not graphs of functions. One such curve is the circle in Figure 1.59. While y is not related to x as a function in this instance, there is certainly some sort of relationship going on. In fact, not only does the shape of the graph show a significant geometric relationship among the points, but the ordered pairs 1x, y2 exhibit a significant algebraic relationship as well: They consist exactly of the solutions to the equation x 2 + y 2 = 4.

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 115

SECTION 1.4

Graphing Relations Relations that are not functions are often not easy to graph. We will study some special cases later in the course (circles, ellipses, etc.), but some simple-looking relations like those in Example 6 are difficult to graph. Nor do our calculators help much, because the equation cannot be put into “Y1 = ” form. Interestingly, we do know that the graph of the relation in Example 6, whatever it looks like, fails the vertical line test.

Building Functions from Functions

115

The general term for a set of ordered pairs 1x, y2 is a relation. If the relation happens to relate a single value of y to each value of x, then the relation is also a function and its graph will pass the vertical line test. In the case of the circle with equation x 2 + y 2 = 4, both 10, 22 and 10, -22 are in the relation, so y is not a function of x.

EXAMPLE 6 Verifying Pairs in a Relation Determine which of the ordered pairs 12, -52, 11, 32, and 12, 12 are in the relation defined by x 2y + y 2 = 5. Is the relation a function? SOLUTION We simply substitute the x- and y-coordinates of the ordered pairs into

x 2y + y 2 and see if we get 5.

12221-52 + 1- 522 = 5 1122132 + 1322 = 12 Z 5 1222112 + 1122 = 5

12, -52:

11, 32: 12, 12:

Substitute x = 2, y = -5. Substitute x = 1, y = 3. Substitute x = 2, y = 1.

So, 12, -52 and 12, 12 are in the relation, but 11, 32 is not. Since the equation relates two different y-values 1 -5 and 12 to the same x-value 122, the relation cannot be a function. Now try Exercise 35. Let us revisit the circle x 2 + y 2 = 4. While it is not a function itself, we can split it into two equations that do define functions, as follows: x 2 + y2 = 4 y2 = 4 - x 2 y = + 24 - x 2 or y = - 24 - x 2

The graphs of these two functions are, respectively, the upper and lower semicircles of the circle in Figure 1.59. They are shown in Figure 1.60. Since all the ordered pairs in either of these functions satisfy the equation x 2 + y 2 = 4, we say that the relation given by the equation defines the two functions implicitly.

y

y

3

3 2 1

1 –5 –4 –3 –2 –1 –1 –2 –3 (a)

1 2 3 4 5

x

–5 –4 –3

–1 –1

1

3 4 5

x

–3 (b)

FIGURE 1.60 The graphs of (a) y = + 24 - x 2 and (b) y = - 24 - x 2. In each case, y is defined as a function of x. These two functions are defined implicitly by the relation x 2 + y 2 = 4.

6965_CH01_pp063-156.qxd

116

1/14/10

1:01 PM

Page 116

CHAPTER 1 Functions and Graphs

EXAMPLE 7 Using Implicitly Defined Functions Describe the graph of the relation x 2 + 2xy + y 2 = 1. SOLUTION This looks like a difficult task at first, but notice that the expression

on the left of the equal sign is a factorable trinomial. This enables us to split the relation into two implicitly defined functions as follows: x 2 + 2xy + y 2 = 1 1x + y22 = 1

Factor.

x + y = "1 x + y = 1 or x + y = - 1 y = - x + 1 or y = - x - 1

Extract square roots. Solve for y.

The graph consists of two parallel lines (Figure 1.61), each the graph of one of the implicitly defined functions. Now try Exercise 37. y 4 3 2 1 –5 –4 –3 –2 –1 –1 –2 –3 –4

1 2 3 4 5

x

FIGURE 1.61 The graph of the relation x 2 + 2xy + y 2 = 1. (Example 7)

QUICK REVIEW 1.4

(For help, go to Sections P.1, 1.2, and 1.3.)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 1–10, find the domain of the function and express it in interval notation. 1. ƒ1x2 =

x - 2 x + 3

2. g1x2 = ln1x - 12 3. ƒ1t2 = 25 - t 4. g1x2 =

3

5. ƒ1x2 = 2ln1x2

6. h1x2 = 21 - x 2 7. ƒ1t2 =

t + 5

t2 + 1 8. g1t2 = ln1 ƒ t ƒ 2 9. ƒ1x2 =

1

21 - x 2 10. g1x2 = 2

22x - 1

SECTION 1.4 EXERCISES In Exercises 1–4, find formulas for the functions ƒ + g, ƒ - g, and ƒg. Give the domain of each. 1. ƒ1x2 = 2x - 1; g1x2 = x 2 2. ƒ1x2 = 1x - 122; g1x2 = 3 - x 3. ƒ1x2 = 2x; g1x2 = sin x

4. ƒ1x2 = 2x + 5; g1x2 = ƒ x + 3 ƒ

In Exercises 5–8, find formulas for ƒ/g and g/ƒ. Give the domain of each. 5. ƒ1x2 = 2x + 3; g1x2 = x 2

6. ƒ1x2 = 2x - 2; g1x2 = 2x + 4 7. ƒ1x2 = x 2; g1x2 = 21 - x 2 3 1 - x3 8. ƒ1x2 = x 3; g1x2 = 2

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 117

SECTION 1.4

9. ƒ1x2 = x 2 and g1x2 = 1/x are shown below in the viewing window 30, 54 by 30, 54. Sketch the graph of the sum 1ƒ + g21x2 by adding the y-coordinates directly from the graphs. Then graph the sum on your calculator and see how close you came.

Building Functions from Functions

In Exercises 23–30, find ƒ1x2 and g1x2 so that the function can be described as y = ƒ1g1x22. (There may be more than one possible decomposition.) 23. y = 2x 2 - 5x 25. y = ƒ 3x - 2 ƒ

27. y = 1x - 325 + 2 29. y = cos1 2x2

[0, 5] by [0, 5]

10. The graphs of ƒ1x2 = x 2 and g1x2 = 4 - 3x are shown in the viewing window 3 - 5, 54 by 3 - 10, 254. Sketch the graph of the difference 1ƒ - g21x2 by subtracting the y-coordinates directly from the graphs. Then graph the difference on your calculator and see how close you came.

117

24. y = 1x 3 + 122 26. y =

1

3

x - 5x + 3 28. y = esin x 30. y = 1tan x22 + 1

31. Weather Balloons A high-altitude spherical weather balloon expands as it rises due to the drop in atmospheric pressure. Suppose that the radius r increases at the rate of 0.03 inch per second and that r = 48 inches at time t = 0. Determine an equation that models the volume V of the balloon at time t and find the volume when t = 300 seconds.

32. A Snowball’s Chance Jake stores a small cache of 4-inch-diameter snowballs in the basement freezer, unaware that the freezer’s self-defrosting feature will cause each snowball to lose about 1 cubic inch of volume every 40 days. He remembers them a year later (call it 360 days) and goes to retrieve them. What is their diameter then? 33. Satellite Photography A satellite camera takes a rectangle-shaped picture. The smallest region that can be photographed is a 5-km by 7-km rectangle. As the camera zooms out, the length l and width w of the rectangle increase at a rate of 2 km/sec. How long does it take for the area A to be at least 5 times its original size?

[–5, 5] by [–10, 25]

In Exercises 11–14, find 1ƒ ! g2132 and 1g ! ƒ21 -22. 11. ƒ1x2 = 2x - 3; g1x2 = x + 1

12. ƒ1x2 = x 2 - 1; g1x2 = 2x - 3

13. ƒ1x2 = x 2 + 4; g1x2 = 2x + 1 14. ƒ1x2 =

x ; g1x2 = 9 - x 2 x + 1

In Exercises 15–22, find ƒ1g1x22 and g1ƒ1x22. State the domain of each. 15. ƒ1x2 = 3x + 2; g1x2 = x - 1 16. ƒ1x2 = x 2 - 1; g1x2 =

1 x - 1

17. ƒ1x2 = x 2 - 2; g1x2 = 2x + 1 18. ƒ1x2 =

1 ; g1x2 = 2x x - 1

19. ƒ1x2 = x 2; g1x2 = 21 - x 2 3 1 - x3 20. ƒ1x2 = x 3; g1x2 = 2 21. ƒ1x2 =

1 1 ; g1x2 = 2x 3x

1 1 ; g1x2 = 22. ƒ1x2 = x + 1 x - 1

34. Computer Imaging New Age Special Effects, Inc., prepares computer software based on specifications prepared by film directors. To simulate an approaching vehicle, they begin with a computer image of a 5-cm by 7-cm by 3-cm box. The program increases each dimension at a rate of 2 cm/sec. How long does it take for the volume V of the box to be at least 5 times its initial size? 35. Which of the ordered pairs 11, 12, 14, -22, and 13, -12 are in the relation given by 3x + 4y = 5? 36. Which of the ordered pairs 15, 12, 13, 42, and 10, - 52 are in the relation given by x 2 + y 2 = 25?

In Exercises 37–44, find two functions defined implicitly by the given relation. 37. x 2 + y 2 = 25

38. x + y 2 = 25

39. x 2 - y 2 = 25

40. 3x 2 - y 2 = 25

41. x + ƒ y ƒ = 1

42. x - ƒ y ƒ = 1

43. y 2 = x 2

44. y 2 = x

Standardized Test Questions 45. True or False The domain of the quotient function 1ƒ/g21x2 consists of all numbers that belong to both the domain of ƒ and the domain of g. Justify your answer. 46. True or False The domain of the product function 1ƒg21x2 consists of all numbers that belong to either the domain of ƒ or the domain of g. Justify your answer.

6965_CH01_pp063-156.qxd

118

1/14/10

1:01 PM

Page 118

CHAPTER 1 Functions and Graphs

You may use a graphing calculator when solving Exercises 47–50. 47. Multiple Choice Suppose ƒ and g are functions with domain all real numbers. Which of the following statements is not necessarily true? (A) (ƒ + g2(x2 = (g + ƒ2(x2 (B) (ƒg2(x2 = (gƒ2(x2 (C) ƒ(g(x22 = g(ƒ(x22

(D) (ƒ - g2(x2 = -(g - ƒ2(x2

(E) (ƒ ! g2(x2 = ƒ(g(x22 48. Multiple Choice If ƒ(x2 = x - 7 and g(x2 = 24 - x, what is the domain of the function ƒ/g? (A) ( - q , 42

(B) (- q , 44

(D) [4, q 2

(E) (4, 72 ´ (7, q 2

(D) x 4 + 2x 2 + 1

(E) x 4 + 2x 2 + 2

(B) 2x 2 + 1

(B) y 2 = x 2

2

2

(D) x + y = 1

(C) y 3 = x 3

51. Three on a Match Match each function ƒ with a function g and a domain D so that 1ƒ ! g21x2 = x 2 with domain D. ƒ

g

D

ex 1x + 222

22 - x x + 1

x Z 0 x Z 1 10, q 2

2 ln x 1 x - 1

x 2 - 2x + 1 x + 1 2 a b x

2x - 2 x + 1 x

(x - 1)2

(c) 1ƒ/g21x2 = 1

(d) ƒ1g1x22 = 9x 4 + 1 (e) g1ƒ1x22 = 9x 4 + 1

Extending the Ideas

(a) Function addition. That is, find a function g such that 1ƒ + g21x2 = 1g + ƒ21x2 = ƒ1x2.

(E) x = ƒ y ƒ

1x 2 - 222 1

(b) 1ƒ + g21x2 = 3x 2

(C) x 4 + 1

Explorations

2

(a) 1ƒg21x2 = x 4 - 1

53. Identifying Identities An identity for a function operation is a function that combines with a given function ƒ to return the same function ƒ. Find the identity functions for the following operations:

50. Multiple Choice Which of the following relations defines the function y = ƒ x ƒ implicitly? (A) y = x

Let ƒ1x2 = x 2 + 1. Find a function g so that

(C) (4, q 2

49. Multiple Choice If ƒ(x2 = x 2 + 1, then (ƒ ! ƒ2(x2 = (A) 2x 2 + 2

52. Be a g Whiz

32, q 2

1 - q , 24

1- q, q2

(b) Function multiplication. That is, find a function g such that 1ƒg21x2 = 1gƒ21x2 = ƒ1x2. (c) Function composition. That is, find a function g such that 1ƒ ! g21x2 = 1g ! ƒ21x2 = ƒ1x2.

54. Is Function Composition Associative? You already know that function composition is not commutative; that is, 1ƒ ! g21x2 Z 1g ! ƒ21x2. But is function composition associative? That is, does 1ƒ ! 1g ! h221x2 = 11ƒ ! g2 ! h22 1x2? Explain your answer.

55. Revisiting Example 6 Solve x 2y + y 2 = 5 for y using the quadratic formula and graph the pair of implicit functions.

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 119

SECTION 1.5

What you’ll learn about • Relations Defined Parametrically • Inverse Relations and Inverse Functions

... and why Some functions and graphs can best be defined parametrically, while some others can be best understood as inverses of functions we already know.

Parametric Relations and Inverses

119

1.5 Parametric Relations and Inverses Relations Defined Parametrically Another natural way to define functions or, more generally, relations, is to define both elements of the ordered pair 1x, y2 in terms of another variable t, called a parameter. We illustrate with an example.

EXAMPLE 1 Defining a Function Parametrically Consider the set of all ordered pairs 1x, y2 defined by the equations x = t + 1 y = t 2 + 2t

where t is any real number. (a) Find the points determined by t = - 3, -2, - 1, 0, 1, 2, and 3. (b) Find an algebraic relationship between x and y. (This is often called “eliminating the parameter.”) Is y a function of x? (c) Graph the relation in the 1x, y2 plane. SOLUTION

(a) Substitute each value of t into the formulas for x and y to find the point that it determines parametrically: t

x = t + 1

y = t 2 + 2t

-3 -2 -1 0 1 2 3

-2 -1 0 1 2 3 4

3 0 -1 0 3 8 15

1x, y2

1- 2, 32 1- 1, 02 10, - 12 11, 02 12, 32 13, 82 14, 152

(b) We can find the relationship between x and y algebraically by the method of substitution. First solve for t in terms of x to obtain t = x - 1. y = t 2 + 2t

Given 2

y = 1x - 12 + 21x - 12 = x 2 - 2x + 1 + 2x - 2 = x2 - 1

t = x - 1 Expand. Simplify.

This is consistent with the ordered pairs we had found in the table. As t varies over all real numbers, we will get all the ordered pairs in the relation y = x 2 - 1, which does indeed define y as a function of x. (c) Since the parametrically defined relation consists of all ordered pairs in the relation y = x 2 - 1, we can get the graph by simply graphing the parabola Now try Exercise 5. y = x 2 - 1. See Figure 1.62.

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 120

CHAPTER 1 Functions and Graphs

120

EXAMPLE 2 Using a Graphing Calculator in Parametric Mode

y 5 t = –3

t=1

t = –2 t = –1

t=0

5

x

Consider the set of all ordered pairs 1x, y2 defined by the equations x = t 2 + 2t y = t + 1

where t is any real number. (a) Use a graphing calculator to find the points determined by t = - 3, 2, - 1, 0, 1, 2, and 3.

–5

FIGURE 1.62 (Example 1)

(b) Use a graphing calculator to graph the relation in the 1x, y2 plane. (c) Is y a function of x? (d) Find an algebraic relationship between x and y. SOLUTION

t -3 -2 -1 0 1 2 3

1x, y)

13, - 22 10, - 12 1 - 1, 02 10, 12 13, 22 18, 32 115, 42

(a) When the calculator is in parametric mode, the “Y = ” screen provides a space to enter both X and Y as functions of the parameter T (Figure 1.63a). After entering the functions, use the table setup in Figure 1.63b to obtain the table shown in Figure 1.63c. The table shows, for example, that when T = -3 we have X1T = 3 and Y1T = - 2, so the ordered pair corresponding to t = -3 is (3, - 2). (b) In parametric mode, the “WINDOW” screen contains the usual x-axis information, as well as “Tmin,” “Tmax,” and “Tstep” (Figure 1.64a). To include most of the points listed in part (a), we set Xmin = - 5, Xmax = 5, Ymin = -3, and Ymax = 3. Since t = y - 1, we set Tmin and Tmax to values one less than those for Ymin and Ymax. The value of Tstep determines how far the grapher will go from one value of t to the next as it computes the ordered pairs. With Tmax - Tmin = 6 and Tstep = 0.1, the grapher will compute 60 points, which is sufficient. (The more points, the smoother the graph. See Exploration 1.) The graph is shown in Figure 1.64b. Use TRACE to find some of the points found in (a). (c) No, y is not a function of x. We can see this from part (a) because 10, -12 and 10, 12 have the same x-value but different y-values. Alternatively, notice that the graph in (b) fails the vertical line test. (d) We can use the same algebraic steps as in Example 1 to get the relation in terms of x and y: x = y 2 - 1. Now try Exercise 7.

Plot1

Plot2

\X1T=T2+2T Y1T=T+1 \X2T= Y2T= \X3T= Y3T= \X4T=

Plot3

T

TABLE SETUP TblStart=–3 ∆Tbl=1 Indpnt: Auto Ask Depend: Auto Ask

–3 –2 –1 0 1 2 3

X1T

Y1T = T+1 (a)

(b)

FIGURE 1.63 Using the table feature of a grapher set in parametric mode. (Example 2)

Y1T –2 –1 0 1 2 3 4

3 0 –1 0 3 8 15

(c)

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 121

SECTION 1.5

WINDOW Tmin=–4 Tmax=2 Tstep=.1 Xmin=–5 Xmax=5 Xscl=1 Ymin=–3

EXPLORATION 1

Parametric Relations and Inverses

121

Watching your Tstep

1. Graph the parabola in Example 2 in parametric mode as described in the solu-

tion. Press TRACE and observe the values of T, X, and Y. At what value of T does the calculator begin tracing? What point on the parabola results? (It’s off the screen.) At what value of T does it stop tracing? What point on the parabola results? How many points are computed as you TRACE from start to finish? (a)

2. Leave everything else the same and change the Tstep to 0.01. Do you get a

smoother graph? Why or why not? 3. Leave everything else the same and change the Tstep to 1. Do you get a

smoother graph? Why or why not? 4. What effect does the Tstep have on the speed of the grapher? Is this easily

explained? 5. Now change the Tstep to 2. Why does the left portion of the parabola disap-

pear? (It may help to TRACE along the curve.) 6. Change the Tstep back to 0.1 and change the Tmin to -1. Why does the bot[–5, 5] by [–3, 3]

(b)

FIGURE 1.64 The graph of a parabola in

tom side of the parabola disappear? (Again, it may help to TRACE.)

7. Make a change to the window that will cause the grapher to show the bottom

side of the parabola but not the top.

parametric mode on a graphing calculator. (Example 2)

Time For T

Inverse Relations and Inverse Functions

Functions defined by parametric equations are frequently encountered in problems of motion, where the x- and y-coordinates of a moving object are computed as functions of time. This makes time the parameter, which is why you almost always see parameters given as “t” in parametric equations.

What happens when we reverse the coordinates of all the ordered pairs in a relation? We obviously get another relation, as it is another set of ordered pairs, but does it bear any resemblance to the original relation? If the original relation happens to be a function, will the new relation also be a function? We can get some idea of what happens by examining Examples 1 and 2. The ordered pairs in Example 2 can be obtained by simply reversing the coordinates of the ordered pairs in Example 1. This is because we set up Example 2 by switching the parametric equations for x and y that we used in Example 1. We say that the relation in Example 2 is the inverse relation of the relation in Example 1.

DEFINITION Inverse Relation The ordered pair 1a, b2 is in a relation if and only if the ordered pair 1b, a2 is in the inverse relation. We will study the connection between a relation and its inverse. We will be most interested in inverse relations that happen to be functions. Notice that the graph of the inverse relation in Example 2 fails the vertical line test and is therefore not the graph of a function. Can we predict this failure by considering the graph of the original relation in Example 1? Figure 1.65 suggests that we can. The inverse graph in Figure 1.65b fails the vertical line test because two different y-values have been paired with the same x-value. This is a direct consequence of the fact that the original relation in Figure 1.65a paired two different x-values with the same y-value. The inverse graph fails the vertical line test precisely because the original graph fails the horizontal line test. This gives us a test for relations whose inverses are functions.

6965_CH01_pp063-156.qxd

122

1/14/10

1:01 PM

Page 122

CHAPTER 1 Functions and Graphs

y

(–1, 1)

y

3 2

–5 –4 –3 –2 –1 –1 –2 –3

3 2 1

(1, 1) 1 2 3 4 5

x

–5 –4 –3 –2 –1 –1 –2 –3

(a)

(1, 1) x

2 3 4 5 (–1, 1)

(b)

FIGURE 1.65 The inverse relation in (b) fails the vertical line test because the original relation in (a) fails the horizontal line test.

Horizontal Line Test The inverse of a relation is a function if and only if each horizontal line intersects the graph of the original relation in at most one point.

EXAMPLE 3 Applying the Horizontal Line Test Which of the graphs (1)–(4) in Figure 1.66 are graphs of (a) relations that are functions? (b) relations that have inverses that are functions? SOLUTION

(a) Graphs (1) and (4) are graphs of functions because these graphs pass the vertical line test. Graphs (2) and (3) are not graphs of functions because these graphs fail the vertical line test. (b) Graphs (1) and (2) are graphs of relations whose inverses are functions because these graphs pass the horizontal line test. Graphs (3) and (4) fail the horizontal line test so their inverse relations are not functions. Now try Exercise 9. y 3 2 1 –5 –4 –3

–1

y

y 3

3 2 1 1 2 3 4 5

x

y 3 2 1

1

–5 –4 –3 –2 –1 –1

1 2 3 4 5

x

–5

–3 –2 –1 –1

1 2 3

5

x

–5 –4 –3 –2

–3 (1)

2 3 4 5

x

(4)

(3)

(2)

–1 –2 –3

FIGURE 1.66 (Example 3) A function whose inverse is a function has a graph that passes both the horizontal and vertical line tests (such as graph (1) in Example 3). Such a function is one-to-one, since every x is paired with a unique y and every y is paired with a unique x. Caution about Function Notation -1

The symbol ƒ is read “ƒ inverse” and should never be confused with the reciprocal of ƒ. If ƒ is a function, the symbol ƒ -1, can only mean ƒ inverse. The reciprocal of ƒ must be written as 1/ƒ.

DEFINITION Inverse Function If ƒ is a one-to-one function with domain D and range R, then the inverse function of ƒ, denoted ƒ-1, is the function with domain R and range D defined by ƒ - 11b2 = a

if and only if

ƒ1a2 = b.

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 123

SECTION 1.5

Parametric Relations and Inverses

123

EXAMPLE 4 Finding an Inverse Function Algebraically Find an equation for ƒ-11x2 if ƒ1x2 = x/1x + 12.

SOLUTION The graph of ƒ in Figure 1.67 suggests that ƒ is one-to-one. The origi-

nal function satisfies the equation y = x/1x + 12. If ƒ truly is one-to-one, the inverse function ƒ-1 will satisfy the equation x = y/1y + 12. (Note that we just switch the x and the y.) If we solve this new equation for y we will have a formula for ƒ-11x2:

[–4.7, 4.7] by [–5, 5]

x =

FIGURE 1.67 The graph of

ƒ1x2 = x/1x + 12. (Example 4)

x1y + 12 xy + x xy - y y1x - 12

= = = =

y y + 1

y y -x -x

-x x - 1 x y = 1 - x y =

Multiply by y + 1. Distributive property Isolate the y terms. Factor out y. Divide by x - 1. Multiply numerator and denominator by -1.

Therefore ƒ-11x2 = x/11 - x2.

Now try Exercise 15.

Let us candidly admit two things regarding Example 4 before moving on to a graphical model for finding inverses. First, many functions are not one-to-one and so do not have inverse functions. Second, the algebra involved in finding an inverse function in the manner of Example 4 can be extremely difficult. We will actually find very few inverses this way. As you will learn in future chapters, we will usually rely on our understanding of how ƒ maps x to y to understand how ƒ-1 maps y to x. It is possible to use the graph of ƒ to produce a graph of ƒ-1 without doing any algebra at all, thanks to the following geometric reflection property:

The Inverse Reflection Principle The points 1a, b2 and 1b, a2 in the coordinate plane are symmetric with respect to the line y = x. The points 1a, b2 and 1b, a2 are reflections of each other across the line y = x.

EXAMPLE 5 Finding an Inverse Function Graphically The graph of a function y = ƒ(x) is shown in Figure 1.68. Sketch a graph of the function y = ƒ-1(x). Is ƒ a one-to-one function?

y

SOLUTION We need not find a formula for ƒ-1(x). All we need to do is to find the

4 3 2 –5 –4 –3 –2

–1 –2 –3 –4

reflection of the given graph across the line y = x. This can be done geometrically.

1 2 3 4 5

x

FIGURE 1.68 The graph of a one-to-one function. (Example 5)

Imagine a mirror along the line y = x and draw the reflection of the given graph in the mirror (Figure 1.69). Another way to visualize this process is to imagine the graph to be drawn on a large pane of glass. Imagine the glass rotating around the line y = x so that the positive x-axis switches places with the positive y-axis. (The back of the glass must be rotated to the front for this to occur.) The graph of ƒ will then become the graph of ƒ-1. Since the inverse of ƒ has a graph that passes the horizontal and vertical line test, ƒ is a one-to-one function. Now try Exercise 23.

6965_CH01_pp063-156.qxd

124

1/14/10

1:01 PM

Page 124

CHAPTER 1 Functions and Graphs

y 3 2 –5 –4 –3 –2

–1 –2 –3

x

The graph of f.

–5 –4 –3 –2

3 2 1

3 2

3 2 1 2 3 4 5

y

y

y

1 2 3 4 5

–1 –2 –3

x

–5 –4 –3 –2

2 3 4 5

x

–5 –4 –3 –2 –1

x

–2 –3

–2 –3

The mirror y = x.

2 3 4 5

The graph of f –1.

The reflection.

FIGURE 1.69 The mirror method. The graph of ƒ is reflected in an imaginary mirror along the line y = x to produce the graph of ƒ -1.

(Example 5)

There is a natural connection between inverses and function composition that gives further insight into what an inverse actually does: It “undoes” the action of the original function. This leads to the following rule:

The Inverse Composition Rule A function ƒ is one-to-one with inverse function g if and only if ƒ1g1x22 = x for every x in the domain of g, and g1ƒ1x22 = x for every x in the domain of ƒ.

EXAMPLE 6 Verifying Inverse Functions 3 x - 1 are inverse functions. Show algebraically that ƒ1x2 = x 3 + 1 and g1x2 = 2 SOLUTION We use the Inverse Composition Rule.

ƒ1g1x22 = ƒ12 3 x - 12 = 12 3 x - 123 + 1 = x - 1 + 1 = x

g1ƒ1x22 = g1x 3 + 12 = 2 3 1x 3 + 12 - 1 = 2 3 x3 = x

Since these equations are true for all x, the Inverse Composition Rule guarantees that ƒ and g are inverses. You do not have far to go to find graphical support of this algebraic verification, since these are the functions whose graphs are shown in Example 5! Now try Exercise 27. Some functions are so important that we need to study their inverses even though they are not one-to-one. A good example is the square root function, which is the “inverse” of the square function. It is not the inverse of the entire squaring function, because the full parabola fails the horizontal line test. Figure 1.70 shows that the function y = 2x is really the inverse of a “restricted-domain” version of y = x 2 defined only for x Ú 0.

y

y 4 3 2 1

4 3 2 1 –5 –4 –3 –2 –1 –1 –2

1 2 3 4 5 6

x

The graph of y ! x 2 (not one-to-one).

–5 –4 –3 –2 –1 –1 –2

y

y

1 2 3 4 5 6

The inverse relation of y ! x 2 (not a function).

x

4 3 2 1

4 3 2 1 –5 –4 –3 –2 –1 –1 –2

1 2 3 4 5 6

x

The graph of y ! !x" (a function).

–5 –4 –3 –2 –1 –1 –2

1 2 3 4 5 6

The graph of the function whose inverse is y !!x".

FIGURE 1.70 The function y = x 2 has no inverse function, but y = 2x is the inverse function of y = x 2 on the restricted domain 30, q 2.

x

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 125

SECTION 1.5

Parametric Relations and Inverses

125

The consideration of domains adds a refinement to the algebraic inverse-finding method of Example 4, which we now summarize: How to Find an Inverse Function Algebraically

Given a formula for a function ƒ, proceed as follows to find a formula for ƒ -1. 1. Determine that there is a function ƒ -1 by checking that ƒ is one-to-one. State any restrictions on the domain of ƒ. (Note that it might be necessary to impose some to get a one-to-one version of ƒ.) 2. Switch x and y in the formula y = ƒ1x2. 3. Solve for y to get the formula y = ƒ -11x2. State any restrictions on the domain of ƒ -1.

EXAMPLE 7 Finding an Inverse Function Show that ƒ1x2 = 2x + 3 has an inverse function and find a rule for ƒ-11x2. State any restrictions on the domains of ƒ and ƒ-1. SOLUTION

Solve Algebraically The graph of ƒ passes the horizontal line test, so ƒ has an inverse function (Figure 1.71). Note that ƒ has domain 3-3, q 2 and range 30, q 2. To find ƒ-1 we write y = 2x + 3 where x x = 2y + 3 where y x2 = y + 3 where y 2 y = x - 3 where y

Ú Ú Ú Ú

- 3, y - 3, x - 3, x - 3, x

Ú Ú Ú Ú

0 0 0 0

Interchange x and y. Square. Solve for y.

Thus ƒ-11x2 = x 2 - 3, with an “inherited” domain restriction of x Ú 0. Figure 1.71 shows the two functions. Note the domain restriction of x Ú 0 imposed on the parabola y = x 2 - 3. Support Graphically

[–4.7, 4.7] by [–3.1, 3.1]

FIGURE 1.71 The graph of

ƒ1x2 = 2x + 3 and its inverse, a restricted version of y = x 2 - 3. (Example 7)

QUICK REVIEW 1.5

Use a grapher in parametric mode and compare the graphs of the two sets of parametric equations with Figure 1.71: x = t y = 2t + 3

and

x = 2t + 3 y = t Now try Exercise 17.

(For help, go to Section P.3 and P.4.)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 1–10, solve the equation for y. 1. x = 3y - 6

2. x = 0.5y + 1

3. x = y 2 + 4 y - 2 5. x = y + 3

4. x = y 2 - 6 3y - 1 6. x = y + 2

7. x =

2y + 1 y - 4

9. x = 2y + 3, y Ú -3

10. x = 2y - 2, y Ú 2

8. x =

4y + 3 3y - 1

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 126

CHAPTER 1 Functions and Graphs

126

SECTION 1.5 EXERCISES 25.

In Exercises 1–4, find the 1x, y2 pair for the value of the parameter.

26.

y

1. x = 3t and y = t 2 + 5 for t = 2

4 3 2 1

2. x = 5t - 7 and y = 17 - 3t for t = - 2 3. x = t 3 - 4t and y = 2t + 1 for t = 3

–5 –4 –3 –2 –1 –1

4. x = ƒ t + 3 ƒ and y = 1/t for t = - 8

In Exercises 5–8, complete the following. (a) Find the points determined by t = -3, -2, - 1, 0, 1, 2, and 3. (b) Find a direct algebraic relationship between x and y and determine whether the parametric equations determine y as a function of x. (c) Graph the relationship in the xy-plane.

4 3 1 1 2 3 4 5

x

27. ƒ1x2 = 3x - 2 and g1x2 =

7. x = t 2 and y = t - 2

8. x = 2t and y = 2t - 5

28. ƒ1x2 =

9.

10.

y

x

11.

1 2 3 4 5

x

x + 2 3

x + 3 and g1x2 = 4x - 3 4

3x - 1 29. ƒ1x2 = x 3 + 1 and g1x2 = 2 30. ƒ1x2 =

7 7 and g1x2 = x x

31. ƒ1x2 =

1 x + 1 and g1x2 = x x - 1

32. ƒ1x2 =

2x + 3 x + 3 and g1x2 = x - 2 x - 1

x

12.

y

–1 –1 –2 –3 –4

In Exercises 27–32, confirm that ƒ and g are inverses by showing that ƒ1g1x22 = x and g1ƒ1x22 = x.

6. x = t + 1 and y = t 2 - 2t

In Exercises 9–12, the graph of a relation is shown. (a) Is the relation a function? (b) Does the relation have an inverse that is a function?

–5 –4 –3

–3 –4

5. x = 2t and y = 3t - 1

y

y

33. Currency Conversion In May of 2002 the exchange rate for converting U.S. dollars 1x2 to euros 1y2 was y = 1.08x.

y

(a) How many euros could you get for $100 U.S.?

(b) What is the inverse function, and what conversion does it represent? x

x

In Exercises 13–22, find a formula for ƒ -11x2. Give the domain of ƒ -1, including any restrictions “inherited” from ƒ. 13. ƒ1x2 = 3x - 6 15. ƒ1x2 =

14. ƒ1x2 = 2x + 5

2x - 3 x + 1

16. ƒ1x2 =

19. ƒ1x2 = x 3

18. ƒ1x2 = 2x + 2

3x + 5 21. ƒ1x2 = 2

22. ƒ1x2 = 2 3x - 2

36. Which basic functions (Section 1.3) are their own inverses? 37. Which basic function can be defined parametrically as follows?

In Exercises 23–26, determine whether the function is one-to-one. If it is one-to-one, sketch the graph of the inverse.

4 3 2 1 –5 –4 –3 –2 –1 –2 –3 –4

4 3 2 1 1 2 3 4 5

x

–5 –4 –3 –2

–1 –2 –3 –4

x = t 3 and y = 2t 6 for - q 6 t 6 q

38. Which basic function can be defined parametrically as follows? x = 8t 3 and y = 12t23 for - q 6 t 6 q

y

24.

(a) Find a formula for c -11x2. What is this formula used for?

35. Which pairs of basic functions (Section 1.3) are inverses of each other?

20. ƒ1x2 = 2x 3 + 5

y

34. Temperature Conversion The formula for converting Celsius temperature 1x2 to Kelvin temperature is k1x2 = x + 273.16. The formula for converting Fahrenheit temperature 1x2 to Celsius temperature is c1x2 = 15/921x - 322. (b) Find 1k ! c21x2. What is this formula used for?

x + 3 x - 2

17. ƒ1x2 = 2x - 3

23.

(c) In the spring of 2002, a tourist had an elegant lunch in Provence, France, ordering from a “fixed price” 48-euro menu. How much was that in U.S. dollars?

1

3 4 5

x

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 127

SECTION 1.5

Standardized Test Questions 39. True or False If ƒ is a one-to-one function with domain D and range R, then ƒ-1 is a one-to-one function with domain R and range D. Justify your answer. 40. True or False The set of points 1t + 1, 2t + 32 for all real numbers t form a line with slope 2. Justify your answer.

In Exercises 41–44, answer the questions without using a calculator.

41. Multiple Choice Which ordered pair is in the inverse of the relation given by x 2y + 5y = 9? (A) 12, 12

(B) 1-2, 12

(E) 11, - 22

(C) 1 -1, 22

(D) 12, -12

42. Multiple Choice Which ordered pair is not in the inverse of the relation given by xy 2 - 3x = 12? (A) 10, - 42

(B) 14, 12

(E) 11, - 62

(C) 13, 22

(D) 12, 122

43. Multiple Choice Which function is the inverse of the function ƒ1x2 = 3x - 2? (A) g1x2 =

x + 2 3

(B) g1x2 = 2 - 3x (C) g1x2 =

x + 2 3

(D) g1x2 =

x - 3 2

(E) g1x2 =

x - 2 3

44. Multiple Choice Which function is the inverse of the function ƒ1x2 = x 3 + 1?

Parametric Relations and Inverses

127

46. Function Properties Not Inherited by Inverses There are some properties of functions that are not necessarily shared by inverse functions, even if the inverses exist. Suppose that ƒ has an inverse function ƒ-1. For each of the following properties, give an example to show that ƒ can have the property while ƒ-1 does not. (a) ƒ has a graph with a horizontal asymptote. (b) ƒ has domain all real numbers. (c) ƒ has a graph that is bounded above. (d) ƒ has a removable discontinuity at x = 5. 47. Scaling Algebra Grades A teacher gives a challenging algebra test to her class. The lowest score is 52, which she decides to scale to 70. The highest score is 88, which she decides to scale to 97. (a) Using the points 152, 702 and 188, 972, find a linear equation that can be used to convert raw scores to scaled grades.

(b) Find the inverse of the function defined by this linear equation. What does the inverse function do? 48. Writing to Learn (Continuation of Exercise 47) Explain why it is important for fairness that the scaling function used by the teacher be an increasing function. (Caution: It is not because “everyone’s grade must go up.” What would the scaling function in Exercise 47 do for a student who does enough “extra credit” problems to get a raw score of 136?)

Extending the Ideas 49. Modeling a Fly Ball Parametrically A baseball that leaves the bat at an angle of 60° from horizontal traveling 110 feet per second follows a path that can be modeled by the following pair of parametric equations. (You might enjoy verifying this if you have studied motion in physics.)

3x - 1 (A) g1x2 = 2

x = 1101t2cos160°2

(C) g1x2 = x 3 - 1

You can simulate the flight of the ball on a grapher. Set your grapher to parametric mode and put the functions above in for X2T and Y2T. Set X1T = 325 and Y1T = 5T to draw a 30-foot fence 325 feet from home plate. Set Tmin = 0, Tmax = 6, Tstep = 0.1, Xmin = 0, Xmax = 350, Xscl = 0, Ymin = 0, Ymax = 300, and Yscl = 0.

3x - 1 (B) g1x2 = 2 3x + 1 (D) g1x2 = 2 (E) g1x2 = 1 - x 3

Explorations 45. Function Properties Inherited by Inverses There are some properties of functions that are automatically shared by inverse functions (when they exist) and some that are not. Suppose that ƒ has an inverse function ƒ-1. Give an algebraic or graphical argument (not a rigorous formal proof) to show that each of these properties of ƒ must necessarily be shared by ƒ-1. (a) ƒ is continuous. (b) ƒ is one-to-one. (c) ƒ is odd (graphically, symmetric with respect to the origin). (d) ƒ is increasing.

y = 1101t2sin160°2 - 16t 2

(a) Now graph the function. Does the fly ball clear the fence? (b) Change the angle to 30° and run the simulation again. Does the ball clear the fence? (c) What angle is optimal for hitting the ball? Does it clear the fence when hit at that angle?

6965_CH01_pp063-156.qxd

128

1/14/10

1:01 PM

Page 128

CHAPTER 1 Functions and Graphs

50. The Baylor GPA Scale Revisited (See Problem 78 in Section 1.2.) The function used to convert Baylor School percentage grades to GPAs on a 4-point scale is 1

1.7 31.7 y = a (x - 65)b + 1. 30

The function has domain [65, 100]. Anything below 65 is a failure and automatically converts to a GPA of 0. (a) Find the inverse function algebraically. What can the inverse function be used for? (b) Does the inverse function have any domain restrictions? (c) Verify with a graphing calculator that the function found in (a) and the given function are really inverses.

51. Group Activity (Continuation of Exercise 50) The number 1.7 that appears in two places in the GPA scaling formula is called the scaling factor 1k2. The value of k can be changed to alter the curvature of the graph while keeping the points 165, 12 and 195, 42 fixed. It was felt that the lowest D 1652 needed to be scaled to 1.0, while the middle A 1952 needed to be scaled to 4.0. The faculty’s Academic Council considered several values of k before settling on 1.7 as the number that gives the “fairest” GPAs for the other percentage grades. Try changing k to other values between 1 and 2. What kind of scaling curve do you get when k = 1? Do you agree with the Baylor decision that k = 1.7 gives the fairest GPAs?

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 129

SECTION 1.6

Graphical Transformations

129

1.6 Graphical Transformations What you’ll learn about • Transformations

Transformations The following functions are all different: y = x2 y = 1x - 322 y = 1 - x2 y = x 2 - 4x + 5

• Vertical and Horizontal Translations • Reflections Across Axes • Vertical and Horizontal Stretches and Shrinks • Combining Transformations

... and why Studying transformations will help you to understand the relationships between graphs that have similarities but are not the same.

However, a look at their graphs shows that, while no two are exactly the same, all four have the same identical shape and size. Understanding how algebraic alterations change the shapes, sizes, positions, and orientations of graphs is helpful for understanding the connection between algebraic and graphical models of functions. In this section we relate graphs using transformations, which are functions that map real numbers to real numbers. By acting on the x-coordinates and y-coordinates of points, transformations change graphs in predictable ways. Rigid transformations, which leave the size and shape of a graph unchanged, include horizontal translations, vertical translations, reflections, or any combination of these. Nonrigid transformations, which generally distort the shape of a graph, include horizontal or vertical stretches and shrinks.

Vertical and Horizontal Translations A vertical translation of the graph of y = ƒ1x2 is a shift of the graph up or down in the coordinate plane. A horizontal translation is a shift of the graph to the left or the right. The following exploration will give you a good feel for what translations are and how they occur. EXPLORATION 1

Introducing Translations

Set your viewing window to 3-5, 54 by 3- 5, 154 and your graphing mode to sequential as opposed to simultaneous. 1. Graph the functions

y1 = x 2 y2 = y11x2 + 3 = x 2 + 3 y3 = y11x2 + 1 = x 2 + 1

y4 = y11x2 - 2 = x 2 - 2 y5 = y11x2 - 4 = x 2 - 4

on the same screen. What effect do the +3, +1, - 2, and -4 seem to have? 2. Graph the functions

y1 = x 2 y2 = y11x + 32 = 1x + 322 y3 = y11x + 12 = 1x + 122

y4 = y11x - 22 = 1x - 222 y5 = y11x - 42 = 1x - 422

on the same screen. What effect do the +3, +1, - 2, and -4 seem to have? 3. Repeat steps 1 and 2 for the functions y1 = x 3, y1 = ƒ x ƒ , and y1 = 2x.

Do your observations agree with those you made after steps 1 and 2?

Technology Alert In Exploration 1, the notation y11x + 32 means the function y1, evaluated at x + 3. It does not mean multiplication.

In general, replacing x by x - c shifts the graph horizontally c units. Similarly, replacing y by y - c shifts the graph vertically c units. If c is positive the shift is to the right or up; if c is negative the shift is to the left or down.

6965_CH01_pp063-156.qxd

130

1/14/10

1:01 PM

Page 130

CHAPTER 1 Functions and Graphs

This is a nice, consistent rule that unfortunately gets complicated by the fact that the c for a vertical shift rarely shows up being subtracted from y. Instead, it usually shows up on the other side of the equal sign being added to ƒ1x2. That leads us to the following rule, which only appears to be different for horizontal and vertical shifts:

Translations Let c be a positive real number. Then the following transformations result in translations of the graph of y = ƒ1x2: Horizontal Translations

y = ƒ1x - c2 y = ƒ1x + c2

a translation to the right by c units a translation to the left by c units

Vertical Translations

y = ƒ1x2 + c y = ƒ1x2 - c

a translation up by c units a translation down by c units

EXAMPLE 1 Vertical Translations Describe how the graph of y = ƒ x ƒ can be transformed to the graph of the given equation. (a) y = ƒ x ƒ - 4

(b) y = ƒ x + 2 ƒ

SOLUTION

(a) The equation is in the form y = ƒ1x2 - 4, a translation down by 4 units. See Figure 1.72. (b) The equation is in the form y = ƒ1x + 22, a translation left by 2 units. See Figure 1.73. Now try Exercise 3. y

y 5

5

5

x

5

–5

–5

FIGURE 1.72 y = ƒ x ƒ - 4.

FIGURE 1.73 y = ƒ x + 2 ƒ .

(Example 1)

x

(Example 1)

EXAMPLE 2 Finding Equations for Translations Each view in Figure 1.74 shows the graph of y1 = x 3 and a vertical or horizontal translation y2. Write an equation for y2 as shown in each graph. SOLUTION

(a) y2 = x 3 - 3 = y1 1x2 - 3 (a vertical translation down by 3 units) (b) y2 = 1x + 223 = y1 1x + 22 (a horizontal translation left by 2 units) (c) y2 = 1x - 323 = y1 1x - 32 (a horizontal translation right by 3 units) Now try Exercise 25.

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 131

SECTION 1.6

y

y

6 5 4 3 2 1 –5 –4 –3 –2 –1 –1 –2

Graphical Transformations

y

6

6 5 4 3 2 1

2 1 1 2 3 4 5

x

–5 –4

–2

–4 –5 –6

1 2 3 4 5

–1 –2 –3 –4 –5 –6

x

–5 –4 –3 –2

–1 –2 –3 –4 –5 –6

(b)

(a)

131

1

3 4 5

x

(c)

FIGURE 1.74 Translations of y1 = x 3. (Example 2)

Reflections Across Axes Points 1x, y2 and 1x, - y2 are reflections of each other across the x-axis. Points 1x, y2 and 1 -x, y2 are reflections of each other across the y-axis. (See Figure 1.75.) Two points (or graphs) that are symmetric with respect to a line are reflections of each other across that line. y

(–x, y)

(x, y) x (x, –y)

FIGURE 1.75 The point 1x, y2 and its reflections across the x-and y-axes.

Figure 1.75 suggests that a reflection across the x-axis results when y is replaced by -y, and a reflection across the y-axis results when x is replaced by -x. Double Reflection

Reflections

Note that a reflection through the origin is the result of reflections in both axes, performed in either order.

The following transformations result in reflections of the graph of y = ƒ1x2: Across the x-axis

y = - ƒ1x2 Across the y-axis

y = ƒ1-x2 Through the origin

y = - ƒ1- x2

EXAMPLE 3 Finding Equations for Reflections Find an equation for the reflection of ƒ1x2 =

5x - 9 across each axis. x2 + 3 (continued)

6965_CH01_pp063-156.qxd

132

2/2/10

11:22 AM

Page 132

CHAPTER 1 Functions and Graphs

SOLUTION

Solve Algebraically 9 - 5x 5x - 9 = 2 2 x + 3 x + 3 51-x2 - 9 -5x - 9 Across the y-axis: y = ƒ1- x2 = = 2 2 1-x2 + 3 x + 3 Support Graphically Across the x-axis: y = -ƒ1x2 = -

The graphs in Figure 1.76 support our algebraic work.

[–5, 5] by [–4, 4] (a)

[–5, 5] by [–4, 4] (b)

FIGURE 1.76 Reflections of ƒ1x2 = 15x - 92/1x 2 + 32 across (a) the x-axis and (b) the y-axis. (Example 3) Now try Exercise 29. You might expect that odd and even functions, whose graphs already possess special symmetries, would exhibit special behavior when reflected across the axes. They do, as shown by Example 4 and Exercises 33 and 34.

EXAMPLE 4 Reflecting Even Functions Prove that the graph of an even function remains unchanged when it is reflected across the y-axis. SOLUTION Note that we can get plenty of graphical support for these statements by reflecting the graphs of various even functions, but what is called for here is proof, which will require algebra.

Let ƒ be an even function; that is, ƒ1-x2 = ƒ1x2 for all x in the domain of ƒ. To reflect the graph of y = ƒ1x2 across the y-axis, we make the transformation y = ƒ1- x2. But ƒ1-x2 = ƒ1x2 for all x in the domain of ƒ, so this transformation results in y = ƒ1x2. The graph of ƒ therefore remains unchanged. Now try Exercise 33.

Graphing Absolute Value Compositions Given the graph of y = ƒ1x2, the graph of y = ƒ ƒ1x2 ƒ can be obtained by reflecting the portion of the graph below the x-axis across the x-axis, leaving the portion above the x-axis unchanged; the graph of y = ƒ1 ƒ x ƒ 2 can be obtained by replacing the portion of the graph to the left of the y-axis by a reflection of the portion to the right of the y-axis across the y-axis, leaving the portion to the right of the y-axis unchanged. (The result will show even symmetry.) Function compositions with absolute value can be realized graphically by reflecting portions of graphs, as you will see in the following Exploration.

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 133

SECTION 1.6

EXPLORATION 2

133

Graphical Transformations

Compositions with Absolute Value

The graph of y = ƒ1x2 is shown at the right. Match each of the graphs below with one of the following equations and use the language of function reflection to defend your match. Note that two of the graphs will not be used.

y

x

1. y = ƒ ƒ1x2 ƒ 2. y = ƒ1 ƒ x ƒ 2

3. y = - ƒ ƒ1x2 ƒ 4. y = ƒ ƒ1 ƒ x ƒ 2 ƒ

(A)

(B) y

y

x

(C)

x

(D) y

y

x

x

(E)

(F) y

y

x

x

Vertical and Horizontal Stretches and Shrinks We now investigate what happens when we multiply all the y-coordinates (or all the x-coordinates) of a graph by a fixed real number.

6965_CH01_pp063-156.qxd

134

1/14/10

1:01 PM

Page 134

CHAPTER 1 Functions and Graphs

Introducing Stretches and Shrinks

EXPLORATION 3

Set your viewing window to [ -4.7, 4.7] by [- 1.1, 5.1] and your graphing mode to sequential as opposed to simultaneous. 1. Graph the functions

y1 y2 y3 y4 y5

= = = = =

24 - x 2 1.5y11x2 = 1.524 - x 2 2y11x2 = 224 - x 2 0.5y11x2 = 0.524 - x 2 0.25y11x2 = 0.2524 - x 2

on the same screen. What effect do the 1.5, 2, 0.5, and 0.25 seem to have? 2. Graph the functions

y1 = 24 - x 2 y2 = y111.5x2 = 24 - 11.5x22 y3 = y112x2 = 24 - 12x22

y4 = y110.5x2 = 24 - 10.5x22 y5 = y110.25x2 = 24 - 10.25x22

on the same screen. What effect do the 1.5, 2, 0.5, and 0.25 seem to have?

Exploration 3 suggests that multiplication of x or y by a constant results in a horizontal or vertical stretching or shrinking of the graph. In general, replacing x by x/c distorts the graph horizontally by a factor of c. Similarly, replacing y by y/c distorts the graph vertically by a factor of c. If c is greater than 1 the distortion is a stretch; if c is less than 1 the distortion is a shrink. As with translations, this is a nice, consistent rule that unfortunately gets complicated by the fact that the c for a vertical stretch or shrink rarely shows up as a divisor of y. Instead, it usually shows up on the other side of the equal sign as a factor multiplied by ƒ1x2. That leads us to the following rule:

Stretches and Shrinks Let c be a positive real number. Then the following transformations result in stretches or shrinks of the graph of y = ƒ1x2: [–7, 7] by [–80, 80] (a)

Horizontal Stretches or Shrinks

x y = ƒa b c

e

a stretch by a factor of c a shrink by a factor of c

if c 7 1 if c 6 1

Vertical Stretches or Shrinks

y = c # ƒ1x2

e

a stretch by a factor of c a shrink by a factor of c

if c 7 1 if c 6 1

EXAMPLE 5 Finding Equations for Stretches and Shrinks [–7, 7] by [–80, 80] (b)

FIGURE 1.77 The graph of y1 = ƒ1x2 = x 3 - 16x, shown with (a) a vertical stretch and (b) a horizontal shrink. (Example 5)

Let C1 be the curve defined by y1 = ƒ1x2 = x 3 - 16x. Find equations for the following nonrigid transformations of C1: (a) C2 is a vertical stretch of C1 by a factor of 3. (b) C3 is a horizontal shrink of C1 by a factor of 1/2.

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 135

SECTION 1.6

Graphical Transformations

135

SOLUTION

Solve Algebraically (a) Denote the equation for C2 by y2. Then

y2 = 3 # ƒ1x2 = 31x 3 - 16x2 = 3x 3 - 48x

(b) Denote the equation for C3 by y3. Then x b 1/2 = ƒ12x2 = 12x23 - 1612x2 = 8x 3 - 32x

y3 = ƒa

Support Graphically

The graphs in Figure 1.77 support our algebraic work.

Now try Exercise 39.

Combining Transformations Transformations may be performed in succession—one after another. If the transformations include stretches, shrinks, or reflections, the order in which the transformations are performed may make a difference. In those cases, be sure to pay particular attention to order.

EXAMPLE 6 Combining Transformations in Order (a) The graph of y = x 2 undergoes the following transformations, in order. Find the equation of the graph that results. • a horizontal shift 2 units to the right • a vertical stretch by a factor of 3 • a vertical translation 5 units up (b) Apply the transformations in (a) in the opposite order and find the equation of the graph that results. SOLUTION

(a) Applying the transformations in order, we have

x 2 Q 1x - 222 Q 31x - 222 Q 31x - 222 + 5.

Expanding the final expression, we get the function y = 3x 2 - 12x + 17. (b) Applying the transformations in the opposite order, we have x 2 Q x 2 + 5 Q 31x 2 + 52 Q 311x - 222 + 52.

Expanding the final expression, we get the function y = 3x 2 - 12x + 27. The second graph is ten units higher than the first graph because the vertical stretch lengthens the vertical translation when the translation occurs first. Order often matters when stretches, shrinks, or reflections are involved. Now try Exercise 47.

6965_CH01_pp063-156.qxd

136

1/14/10

1:01 PM

Page 136

CHAPTER 1 Functions and Graphs

EXAMPLE 7 Transforming a Graph Geometrically

y

y = f(x)

The graph of y = ƒ1x2 is shown in Figure 1.78. Determine the graph of the composite function y = 2ƒ1x + 12 - 3 by showing the effect of a sequence of transformations on the graph of y = ƒ1x2.

3 1

–4 –3 –2 –1 –1 –2 –3 –4

1 2 3 4

SOLUTION

x

The graph of y = 2ƒ1x + 12 - 3 can be obtained from the graph of y = ƒ1x2 by the following sequence of transformations: (a) a vertical stretch by a factor of 2 to get y = 2ƒ1x2 (Figure 1.79a) (b) a horizontal translation 1 unit to the left to get y = 2ƒ1x + 12 (Figure 1.79b) (c) a vertical translation 3 units down to get y = 2ƒ1x + 12 - 3 (Figure 1.79c)

FIGURE 1.78 The graph of the function y = ƒ1x2 in Example 7.

(The order of the first two transformations can be reversed without changing the final graph.) Now try Exercise 51. y

y

3 2 1

4 3 2 1

y = 2f(x)

–4 –3 –2 –1 –1 –2 –3 –4

x

1 2 3 4

–4 –3 –2 –1 –1 –2 –3 –4

y 4 3 2 1

y = 2f(x + 1)

1 2 3 4

x

–4 –3 –2 –1 –1 –2 –3 –4

y = 2f(x + 1) – 3

1 2 3 4

Vertical stretch of factor 2

Horizontal translation left 1 unit

Vertical translation down 3 units

(a)

(b)

(c)

FIGURE 1.79 Transforming the graph of y = ƒ1x2 in Figure 1.78 to get the graph of y = 2ƒ1x + 12 - 3. (Example 7)

QUICK REVIEW 1.6

(For help, go to Section A.2.)

Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 1–6, write the expression as a binomial squared. 2

2

1. x + 2x + 1

2. x - 6x + 9

3. x 2 + 12x + 36

4. 4x 2 + 4x + 1

25 5. x 2 - 5x + 4

6. 4x 2 - 20x + 25

In Exercises 7–10, perform the indicated operations and simplify. 7. 1x - 222 + 31x - 22 + 4

8. 21x + 322 - 51x + 32 - 2 9. 1x - 123 + 31x - 122 - 31x - 12

10. 21x + 123 - 61x + 122 + 61x + 12 - 2

SECTION 1.6 EXERCISES In Exercises 1–8, describe how the graph of y = x 2 can be transformed to the graph of the given equation. 1. y = x 2 - 3

2. y = x 2 + 5.2 2

3. y = 1x + 42

2

2

5. y = 1100 - x2

7. y = 1x - 122 + 3

4. y = 1x - 32 2

6. y = x - 100 8. y = 1x + 5022 - 279

In Exercises 9–12, describe how the graph of y = 2x can be transformed to the graph of the given equation. 9. y = - 1x

11. y = 1 - x

10. y = 1x - 5 12. y = 13 - x

In Exercises 13–16, describe how the graph of y = x 3 can be transformed to the graph of the given equation. 13. y = 2x 3 15. y = 10.2x23

14. y = 12x23 16. y = 0.3x 3

x

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 137

SECTION 1.6

In Exercises 17–20, describe how to transform the graph of ƒ into the graph of g.

Graphical Transformations

137

y

17. ƒ1x2 = 1x + 2 and g1x2 = 1x - 4

18. ƒ1x2 = 1x - 122 and g1x2 = - 1x + 322 19. ƒ1x2 = 1x - 223 and g1x2 = - 1x + 223 20. ƒ1x2 = ƒ 2x ƒ and g1x2 = 4 ƒ x ƒ

x

In Exercises 21–24, sketch the graphs of ƒ, g, and h by hand. Support your answers with a grapher. 21. ƒ1x2 = 1x + 222

22. ƒ1x2 = x 3 - 2

3x + 1 23. ƒ1x2 = 2

24. ƒ1x2 = -2 ƒ x ƒ - 3

g1x2 = 3x 2 - 2

g1x2 = 1x + 423 - 1

h1x2 = - 21x - 322

h1x2 = 21x - 123

g1x2 = 2 2 3x - 2

g1x2 = 3 ƒ x + 5 ƒ + 4

h1x2 = - 2 3x - 3

h1x2 = ƒ 3x ƒ

In Exercises 25–28, the graph is that of a function y = ƒ1x2 that can be obtained by transforming the graph of y = 2x. Write a formula for the function ƒ. 25.

26.

In Exercises 39–42, transform the given function by (a) a vertical stretch by a factor of 2, and (b) a horizontal shrink by a factor of 1/3. 39. ƒ1x2 = x 3 - 4x

40. ƒ1x2 = ƒ x + 2 ƒ

41. ƒ1x2 = x 2 + x - 2

42. ƒ1x2 =

In Exercises 43–46, describe a basic graph and a sequence of transformations that can be used to produce a graph of the given function. 43. y = 21x - 322 - 4

44. y = - 32x + 1

2

46. y = - 2 ƒ x + 4 ƒ + 1

45. y = 13x2 - 4

[–10, 10] by [–5, 5]

27.

[–10, 10] by [–5, 5]

1 x + 2

In Exercises 47–50, a graph G is obtained from a graph of y by the sequence of transformations indicated. Write an equation whose graph is G. 47. y = x 2: a vertical stretch by a factor of 3, then a shift right 4 units.

28.

48. y = x 2: a shift right 4 units, then a vertical stretch by a factor of 3. [–10, 10] by [–5, 5]

[–10, 10] by [–5, 5] Vertical stretch = 2

In Exercises 29–32, find the equation of the reflection of ƒ across (a) the x-axis and (b) the y-axis. 3

49. y = ƒ x ƒ : a shift left 2 units, then a vertical stretch by a factor of 2, and finally a shift down 4 units. 50. y = ƒ x ƒ : a shift left 2 units, then a horizontal shrink by a factor of 1/2, and finally a shift down 4 units. Exercises 51–54 refer to the function ƒ whose graph is shown below. y

2

29. ƒ1x2 = x - 5x - 3x + 2 30. ƒ1x2 = 2 2x + 3 - 4

3 2 1

3 8x 31. ƒ1x2 = 2

32. ƒ1x2 = 3 ƒ x + 5 ƒ 33. Reflecting Odd Functions Prove that the graph of an odd function is the same when reflected across the x-axis as it is when reflected across the y-axis. 34. Reflecting Odd Functions Prove that if an odd function is reflected about the y-axis and then reflected again about the x-axis, the result is the original function. Exercises 35–38 refer to the graph of y = ƒ1x2 shown at the top of the next column. In each case, sketch a graph of the new function. 35. y = ƒ ƒ1x2 ƒ 36. y = ƒ1|x|2 37. y = - ƒ1 ƒ x ƒ 2 38. y = ƒ ƒ1 ƒ x ƒ 2 ƒ

–3 –2 –1

1 2 3 4

x

–2 –3 –4

51. Sketch the graph of y = 2 + 3ƒ1x + 12. 52. Sketch the graph of y = -ƒ1x + 12 + 1. 53. Sketch the graph of y = ƒ12x2. 54. Sketch the graph of y = 2ƒ1x - 12 + 2. 55. Writing to Learn Graph some examples to convince yourself that a reflection and a translation can have a different effect when combined in one order than when combined in the opposite order. Then explain in your own words why this can happen.

6965_CH01_pp063-156.qxd

138

1/14/10

1:01 PM

Page 138

CHAPTER 1 Functions and Graphs

56. Writing to Learn Graph some examples to convince yourself that vertical stretches and shrinks do not affect a graph’s x-intercepts. Then explain in your own words why this is so. 57. Celsius vs. Fahrenheit The graph shows the temperature in degrees Celsius in Windsor, Ontario, for one 24-hour period. Describe the transformations that convert this graph to one showing degrees Fahrenheit. [Hint: F1t2 = 19/52C1t2 + 32.] y

C(t)

63. Multiple Choice Given a function ƒ, which of the following represents a vertical translation of 2 units upward, followed by a reflection across the y-axis? (A) y = ƒ1- x2 + 2

(B) y = 2 - ƒ1x2

(C) y = ƒ12 - x2

(D) y = -ƒ1x - 22

(E) y = ƒ1x2 - 2 64. Multiple Choice Given a function ƒ, which of the following represents reflection across the x-axis, followed by a horizontal shrink by a factor of 1/2? (A) y = - 2ƒ1x2

(B) y = -ƒ1x2/2

(C) y = ƒ1 - 2x2

(D) y = -ƒ1x/22

(E) y = -ƒ12x2

Explorations 24

t

58. Fahrenheit vs. Celsius The graph shows the temperature in degrees Fahrenheit in Mt. Clemens, Michigan, for one 24-hour period. Describe the transformations that convert this graph to one showing degrees Celsius. [Hint: F1t2 = 19/52C1t2 + 32.] y

F(t)

24

t

Standardized Test Questions 59. True or False The function y = ƒ1x + 32 represents a translation to the right by 3 units of the graph of y = ƒ1x2. Justify your answer. 60. True or False The function y = ƒ1x2 - 4 represents a translation down 4 units of the graph of y = ƒ1x2. Justify your answer. In Exercises 61–64, you may use a graphing calculator to answer the question. 61. Multiple Choice Given a function ƒ, which of the following represents a vertical stretch by a factor of 3? (A) y = ƒ13x2

(B) y = ƒ1x/32

(C) y = 3ƒ1x2

(D) y = ƒ1x2/3

(E) y = ƒ1x2 + 3 62. Multiple Choice Given a function ƒ, which of the following represents a horizontal translation of 4 units to the right? (A) y = ƒ1x2 + 4

(B) y = ƒ1x2 - 4

(C) y = ƒ1x + 42

(D) y = ƒ1x - 42

(E) y = 4ƒ1x2

65. International Finance Table 1.11 shows the (adjusted closing) price of a share of stock in Dell Computer for each month of 2008.

Table 1.11

Dell Computer

Month

Price ($)

1 2 3 4 5 6 7 8 9 10 11 12

20.04 19.90 19.92 18.63 23.06 21.88 24.57 21.73 16.48 12.20 11.17 10.24

Source: Yahoo! Finance.

(a) Graph price 1y2 as a function of month 1x2 as a line graph, connecting the points to make a continuous graph. (b) Explain what transformation you would apply to this graph to produce a graph showing the price of the stock in Japanese yen. 66. Group Activity Get with a friend and graph the function y = x 2 on both your graphers. Apply a horizontal or vertical stretch or shrink to the function on one of the graphers. Then change the window of that grapher to make the two graphs look the same. Can you formulate a general rule for how to find the window?

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 139

SECTION 1.6

Extending the Ideas 67. The Absolute Value Transformation Graph the function ƒ1x2 = x 4 - 5x 3 + 4x 2 + 3x + 2 in the viewing window [ -5, 5] by [-10, 10]. (Put the equation in Y1.) (a) Study the graph and try to predict what the graph of y = ƒ ƒ1x2 ƒ will look like. Then turn Y1 off and graph Y2 = abs (Y1). Did you predict correctly? (b) Study the original graph again and try to predict what the graph of y = ƒ1 ƒ x ƒ 2 will look like. Then turn Y1 off and graph Y2 = Y11abs 1X22. Did you predict correctly?

(c) Given the graph of y = g1x2 shown below, sketch a graph of y = ƒ g1x2 ƒ . (d) Given the graph of y = g1x2 shown below, sketch a graph of y = g1 ƒ x ƒ 2. y

6 5 4 3 2 1 –4 –3 –2 –1 –1 –3 –4 –5 –6

1 2 3 4 5

x

Graphical Transformations

139

68. Parametric Circles and Ellipses Set your grapher to parametric and radian mode and your window as follows: Tmin = 0, Tmax = 7, Tstep = 0.1 Xmin = - 4.7, Xmax = 4.7, Xscl = 1 Ymin = -3.1, Ymax = 3.1, Yscl = 1 (a) Graph the parametric equations x = cos t and y = sin t. You should get a circle of radius 1. (b) Use a transformation of the parametric function of x to produce the graph of an ellipse that is 4 units wide and 2 units tall. (c) Use a transformation of both parametric functions to produce a circle of radius 3. (d) Use a transformation of both functions to produce an ellipse that is 8 units wide and 4 units tall. (You will learn more about ellipses in Chapter 8.)

6965_CH01_pp063-156.qxd

140

2/3/10

3:06 PM

Page 140

CHAPTER 1 Functions and Graphs

1.7 Modeling with Functions What you’ll learn about • Functions from Formulas • Functions from Graphs • Functions from Verbal Descriptions • Functions from Data

... and why Using a function to model a variable under observation in terms of another variable often allows one to make predictions in practical situations, such as predicting the future growth of a business based on known data.

Functions from Formulas Now that you have learned more about what functions are and how they behave, we want to return to the modeling theme of Section 1.1. In that section we stressed that one of the goals of this course was to become adept at using numerical, algebraic, and graphical models of the real world in order to solve problems. We now want to focus your attention more precisely on modeling with functions. You have already seen quite a few formulas in the course of your education. Formulas involving two variable quantities always relate those variables implicitly, and quite often the formulas can be solved to give one variable explicitly as a function of the other. In this book we will use a variety of formulas to pose and solve problems algebraically, although we will not assume prior familiarity with those formulas that we borrow from other subject areas (like physics or economics). We will assume familiarity with certain key formulas from mathematics.

EXAMPLE 1 Forming Functions from Formulas Write the area A of a circle as a function of its (a) radius r. (b) diameter d. (c) circumference C. SOLUTION

(a) The familiar area formula from geometry gives A as a function of r: A = pr 2 (b) This formula is not so familiar. However, we know that r = d/2, so we can substitute that expression for r in the area formula: A = pr 2 = p1d/222 = 1p/42d 2

(c) Since C = 2pr, we can solve for r to get r = C/12p2. Then substitute to get A: A = pr 2 = p1C/12p222 = pC 2/14p22 = C 2/14p2. Now try Exercise 19.

EXAMPLE 2 A Maximum Value Problem A square of side x inches is cut out of each corner of an 8 in. by 15 in. piece of cardboard and the sides are folded up to form an open-topped box (Figure 1.80).

FIGURE 1.80 An open-topped box made by cutting the corners from a piece of cardboard and folding up the sides. (Example 2)

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 141

SECTION 1.7

Modeling with Functions

141

(a) Write the volume V of the box as a function of x. (b) Find the domain of V as a function of x. (Note that the model imposes restrictions on x.) (c) Graph V as a function of x over the domain found in part (b) and use the maximum finder on your grapher to determine the maximum volume such a box can hold. (d) How big should the cut-out squares be in order to produce the box of maximum volume? [0, 4] by [0, 100] (a)

Maximum X=1.666666 Y=90.740741 [0, 4] by [0, 100] (b)

FIGURE 1.81 The graph of the volume of

SOLUTION

(a) The box will have a base with sides of width 8 - 2x and length 15 - 2x. The depth of the box will be x when the sides are folded up. Therefore V = x18 - 2x2 115 - 2x2. (b) The formula for V is a polynomial with domain all reals. However, the depth x must be nonnegative, as must the width of the base, 8 - 2x. Together, these two restrictions yield a domain of [0, 4]. (The endpoints give a box with no volume, which is as mathematically feasible as other zero concepts.) (c) The graph is shown in Figure 1.81. The maximum finder shows that the maximum occurs at the point 15/3, 90.742. The maximum volume is about 90.74 in.3. (d) Each square should have sides of one-and-two-thirds inches. Now try Exercise 33.

the box in Example 2.

Functions from Graphs When “thinking graphically” becomes a genuine part of your problem-solving strategy, it is sometimes actually easier to start with the graphical model than it is to go straight to the algebraic formula. The graph provides valuable information about the function.

EXAMPLE 3 Protecting an Antenna A small satellite dish is packaged with a cardboard cylinder for protection. The parabolic dish is 24 in. in diameter and 6 in. deep, and the diameter of the cardboard cylinder is 12 in. How tall must the cylinder be to fit in the middle of the dish and be flush with the top of the dish? (See Figure 1.82.) SOLUTION

Solve Algebraically The diagram in Figure 1.82a showing the cross section of this 3-dimensional problem is also a 2-dimensional graph of a quadratic function. We can transform our basic function y = x 2 with a vertical shrink so that it goes through the points 112, 62 and 1- 12, 62, thereby producing a graph of the parabola in the coordinate plane (Figure 1.82b). y = kx 2 6 = k1!1222 1 6 = k = 144 24

Thus, y =

1 2 x . 24

Vertical shrink Substitute x = !12, y = 6. Solve for k.

(continued)

6965_CH01_pp063-156.qxd

142

1/14/10

1:01 PM

Page 142

CHAPTER 1 Functions and Graphs

To find the height of the cardboard cylinder, we first find the y-coordinate of the parabola 6 inches from the center, that is, when x = 6: 1 1622 = 1.5 24

y =

From that point to the top of the dish is 6 - 1.5 = 4.5 in.

Now try Exercise 35.

24 6

(a) y 14 10 (–12, 6)

(12, 6)

6 2

–14 –10

–6

–2

2

6

10

14

x

(b)

FIGURE 1.82 (a) A parabolic satellite dish with a protective cardboard cylinder in the middle for packaging. (b) The parabola in the coordinate plane. (Example 3)

Although Example 3 serves nicely as a “functions from graphs” example, it is also an example of a function that must be constructed by gathering relevant information from a verbal description and putting it together in the right way. People who do mathematics for a living are accustomed to confronting that challenge regularly as a necessary first step in modeling the real world. In honor of its importance, we have saved it until last to close out this chapter in style.

Functions from Verbal Descriptions There is no fail-safe way to form a function from a verbal description. It can be hard work, frequently a good deal harder than the mathematics required to solve the problem once the function has been found. The 4-step problem-solving process in Section 1.1 gives you several valuable tips, perhaps the most important of which is to read the problem carefully. Understanding what the words say is critical if you hope to model the situation they describe.

EXAMPLE 4 Finding the Model and Solving Grain is leaking through a hole in a storage bin at a constant rate of 8 cubic inches per minute. The grain forms a cone-shaped pile on the ground below. As it grows, the height of the cone always remains equal to its radius. If the cone is one foot tall now, how tall will it be in one hour? SOLUTION Reading the problem carefully, we realize that the formula for the volume of the cone is needed (Figure 1.83). From memory or by looking it up, we get the formula V = 11/32pr 2 h. A careful reading also reveals that the height and the radius are always equal, so we can get volume directly as a function of height: V = 11/32ph3.

When h = 12 in., the volume is V = 1p/3211223 = 576p in.3.

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 143

SECTION 1.7

Modeling with Functions

143

One hour later, the volume will have grown by 160 min 218 in.3/min 2 = 480 in.3. The total volume of the pile at that point will be 1576p + 4802 in.3. Finally, we use the volume formula once again to solve for h: FIGURE 1.83 A cone with equal height and radius. (Example 4)

1 ph3 = 576p + 480 3 31576p + 4802 h3 = p 31576p + 4802 h = 3 p B h L 12.98 inches

Now try Exercise 37.

EXAMPLE 5 Letting Units Work for You How many rotations does a 15-in. (radius) tire make per second on a sport utility vehicle traveling 70 mph? SOLUTION It is the perimeter of the tire that comes in contact with the road, so we first find the circumference of the tire:

C = 2pr = 2p1152 = 30p in. This means that 1 rotation = 30p in. From this point we proceed by converting “miles per hour” to “rotations per second” by a series of conversion factors that are really factors of 1: 70 miles 1 hour 1 min 5280 feet 12 inches 1 rotation * * * * * 1 hour 60 min 60 sec 1 mile 1 foot 30p inches 70 * 5280 * 12 rotations = L 13.07 rotations per second 60 * 60 * 30p sec Now try Exercise 39.

Functions from Data In this course we will use the following 3-step strategy to construct functions from data.

Constructing a Function from Data

Given a set of data points of the form 1x, y2, to construct a formula that approximates y as a function of x:

1. Make a scatter plot of the data points. The points do not need to pass the verti-

cal line test. 2. Determine from the shape of the plot whether the points seem to follow the graph of a familiar type of function (line, parabola, cubic, sine curve, etc.). 3. Transform a basic function of that type to fit the points as closely as possible.

Step 3 might seem like a lot of work, and for earlier generations it certainly was; it required all of the tricks of Section 1.6 and then some. We, however, will gratefully use technology to do this “curve-fitting” step for us, as shown in Example 6.

6965_CH01_pp063-156.qxd

144

1/14/10

1:01 PM

Page 144

CHAPTER 1 Functions and Graphs

EXAMPLE 6 Curve-Fitting with Technology Table 1.12 records the low and high daily temperatures observed on 9/9/1999 in 20 major American cities. Find a function that approximates the high temperature 1y2 as a function of the low temperature 1x2. Use this function to predict the high temperature that day for Madison, WI, given that the low was 46. SOLUTION The scatter plot is shown in Figure 1.84.

[45, 90] by [60, 115]

FIGURE 1.84 The scatter plot of the temperature data in Example 6.

Table 1.12

Temperature on 9/9/99

City New York, NY Los Angeles, CA Chicago, IL Houston, TX Philadelphia, PA Albuquerque, NM Phoenix, AZ Atlanta, GA Dallas, TX Detroit, MI

Low

High

City

70 62 52 70 68 61 82 64 65 54

86 80 72 94 86 86 106 90 87 76

Miami, FL Honolulu, HI Seattle, WA Jacksonville, FL Baltimore, MD St. Louis, MO El Paso, TX Memphis, TN Milwaukee, WI Wilmington, DE

Low

High

76 70 50 67 64 57 62 60 52 66

92 85 70 89 88 79 90 86 68 84

Source: AccuWeather, Inc.

[45, 90] by [60, 115]

FIGURE 1.85 The temperature scatter plot with the regression line shown. (Example 6)

Notice that the points do not fall neatly along a well-known curve, but they do seem to fall near an upwardly sloping line. We therefore choose to model the data with a function whose graph is a line. We could fit the line by sight (as we did in Example 5 in Section 1.1), but this time we will use the calculator to find the line of “best fit,” called the regression line. (See your grapher’s owner’s manual for how to do this.) The regression line is found to be approximately y = 0.97x + 23. As Figure 1.85 shows, the line fits the data as well as can be expected. If we use this function to predict the high temperature for the day in Madison, WI, we get y = 0.971462 + 23 = 67.62. (For the record, the high that day was 67.) Now try Exercise 47, parts (a) and (b).

Professional statisticians would be quick to point out that this function should not be trusted as a model for all cities, despite the fairly successful prediction for Madison. (For example, the prediction for San Francisco, with a low of 54 and a high of 64, is off by more than 11 degrees.) The effectiveness of a data-based model is highly dependent on the number of data points and on the way they were selected. The functions we construct from data in this book should be analyzed for how well they model the data, not for how well they model the larger population from which the data came. In addition to lines, we can model scatter plots with several other curves by choosing the appropriate regression option on a calculator or computer. The options to which we will refer in this book (and the chapters in which we will study them) are shown in the following table:

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 145

SECTION 1.7

145

Regression Type

Equation

Linear (Chapter 2)

y = ax + b

Fixed cost plus variable cost, linear growth, free-fall velocity, simple interest, linear depreciation, many others

Quadratic (Chapter 2)

y = ax 2 + bx + c (requires at least 3 points)

Position during free fall, projectile motion, parabolic reflectors, area as a function of linear dimension, quadratic growth, etc.

Cubic (Chapter 2)

y = ax 3 + bx 2 + cx + d (requires at least 4 points)

Volume as a function of linear dimension, cubic growth, miscellaneous applications where quadratic regression does not give a good fit

Quartic (Chapter 2)

y = ax 4 + bx 3 + cx 2 + dx + e (requires at least 5 points)

Quartic growth, miscellaneous applications where quadratic and cubic regression do not give a good fit

Natural logarithmic (ln) (Chapter 3)

y = a + b ln x (requires x 7 0)

Logarithmic growth, decibels (sound), Richter scale (earthquakes), inverse exponential models

Exponential 1b 7 12 (Chapter 3)

Exponential 10 6 b 6 12 (Chapter 3)

Power 1requires x, y 7 02 (Chapter 2)

Logistic (Chapter 3)

Sinusoidal (Chapter 4)

y = a # bx

y = a # bx

y = a # xb

y =

c

1 + a # e - bx

y = a sin 1bx + c2 + d

Graph

Modeling with Functions

Applications

Exponential growth, compound interest, population models

Exponential decay, depreciation, temperature loss of a cooling body, etc.

Inverse-square laws, Kepler’s Third Law

Logistic growth: spread of a rumor, population models

Periodic behavior: harmonic motion, waves, circular motion, etc.

6965_CH01_pp063-156.qxd

146

1/14/10

1:01 PM

Page 146

CHAPTER 1 Functions and Graphs

Displaying Diagnostics If your calculator is giving regression formulas but not displaying the values of r or r 2 or R2, you may be able to fix that. Go to the CATALOG menu and choose a command called “DiagnosticOn.” Enter the command on the home screen and see the reply “Done.” Your next regression should display the diagnostic values.

These graphs are only examples, as they can vary in shape and orientation. (For example, any of the curves could appear upside-down.) The grapher uses various strategies to fit these curves to the data, most of them based on combining function composition with linear regression. Depending on the regression type, the grapher may display a number r called the correlation coefficient or a number r 2 or R 2 called the coefficient of determination. In either case, a useful “rule of thumb” is that the closer the absolute value of this number is to 1, the better the curve fits the data. We can use this fact to help choose a regression type, as in Exploration 1.

EXPLORATION 1

n = 3; d = 0

n = 4; d = !

Diagonals of a Regular Polygon

How many diagonals does a regular polygon have? Can the number be expressed as a function of the number of sides? Try this Exploration. 1. Draw in all the diagonals (i.e., segments connecting nonadjacent points) in

n = 5; d = !

n = 6; d = !

each of the regular polygons shown and fill in the number (d) of diagonals in the space below the figure. The values of d for the triangle 1n = 32 and the decagon 1n = 102 are filled in for you.

2. Put the values of n in list L1, beginning with n = 4. (We want to avoid that

y = 0 value for some of our regressions later.) Put the corresponding values of d in list L2. Display a scatter plot of the ordered pairs.

3. The graph shows an increasing function with some curvature, but it is not clear n = 7; d = !

n = 8; d = !

which kind of growth would fit it best. Try these regressions (preferably in the given order) and record the value of r 2 or R2 for each: linear, power, quadratic, cubic, quartic. (Note that the curvature is not right for logarithmic, logistic, or sinusoidal curve-fitting, so it is not worth it to try those.) 4. What kind of curve is the best fit? (It might appear at first that there is a tie,

but look more closely at the functions you get.) How good is the fit? n = 9; d = !

n = 10; d = 35

FIGURE 1.86 Some polygons. (Exploration 1)

5. Looking back, could you have predicted the results of the cubic and quartic

regressions after seeing the result of the quadratic regression? 6. The “best-fit” curve gives the actual formula for d as a function of n. (In

Chapter 9 you will learn how to derive this formula for yourself.) Use the formula to find the number of diagonals of a 128-gon.

We will have more to say about curve fitting as we study the various function types in later chapters.

Chapter Opener Problem (from page 63) Problem: The table below shows the growth in the Consumer Price Index (CPI) for housing for selected years between 1990 and 2007 (based on 1983 dollars). How can we construct a function to predict the housing CPI for the years 2008–2015?

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 147

SECTION 1.7

Modeling with Functions

147

Consumer Price Index (Housing) Year

Housing CPI

1990 1995 2000 2002 2003 2004 2005 2006 2007

128.5 148.5 169.6 180.3 184.8 189.5 195.7 203.2 209.6

Source: Bureau of Labor Statistics, quoted in The World Almanac and Book of Facts 2009.

Solution: A scatter plot of the data is shown in Figure 1.87, where x is the number of years since 1990. A linear model would work pretty well, but the slight upward curve of the scatter plot suggests that a quadratic model might work better. Using a calculator to compute the quadratic regression curve, we find its equation to be y = 0.089x 2 + 3.17x + 129. As Figure 1.88 shows, the parabola fits the data impressively well.

[–2, 19] by [115, 225]

[–2, 19] by [115, 225]

FIGURE 1.87 Scatter plot of the

FIGURE 1.88 Scatter plot with the

data for the housing CPI.

regression curve shown.

To predict the housing CPI for 2008, use x = 18 in the regression equation. Similarly, we can predict the housing CPI for each of the years 2008–2015, as shown below: Predicted CPI (Housing) Year

Predicted Housing CPI

2008 2009 2010 2011 2012 2013 2014 2015

214.9 221.4 228.0 234.8 241.8 249.0 256.3 263.9

Even with a regression curve that fits the data as beautifully as in Figure 1.88, it is risky to predict this far beyond the data set. Statistics like the CPI are dependent on many volatile factors that can quickly render any mathematical model obsolete. In fact, the mortgage model that fueled the housing growth up to 2007 proved to be unsustainable, and when it broke down it took many well-behaved economic curves (like this one) down with it. In light of that fact, you might enjoy comparing these “predictions” with the actual housing CPI numbers as the years go by!

6965_CH01_pp063-156.qxd

148

1/14/10

1:01 PM

Page 148

CHAPTER 1 Functions and Graphs

QUICK REVIEW 1.7

(For help, go to Section P.3 and P.4.)

In Exercises 1–10, solve the given formula for the given variable. 1. Area of a Triangle Solve for h: A =

1 bh 2

2. Area of a Trapezoid Solve for h: A =

1 1b + b22h 2 1

3. Volume of a Right Circular Cylinder Solve for h: V = pr 2h 4. Volume of a Right Circular Cone Solve for h: 1 V = pr 2h 3 5. Volume of a Sphere Solve for r: V =

6. Surface Area of a Sphere Solve for r: A = 4pr 2 7. Surface Area of a Right Circular Cylinder Solve for h: A = 2prh + 2pr 2 8. Simple Interest Solve for t: I = Prt 9. Compound Interest Solve for P: A = P a1 +

10. Free-Fall from Height H 1 t: s = H - gt 2 2

Solve for

r nt b n

4 pr 3 3

SECTION 1.7 EXERCISES Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 1–10, write a mathematical expression for the quantity described verbally. 1. Five more than three times a number x 2. A number x increased by 5 and then tripled 3. Seventeen percent of a number x 4. Four more than 5% of a number x 5. Area of a Rectangle The area of a rectangle whose length is 12 more than its width x 6. Area of a Triangle The area of a triangle whose altitude is 2 more than its base length x 7. Salary Increase A salary after a 4.5% increase, if the original salary is x dollars 8. Income Loss Income after a 3% drop in the current income of x dollars 9. Sale Price Sale price of an item marked x dollars, if 40% is discounted from the marked price 10. Including Tax Actual cost of an item selling for x dollars if the sales tax rate is 8.75% In Exercises 11–14, choose a variable and write a mathematical expression for the quantity described verbally. 11. Total Cost The total cost is $34,500 plus $5.75 for each item produced. 12. Total Cost The total cost is $28,000 increased by 9% plus $19.85 for each item produced.

13. Revenue The revenue when each item sells for $3.75 14. Profit The profit consists of a franchise fee of $200,000 plus 12% of all sales. In Exercises 15–20, write the specified quantity as a function of the specified variable. It will help in each case to draw a picture. 15. The height of a right circular cylinder equals its diameter. Write the volume of the cylinder as a function of its radius. 16. One leg of a right triangle is twice as long as the other. Write the length of the hypotenuse as a function of the length of the shorter leg. 17. The base of an isosceles triangle is half as long as the two equal sides. Write the area of the triangle as a function of the length of the base. 18. A square is inscribed in a circle. Write the area of the square as a function of the radius of the circle. 19. A sphere is contained in a cube, tangent to all six faces. Find the surface area of the cube as a function of the radius of the sphere. 20. An isosceles triangle has its base along the x-axis with one base vertex at the origin and its vertex in the first quadrant on the graph of y = 6 - x 2. Write the area of the triangle as a function of the length of the base. In Exercises 21–36, write an equation for the problem and solve the problem. 21. One positive number is 4 times another positive number. The sum of the two numbers is 620. Find the two numbers.

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 149

SECTION 1.7

Modeling with Functions

149

22. When a number is added to its double and its triple, the sum is 714. Find the three numbers. 23. Salary Increase Mark received a 3.5% salary increase. His salary after the raise was $36,432. What was his salary before the raise? 24. Consumer Price Index The Consumer Price Index for food and beverages in 2007 was 203.3 after a hefty 3.9% increase from the previous year. What was the Consumer Price Index for food and beverages in 2006? (Source: U.S. Bureau of Labor Statistics) 25. Travel Time A traveler averaged 52 miles per hour on a 182-mile trip. How many hours were spent on the trip? 26. Travel Time On their 560-mile trip, the Bruins basketball team spent two more hours on the interstate highway than they did on local highways. They averaged 45 mph on local highways and 55 mph on the interstate highways. How many hours did they spend driving on local highways? 27. Sale Prices At a shirt sale, Jackson sees two shirts that he likes equally well. Which is the better bargain, and why?

3 $3

Solution 1 x gallons 10%

#

Solution 2 (100 " x) gallons 45%

$

Combined solution 100 gallons 25%

(a) Write an equation that models this problem. (b) Solve the equation graphically. 32. Mixing Solutions The chemistry lab at the University of Hardwoods keeps two acid solutions on hand. One is 20% acid and the other is 35% acid. How much 20% acid solution and how much 35% acid solution should be used to prepare 25 liters of a 26% acid solution? 33. Maximum Value Problem A square of side x inches is cut out of each corner of a 10 in. by 18 in. piece of cardboard and the sides are folded up to form an open-topped box. (a) Write the volume V of the box as a function of x. (b) Find the domain of your function, taking into account the restrictions that the model imposes in x. (c) Use your graphing calculator to determine the dimensions of the cut-out squares that will produce the box of maximum volume. 34. Residential Construction DDL Construction is building a rectangular house that is 16 feet longer than it is wide. A rain gutter is to be installed in four sections around the 136-foot perimeter of the house. What lengths should be cut for the four sections?

40% off

7 $2

25% off

28. Job Offers Ruth is weighing two job offers from the sales departments of two competing companies. One offers a base salary of $25,000 plus 5% of gross sales; the other offers a base salary of $20,000 plus 7% of gross sales. What would Ruth’s gross sales total need to be to make the second job offer more attractive than the first? 29. Cell Phone Antennas From December 2006 to December 2007, the number of cell phone antennas in the United States grew from 195,613 to 213,299. What was the percentage increase in U.S. cell phone antennas in that one-year period? (Source: CTIA, quoted in The World Almanac and Book of Facts 2009) 30. Cell phone Antennas From December 1996 to December 1997, the number of cell phone antennas in the United States grew from 30,045 to 51,600. What was the percentage increase in U.S. cell phone antennas in that one-year period? (Source: CTIA, quoted in The World Almanac and Book of Facts 2009) 31. Mixing Solutions How much 10% solution and how much 45% solution should be mixed together to make 100 gallons of 25% solution?

35. Protecting an Antenna In Example 3, suppose the parabolic dish has a 32-in. diameter and is 8 in. deep, and the radius of the cardboard cylinder is 8 in. Now how tall must the cylinder be to fit in the middle of the dish and be flush with the top of the dish? 36. Interior Design Renée’s Decorating Service recommends putting a border around the top of the four walls in a dining room that is 3 feet longer than it is wide. Find the dimensions of the room if the total length of the border is 54 feet. 37. Finding the Model and Solving Water is stored in a conical tank with a faucet at the bottom. The tank has depth 24 in. and radius 9 in., and it is filled to the brim. If the faucet is opened to allow the water to flow at a rate of 5 cubic inches per second, what will the depth of the water be after 2 minutes? 38. Investment Returns Reggie invests $12,000, part at 7% annual interest and part at 8.5% annual interest. How much is invested at each rate if Reggie’s total annual interest is $900? 39. Unit Conversion A tire of a moving bicycle has radius 16 in. If the tire is making 2 rotations per second, find the bicycle’s speed in miles per hour.

6965_CH01_pp063-156.qxd

150

1/14/10

1:01 PM

Page 150

CHAPTER 1 Functions and Graphs

40. Investment Returns Jackie invests $25,000, part at 5.5% annual interest and the balance at 8.3% annual interest. How much is invested at each rate if Jackie receives a 1-year interest payment of $1571?

Standardized Test Questions 41. True or False A correlation coefficient gives an indication of how closely a regression line or curve fits a set of data. Justify your answer. 42. True or False Linear regression is useful for modeling the position of an object in free fall. Justify your answer. In Exercises 43–46, tell which type of regression is likely to give the most accurate model for the scatter plot shown without using a calculator. (A) Linear regression (B) Quadratic regression (C) Cubic regression (D) Exponential regression

(a) Find an equation that models the cost of producing x pairs of shoes. (b) Find an equation that models the revenue produced from selling x pairs of shoes. (c) Find how many pairs of shoes must be made and sold in order to break even. (d) Graph the equations in (a) and (b). What is the graphical interpretation of the answer in (c)? 48. Employee Benefits John’s company issues employees a contract that identifies salary and the company’s contributions to pension, health insurance premiums, and disability insurance. The company uses the following formulas to calculate these values. Salary

x (dollars)

Pension Health insurance Disability insurance

12% of salary 3% of salary 0.4% of salary

If John’s total contract with benefits is worth $48,814.20, what is his salary?

(E) Sinusoidal regression

49. Manufacturing Queen, Inc., a tennis racket manufacturer, determines that the annual cost C of making x rackets is $23 per racket plus $125,000 in fixed overhead costs. It costs the company $8 to string a racket.

43. Multiple Choice

[0, 12] by [0, 8]

(a) Find a function y1 = u1x2 that models the cost of producing x unstrung rackets. (b) Find a function y2 = s1x2 that models the cost of producing x strung rackets.

44. Multiple Choice

[0, 12] by [0, 8]

45. Multiple Choice

6 $5

$79 [0, 12] by [0, 8]

46. Multiple Choice (c) Find a function y3 = Ru1x2 that models the revenue generated by selling x unstrung rackets. [0, 12] by [0, 8]

Exploration 47. Manufacturing The Buster Green Shoe Company determines that the annual cost C of making x pairs of one type of shoe is $30 per pair plus $100,000 in fixed overhead costs. Each pair of shoes that is manufactured is sold wholesale for $50.

(d) Find a function y4 = Rs1x2 that models the revenue generated by selling x rackets. (e) Graph y1, y2, y3, and y4 simultaneously in the window [0, 10,000] by [0, 500,000].

(f) Writing to Learn Write a report to the company recommending how they should manufacture their rackets, strung or unstrung. Assume that you can include the viewing window in (e) as a graph in the report, and use it to support your recommendation.

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 151

SECTION 1.7

50. Hourly Earnings of U.S. Production Workers The average hourly earnings of U.S. production workers for 1990–2007 are shown in Table 1.13.

Table 1.13

Average Hourly Earnings

Year

Average Hourly Earnings ($)

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

10.20 10.52 10.77 11.05 11.34 11.65 12.04 12.51 13.01 13.49 14.02 14.54 14.97 15.37 15.69 16.13 16.76 17.42

Source: Bureau of Labor Statistics as quoted in The World Almanac and Book of Facts 2009.

(a) Produce a scatter plot of the hourly earnings 1y2 as a function of years since 1990 1x2. (b) Find the linear regression equation for the years 1990–1998. Round the coefficients to the nearest 0.001. (c) Find the linear regression equation for the years 1990–2007. Round the coefficients to the nearest 0.001. (d) Use both lines to predict the hourly earnings for the year 2010. How different are the estimates? Which do you think is a safer prediction of the true value?

(e) Writing to Learn Use the results of parts (a)–(d) to explain why it is risky to predict y-values for x-values that are not very close to the data points, even when the regression plot fits the data points quite well.

Extending the Ideas 51. Newton’s Law of Cooling A 190° cup of coffee is placed on a desk in a 72° room. According to Newton’s Law of Cooling, the temperature T of the coffee after t minutes will be T = 1190 - 722b t + 72, where b is a constant that depends on how easily the cooling substance loses heat. The data in Table 1.14 are from a simulated experiment of gathering temperature readings from a cup of coffee in a 72° room at 20 one-minute intervals:

Table 1.14

Modeling with Functions

151

Cooling a Cup of Coffee

Time

Temp

Time

Temp

1 2 3 4 5 6 7 8 9 10

184.3 178.5 173.5 168.6 164.0 159.2 155.1 151.8 147.0 143.7

11 12 13 14 15 16 17 18 19 20

140.0 136.1 133.5 130.5 127.9 125.0 122.8 119.9 117.2 115.2

(a) Make a scatter plot of the data, with the times in list L1 and the temperatures in list L2. (b) Store L2 - 72 in list L3. The values in L3 should now be an exponential function 1 y = a * b x2 of the values in L1. (c) Find the exponential regression equation for L3 as a function of L1. How well does it fit the data?

52. Group Activity Newton’s Law of Cooling If you have access to laboratory equipment (such as a CBL or CBR unit for your grapher), gather experimental data such as in Exercise 51 from a cooling cup of coffee. Proceed as follows: (a) First, use the temperature probe to record the temperature of the room. It is a good idea to turn off fans and air conditioners that might affect the temperature of the room during the experiment. It should be a constant. (b) Heat the coffee. It need not be boiling, but it should be at least 160°. (It also need not be coffee.) (c) Make a new list consisting of the temperature values minus the room temperature. Make a scatter plot of this list 1y2 against the time values 1x2. It should appear to approach the x-axis as an asymptote. (d) Find the equation of the exponential regression curve. How well does it fit the data? (e) What is the equation predicted by Newton’s Law of Cooling? (Substitute your initial coffee temperature and the temperature of your room for the 190 and 72 in the equation in Exercise 51.) (f) Group Discussion What sort of factors would affect the value of b in Newton’s Law of Cooling? Discuss your ideas with the group.

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 152

CHAPTER 1 Functions and Graphs

152

CHAPTER 1 Key Ideas Properties, Theorems, and Formulas

Reflections of Graphs Across Axes 131 Stretches and Shrinks of Graphs 134

The Zero Factor Property 69 Vertical Line Test 81 Tests for an Even Function 90 Tests for an Odd Function 91 Horizontal Line Test 122 Inverse Reflection Principle 123 Inverse Composition Rule 124 Translations of Graphs 130

Procedures Root, Zero, x-intercept 70 Problem Solving 70 Agreement about Domain 82 Inverse Notation 122 Finding the Inverse of a Function

123

CHAPTER 1 Review Exercises The collection of exercises marked in red could be used as a chapter test.

3.

y

In Exercises 1–10, match the graph with the corresponding function (a)–( j) from the list below. Use your knowledge of function behavior, not your grapher. (a) ƒ1x2 = x 2 - 1

(b) ƒ1x2 = x 2 + 1 2

x 2

(c) ƒ1x2 = 1x - 22

(d) ƒ1x2 = 1x + 22

(g) ƒ1x2 = ƒ x + 2 ƒ

(h) ƒ1x2 = - sin x

(i) ƒ1x2 = ex - 1

(j) ƒ1x2 = 1 + cos x

(e) ƒ1x2 =

1.

x - 1 2

(f) ƒ1x2 = ƒ x - 2 ƒ

4.

y

y

x x

5. 2.

y

y

x x

6965_CH01_pp063-156.qxd

1/21/10

7:18 AM

Page 153

CHAPTER 1

6.

153

In Exercises 11–18, find (a) the domain and (b) the range of the function.

y

x

11. g1x2 = x 3

12. ƒ1x2 = 35x - 602

13. g1x2 = x 2 + 2x + 1

14. h1x2 = 1x - 222 + 5

15. g1x2 = 3 ƒ x ƒ + 8

y

16. k1x2 = - 24 - x 2

x 1 18. k1x2 = x 2 - 2x 29 - x 2 In Exercises 19 and 20, graph the function, and state whether the function is continuous at x = 0. If it is discontinuous, state whether the discontinuity is removable or nonremovable. 17. ƒ1x2 =

7.

Review Exercises

19. ƒ1x2 =

x2 - 3 x + 2

20. k1x2 = e

2x + 3 if x 7 0 3 - x 2 if x … 0

In Exercises 21–24, find all (a) vertical asymptotes and (b) horizontal asymptotes of the graph of the function. Be sure to state your answers as equations of lines. x

21. y = 23. y =

8.

5 2

x - 5x 7x 2

22. y =

3x x - 4

24. y =

ƒxƒ x + 1

2x + 10 In Exercises 25–28, graph the function and state the intervals on which the function is increasing.

y

25. y = x

27. y =

x3 6

26. y = 2 + ƒ x - 1 ƒ x

1 - x

2

28. y =

x2 - 1 x2 - 4

In Exercises 29–32, graph the function and tell whether the function is bounded above, bounded below, or bounded.

9.

6x

29. ƒ1x2 = x + sin x

30. g1x2 =

31. h1x2 = 5 - ex

32. k1x2 = 1000 +

y

x2 + 1 x 1000

In Exercises 33–36, use a grapher to find all (a) relative maximum values and (b) relative minimum values of the function. Also state the value of x at which each relative extremum occurs. x

33. y = 1x + 122 - 7 35. y =

10.

x2 + 4 2

x - 4

34. y = x 3 - 3x 36. y =

4x 2

x + 4

In Exercises 37–40, graph the function and state whether the function is odd, even, or neither.

y

37. y = 3x 2 - 4 ƒ x ƒ 39. y = x

x ex

38. y = sin x - x 3 40. y = x cos 1x2

In Exercises 41–44, find a formula for ƒ - 11x2. 41. ƒ1x2 = 2x + 3 43. ƒ1x2 =

2 x

3x - 8 42. ƒ1x2 = 2 44. ƒ1x2 =

6 x + 4

6965_CH01_pp063-156.qxd

154

1/14/10

1:01 PM

Page 154

CHAPTER 1 Functions and Graphs

Exercises 45–52 refer to the function y = ƒ1x2 whose graph is given below. y

64. Draining a Cylindrical Tank A cylindrical tank with diameter 20 feet is filled with oil to a depth of 40 feet. The oil begins draining so that the depth of oil in the tank decreases at a constant rate of 2 feet per hour. Write the volume of oil remaining in the tank t hours later as a function of t.

4 3 2 –5 –4

–2 –1 –1 –2 –3 –4

The oil begins draining at a constant rate of 2 cubic feet per second. Write the depth of the oil remaining in the tank t seconds later as a function of t.

1 2 3 4 5

x

45. Sketch the graph of y = ƒ1x2 - 1. 46. Sketch the graph of y = ƒ1x - 12. 47. Sketch the graph of y = ƒ1- x2. 48. Sketch the graph of y = - ƒ1x2. 49. Sketch a graph of the inverse relation. 50. Does the inverse relation define y as a function of x? 51. Sketch a graph of y = ƒ ƒ1x2 ƒ . 52. Define ƒ algebraically as a piecewise function. [Hint: the pieces are translations of two of our “basic” functions.] In Exercises 53–58, let ƒ1x2 = 2x and let g1x2 = x 2 - 4.

53. Find an expression for 1ƒ ! g21x2 and give its domain. 54. Find an expression for 1g ! ƒ21x2 and give its domain. 55. Find an expression for 1ƒg21x2 and give its domain. ƒ 56. Find an expression for a b(x) and give its domain. g

57. Describe the end behavior of the graph of y = ƒ1x2. 58. Describe the end behavior of the graph of y = ƒ1g1x22. In Exercises 59–64, write the specified quantity as a function of the specified variable. Remember that drawing a picture will help. 59. Square Inscribed in a Circle A square of side s is inscribed in a circle. Write the area of the circle as a function of s. 60. Circle Inscribed in a Square A circle is inscribed in a square of side s. Write the area of the circle as a function of s. 61. Volume of a Cylindrical Tank A cylindrical tank with diameter 20 feet is partially filled with oil to a depth of h feet. Write the volume of oil in the tank as a function of h. 62. Draining a Cylindrical Tank A cylindrical tank with diameter 20 feet is filled with oil to a depth of 40 feet. The oil begins draining at a constant rate of 2 cubic feet per second. Write the volume of the oil remaining in the tank t seconds later as a function of t. 63. Draining a Cylindrical Tank A cylindrical tank with diameter 20 feet is filled with oil to a depth of 40 feet.

65. U.S. Crude Oil Imports The imports of crude oil to the United States from Canada in the years 2000–2008 (in thousands of barrels per day) are given in Table 1.15.

Table 1.15

Crude Oil Imports from Canada

Year

Barrels/day * 1000

2000 2001 2002 2003 2004 2005 2006 2007 2008

1267 1297 1418 1535 1587 1602 1758 1837 1869

Source: Energy Information Administration, Petroleum Supply Monthly, as reported in The World Almanac and Book of Facts 2009.

(a) Sketch a scatter plot of import numbers in the right-hand column 1y2 as a function of years since 2000 1x2. (b) Find the equation of the regression line and superimpose it on the scatter plot.

(c) Based on the regression line, approximately how many thousands of barrels of oil would the United States import from Canada in 2015? 66. The winning times in the women’s 100-meter freestyle event at the Summer Olympic Games since 1956 are shown in Table 1.16.

Table 1.16

Women’s 100-Meter Freestyle

Year

Time

Year

Time

1956 1960 1964 1968 1972 1976 1980

62.0 61.2 59.5 60.0 58.59 55.65 54.79

1984 1988 1992 1996 2000 2004 2008

55.92 54.93 54.64 54.50 53.83 53.84 53.12

Source: The World Almanac and Book of Facts 2009.

(a) Sketch a scatter plot of the times 1y2 as a function of the years 1x2 beyond 1900. (The values of x will run from 56 to 108.) (b) Explain why a linear model cannot be appropriate for these times over the long term.

6965_CH01_pp063-156.qxd

1/14/10

1:01 PM

Page 155

CHAPTER 1

(c) The points appear to be approaching a horizontal asymptote of y = 52. What would this mean about the times in this Olympic event? (d) Subtract 52 from all the times so that they will approach an asymptote of y = 0. Redo the scatter plot with the new y-values. Now find the exponential regression curve and superimpose its graph on the vertically shifted scatter plot. (e) According to the regression curve, what will be the winning time in the women’s 100-meter freestyle event at the 2016 Olympics? 67. Inscribing a Cylinder Inside a Sphere A right circular cylinder of radius r and height h is inscribed inside a sphere of radius 23 inches.

Review Exercises

155

(b) Write the volume V of the cylinder as a function of r. (c) What values of r are in the domain of V? (d) Sketch a graph of V1r2 over the domain 30, 234.

(e) Use your grapher to find the maximum volume that such a cylinder can have. 68. Inscribing a Rectangle Under a Parabola A rectangle is inscribed between the x-axis and the parabola y = 36 - x 2 with one side along the x-axis, as shown in the figure below. y

(a) Use the Pythagorean Theorem to write h as a function of r.

x

h

3

r

x

(a) Let x denote the x-coordinate of the point highlighted in the figure. Write the area A of the rectangle as a function of x. (b) What values of x are in the domain of A? (c) Sketch a graph of A1x2 over the domain. (d) Use your grapher to find the maximum area that such a rectangle can have.

6965_CH01_pp063-156.qxd

156

1/14/10

1:01 PM

Page 156

CHAPTER 1 Functions and Graphs

CHAPTER 1

Project

Modeling the Growth of a Business In 1971, Starbucks Coffee opened its first location in Pike Place Market—Seattle’s legendary open-air farmer’s market. By 1987, the number of Starbucks stores had grown to 17 and by 2005 there were 10,241 locations. The data in the table below (obtained from Starbucks Coffee’s Web site, www.starbucks.com) summarize the growth of this company from 1987 through 2005. Year

Number of Locations

1987 1989 1991 1993 1995 1997 1999 2001 2003 2005

17 55 116 272 676 1412 2498 4709 7225 10,241

EXPLORATIONS 1. Enter the data in the table into your grapher or computer.

(Let t = 0 represent 1987.) Draw a scatter plot for the data. 2. Refer to page 145 in this chapter. Look at the types of

graphs displayed and the associated regression types. Notice that the exponential regression model with b 7 1 seems to most closely match the plotted data. Use your grapher or computer to find an exponential regression

equation to model this data set (see your grapher’s guidebook for instructions on how to do this). 3. Use the exponential model you just found to predict the to-

tal number of Starbucks locations for 2007 and 2008. 4. There were 15,011 Starbucks locations in 2007 and 16,680

locations in 2008. (You can verify these numbers and find more up-to-date information in the investors’ section of the Starbucks Web site.) Why is there such a big difference between your predicted values and the actual number of Starbucks locations? What real-world feature of business growth was not accounted for in the exponential growth model? 5. You need to model the data set with an equation that takes

into account the fact that growth of a business eventually levels out or reaches a carrying capacity. Refer to page 145 again. Notice that the logistic regression modeling graph appears to show exponential growth at first, but eventually levels out. Use your grapher or computer to find the logistic regression equation to model this data set (see your grapher’s guidebook for instructions on how to do this). 6. Use the logistic model you just found to predict the total

number of Starbucks locations for 2007 and 2008. How do your predictions compare with the actual number of locations for 2007 and 2008? How many locations do you think there will be in the year 2020?

6965_CH02_pp157-250.qxd

1/14/10

1:13 PM

Page 157

CHAPTER 2

Polynomial, Power, and Rational Functions 2.1 Linear and Quadratic

Functions and Modeling

2.2 Power Functions with

Modeling

2.3 Polynomial Functions

of Higher Degree with Modeling

2.4 Real Zeros of Polynomial

Functions

2.5 Complex Zeros and the

Fundamental Theorem of Algebra

2.6 Graphs of Rational

Functions

2.7 Solving Equations in

One Variable

2.8 Solving Inequalities

in One Variable

Humidity and relative humidity are measures used by weather forecasters. Humidity affects our comfort and our health. If humidity is too low, our skin can become dry and cracked, and viruses can live longer. If humidity is too high, it can make warm temperatures feel even warmer, and mold, fungi, and dust mites can live longer. See page 224 to learn how relative humidity is modeled as a rational function.

157

6965_CH02_pp157-250.qxd

158

1/14/10

1:13 PM

Page 158

CHAPTER 2 Polynomial, Power, and Rational Functions

Chapter 2 Overview Chapter 1 laid a foundation of the general characteristics of functions, equations, and graphs. In this chapter and the next two, we will explore the theory and applications of specific families of functions. We begin this exploration by studying three interrelated families of functions: polynomial, power, and rational functions. These three families of functions are used in the social, behavioral, and natural sciences. This chapter includes a thorough study of the theory of polynomial equations. We investigate algebraic methods for finding both real- and complex-number solutions of such equations and relate these methods to the graphical behavior of polynomial and rational functions. The chapter closes by extending these methods to inequalities in one variable.

What you’ll learn about • Polynomial Functions • Linear Functions and Their Graphs • Average Rate of Change • Linear Correlation and Modeling • Quadratic Functions and Their Graphs • Applications of Quadratic Functions

... and why Many business and economic problems are modeled by linear functions. Quadratic and higher-degree polynomial functions are used in science and manufacturing applications.

2.1 Linear and Quadratic Functions and Modeling Polynomial Functions Polynomial functions are among the most familiar of all functions.

DEFINITION Polynomial Function Let n be a nonnegative integer and let a0, a1, a2, Á , an - 1, an be real numbers with an Z 0. The function given by ƒ(x) = an x n + an - 1x n - 1 + Á + a2x 2 + a1x + a0 is a polynomial function of degree n. The leading coefficient is an. The zero function ƒ(x) = 0 is a polynomial function. It has no degree and no leading coefficient.

Polynomial functions are defined and continuous on all real numbers. It is important to recognize whether a function is a polynomial function.

EXAMPLE 1 Identifying Polynomial Functions Which of the following are polynomial functions? For those that are polynomial functions, state the degree and leading coefficient. For those that are not, explain why not. 1 2 4 (c) h1x2 = 29x + 16x 2 (a) ƒ1x2 = 4x 3 - 5x -

(b) g1x2 = 6x - 4 + 7 (d) k1x2 = 15x - 2x 4

SOLUTION

(a) ƒ is a polynomial function of degree 3 with leading coefficient 4. (b) g is not a polynomial function because of the exponent -4. (c) h is not a polynomial function because it cannot be simplified into polynomial form. Notice that 29x 4 + 16x 2 Z 3x 2 + 4x. (d) k is a polynomial function of degree 4 with leading coefficient - 2. Now try Exercise 1.

6965_CH02_pp157-250.qxd

1/14/10

1:13 PM

Page 159

SECTION 2.1

Linear and Quadratic Functions and Modeling

159

The zero function and all constant functions are polynomial functions. Some other familiar functions are also polynomial functions, as shown below.

Polynomial Functions of No and Low Degree Name

Form

Degree

Zero function

ƒ1x2 = 0

Undefined

Constant function

ƒ1x2 = a 1a Z 02

0

ƒ1x2 = ax 2 + bx + c 1a Z 02

2

ƒ1x2 = ax + b 1a Z 02

Linear function Quadratic function

1

We study polynomial functions of degree 3 and higher in Section 2.3. For the remainder of this section, we turn our attention to the nature and uses of linear and quadratic polynomial functions.

Linear Functions and Their Graphs Linear equations and graphs of lines were reviewed in Sections P.3 and P.4, and some of the examples in Chapter 1 involved linear functions. We now take a closer look at the properties of linear functions. A linear function is a polynomial function of degree 1 and so has the form ƒ1x2 = ax + b, where a and b are constants and a Z 0. If we use m for the leading coefficient instead of a and let y = ƒ1x2, then this equation becomes the familiar slope-intercept form of a line: y = mx + b Vertical lines are not graphs of functions because they fail the vertical line test, and horizontal lines are graphs of constant functions. A line in the Cartesian plane is the graph of a linear function if and only if it is a slant line, that is, neither horizontal nor vertical. We can apply the formulas and methods of Section P.4 to problems involving linear functions.

EXAMPLE 2 Finding an Equation of a Linear Function Write an equation for the linear function ƒ such that ƒ1- 12 = 2 and ƒ132 = -2. Surprising Fact Not all lines in the Cartesian plane are graphs of linear functions.

SOLUTION

Solve Algebraically We seek a line through the points 1-1, 22 and 13, - 22. The slope is m =

1- 22 - 2 y2 - y1 -4 = = = -1. x2 - x1 3 - 1-12 4

Using this slope and the coordinates of (- 1, 2) with the point-slope formula, we have y - y1 y - 2 y - 2 y

= m1x - x 12

= - 11x - 1- 122 = -x - 1 = -x + 1

Converting to function notation gives us the desired form: ƒ1x2 = - x + 1

(continued)

6965_CH02_pp157-250.qxd

160

1/21/10

7:20 AM

Page 160

CHAPTER 2 Polynomial, Power, and Rational Functions

y

Support Graphically We can graph y = - x + 1 and see that it includes the points 1-1, 22 and 13, -22 (Figure 2.1).

3 (–1, 2) 2 1 –5 –4 –3 –2 –1 –1 –2 –3

1 2 3 4 5 (3, –2)

x

FIGURE 2.1 The graph of y = -x + 1 passes through 1 - 1, 22 and 13, - 22. (Example 2)

Confirm Numerically

Using ƒ1x2 = - x + 1 we prove that ƒ1-12 = 2 and ƒ132 = -2: ƒ1-12 = -1-12 + 1 = 1 + 1 = 2, and ƒ132 = - 132 + 1 = -3 + 1 = - 2. Now try Exercise 7.

Average Rate of Change Another property that characterizes a linear function is its rate of change. The average rate of change of a function y = ƒ1x2 between x = a and x = b, a Z b, is ƒ1b2 - ƒ1a2 b - a

.

You are asked to prove the following theorem in Exercise 85.

THEOREM Constant Rate of Change A function defined on all real numbers is a linear function if and only if it has a constant nonzero average rate of change between any two points on its graph.

Because the average rate of change of a linear function is constant, it is called simply the rate of change of the linear function. The slope m in the formula ƒ1x2 = mx + b is the rate of change of the linear function. In Exploration 1, we revisit Example 7 of Section P.4 in light of the rate of change concept. EXPLORATION 1

Modeling Depreciation with a Linear Function

Camelot Apartments bought a $50,000 building and for tax purposes are depreciating it $2000 per year over a 25-yr period using straight-line depreciation. 1. What is the rate of change of the value of the building? 2. Write an equation for the value v1t2 of the building as a linear function of the

time t since the building was placed in service. 3. Evaluate v102 and v1162. 4. Solve v1t2 = 39,000.

Rate and Ratio All rates are ratios, whether expressed as miles per hour, dollars per year, or even rise over run.

The rate of change of a linear function is the signed ratio of the corresponding line’s rise over run. That is, for a linear function ƒ1x2 = mx + b, rate of change = slope = m =

change in y ¢y rise = = . run change in x ¢x

This formula allows us to interpret the slope, or rate of change, of a linear function numerically. For instance, in Exploration 1 the value of the apartment building fell from $50,000 to $0 over a 25-yr period. In Table 2.1 we compute ¢y/¢x for the apartment building’s value (in dollars) as a function of time (in years). Because the average rate of change ¢y/¢x is the nonzero constant -2000, the building’s value is a linear function of time decreasing at a rate of $2000/yr.

6965_CH02_pp157-250.qxd

1/14/10

1:13 PM

Page 161

SECTION 2.1

Linear and Quadratic Functions and Modeling

161

Table 2.1 Rate of Change of the Value of the Apartment Building in Exploration 1: y ! "2000x # 50,000 x (time)

y (value)

0

50,000

1

48,000

2

46,000

3

44,000

4

42,000

¢x

¢y

¢y/¢x

1

-2000

-2000

1

-2000

-2000

1

-2000

-2000

1

-2000

-2000

In Exploration 1, as in other applications of linear functions, the constant term represents the value of the function for an input of 0. In general, for any function ƒ, ƒ102 is the initial value of ƒ. So for a linear function ƒ1x2 = mx + b, the constant term b is the initial value of the function. For any polynomial function ƒ1x2 = anx n + Á + a1x + a0, the constant term ƒ102 = a0 is the function’s initial value. Finally, the initial value of any function—polynomial or otherwise—is the y-intercept of its graph. We now summarize what we have learned about linear functions.

Characterizing the Nature of a Linear Function Point of View

Characterization

Verbal

polynomial of degree 1

Algebraic

ƒ(x) = mx + b 1m Z 02

Graphical

slant line with slope m and y-intercept b

Analytical

function with constant nonzero rate of change m: ƒ is increasing if m 7 0, decreasing if m 6 0; initial value of the function = ƒ102 = b

Linear Correlation and Modeling In Section 1.7 we approached modeling from several points of view. Along the way we learned how to use a grapher to create a scatter plot, compute a regression line for a data set, and overlay a regression line on a scatter plot. We touched on the notion of correlation coefficient. We now go deeper into these modeling and regression concepts. Figure 2.2 on page 162 shows five types of scatter plots. When the points of a scatter plot are clustered along a line, we say there is a linear correlation between the quantities represented by the data. When an oval is drawn around the points in the scatter plot, generally speaking, the narrower the oval, the stronger the linear correlation. When the oval tilts like a line with positive slope (as in Figure 2.2a and b), the data have a positive linear correlation. On the other hand, when it tilts like a line with negative slope (as in Figure 2.2d and e), the data have a negative linear correlation. Some scatter plots exhibit little or no linear correlation (as in Figure 2.2c), or have nonlinear patterns.

6965_CH02_pp157-250.qxd

162

1/14/10

1:13 PM

Page 162

CHAPTER 2 Polynomial, Power, and Rational Functions

y

y

y 50

50

50

40

40

40

30

30

30

20

20

20

10

10 10 20 30 40 50 Strong positive linear correlation

x

10 10 20 30 40 50 Weak positive linear correlation

(a)

x

10 20 30 40 50 Little or no linear correlation

(b) y

x

(c) y

50

50

40

40

30

30

20

20

10

10 10 20 30 40 50 Strong negative linear correlation (d)

x

10 20 30 40 50 Weak negative linear correlation

x

(e)

FIGURE 2.2 Five scatter plots and the types of linear correlation they suggest.

Correlation vs. Causation Correlation does not imply causation. Two variables can be strongly correlated, but that does not necessarily mean that one causes the other.

A number that measures the strength and direction of the linear correlation of a data set is the (linear) correlation coefficient, r.

Properties of the Correlation Coefficient, r 1. -1 … r … 1. 2. When r 7 0, there is a positive linear correlation. 3. When r 6 0, there is a negative linear correlation. 4. When ƒ r ƒ L 1, there is a strong linear correlation. 5. When r L 0, there is weak or no linear correlation.

Correlation informs the modeling process by giving us a measure of goodness of fit. Good modeling practice, however, demands that we have a theoretical reason for selecting a model. In business, for example, fixed cost is modeled by a constant function. (Otherwise, the cost would not be fixed.) In economics, a linear model is often used for the demand for a product as a function of its price. For instance, suppose Twin Pixie, a large supermarket chain, conducts a market analysis on its store brand of doughnut-shaped oat breakfast cereal. The chain sets various prices for its 15-oz box at its different stores over a period of time. Then, using these data, the Twin Pixie researchers predict the weekly sales at the entire chain of stores for each price and obtain the data shown in Table 2.2.

6965_CH02_pp157-250.qxd

1/14/10

1:13 PM

Page 163

SECTION 2.1

Linear and Quadratic Functions and Modeling

163

Table 2.2 Weekly Sales Data Based on Marketing Research Price per Box

Boxes Sold

$2.40 $2.60 $2.80 $3.00 $3.20 $3.40 $3.60

38,320 33,710 28,280 26,550 25,530 22,170 18,260

EXAMPLE 3 Modeling and Predicting Demand Use the data in Table 2.2 to write a linear model for demand (in boxes sold per week) as a function of the price per box (in dollars). Describe the strength and direction of the linear correlation. Then use the model to predict weekly cereal sales if the price is dropped to $2.00 or raised to $4.00 per box. SOLUTION

Model [2, 4] by [10000, 40000] (a)

We enter the data and obtain the scatter plot shown in Figure 2.3a. It appears that the data have a strong negative correlation. We then find the linear regression model to be approximately y = - 15,358.93x + 73,622.50, where x is the price per box of cereal and y the number of boxes sold. Figure 2.3b shows the scatter plot for Table 2.2 together with a graph of the regression line. You can see that the line fits the data fairly well. The correlation coefficient of r L - 0.98 supports this visual evidence. Solve Graphically

[2, 4] by [10000, 40000] (b)

Our goal is to predict the weekly sales for prices of $2.00 and $4.00 per box. Using the value feature of the grapher, as shown in Figure 2.3c, we see that y is about 42,900 when x is 2. In a similar manner we could find that y L 12,190 when x is 4. Interpret If Twin Pixie drops the price for its store brand of doughnut-shaped oat breakfast cereal to $2.00 per box, demand will rise to about 42,900 boxes per week. On the other hand, if they raise the price to $4.00 per box, demand will drop to around 12,190 boxes per week. Now try Exercise 49.

X=2

Y=42904.643

We summarize for future reference the analysis used in Example 3.

[0, 5] by [–10000, 80000] (c)

FIGURE 2.3 Scatter plot and regression line graphs for Example 3.

Regression Analysis 1. Enter and plot the data (scatter plot). 2. Find the regression model that fits the problem situation. 3. Superimpose the graph of the regression model on the scatter plot, and ob-

serve the fit. 4. Use the regression model to make the predictions called for in the problem.

6965_CH02_pp157-250.qxd

164

1/14/10

1:13 PM

Page 164

CHAPTER 2 Polynomial, Power, and Rational Functions

Quadratic Functions and Their Graphs A quadratic function is a polynomial function of degree 2. Recall from Section 1.3 that the graph of the squaring function ƒ1x2 = x 2 is a parabola. We will see that the graph of every quadratic function is an upward- or downward-opening parabola. This is because the graph of any quadratic function can be obtained from the graph of the squaring function ƒ1x2 = x 2 by a sequence of translations, reflections, stretches, and shrinks. y

EXAMPLE 4 Transforming the Squaring Function

5

Describe how to transform the graph of ƒ1x2 = x 2 into the graph of the given function. Sketch its graph by hand.

–5

5

x

(a) g1x2 = - 11/22x 2 + 3

(b) h1x2 = 31x + 222 - 1

SOLUTION

(a) The graph of g1x2 = - 11/22x 2 + 3 is obtained by vertically shrinking the graph of ƒ1x2 = x 2 by a factor of 1/2, reflecting the resulting graph across the x-axis, and translating the reflected graph up 3 units (Figure 2.4a). (b) The graph of h1x2 = 31x + 222 - 1 is obtained by vertically stretching the graph of ƒ1x2 = x 2 by a factor of 3 and translating the resulting graph left 2 units and down 1 unit (Figure 2.4b). Now try Exercise 19.

–5 (a) y 5

–5

5

x

–5

The graph of ƒ1x2 = ax 2, a 7 0, is an upward-opening parabola. When a 6 0, its graph is a downward-opening parabola. Regardless of the sign of a, the y-axis is the line of symmetry for the graph of ƒ1x2 = ax 2. The line of symmetry for a parabola is its axis of symmetry, or axis for short. The point on the parabola that intersects its axis is the vertex of the parabola. Because the graph of a quadratic function is always an upward- or downward-opening parabola, its vertex is always the lowest or highest point of the parabola. The vertex of ƒ1x2 = ax 2 is always the origin, as seen in Figure 2.5.

(b)

FIGURE 2.4 The graph of ƒ1x2 = x 2

axis

axis

(blue) shown with (a) g1x2 = -11/22x 2 + 3 and (b) h1x2 = 31x + 222 - 1. (Example 4)

f(x) = ax2, a < 0 vertex vertex f(x) = ax2, a > 0

(b)

(a) 2

FIGURE 2.5 The graph ƒ1x2 = ax for (a) a 7 0 and (b) a 6 0. Expanding ƒ1x2 = a1x - h22 + k and comparing the resulting coefficients with the standard quadratic form ax 2 + bx + c, where the powers of x are arranged in descending order, we can obtain formulas for h and k. ƒ1x2 = = = =

a1x - h22 + k a1x 2 - 2hx + h22 + k ax 2 + 1-2ah2x + 1ah2 + k2 ax 2 + bx + c

Expand 1x - h23.

Distributive property Let b = -2ah and c = ah2 + k.

6965_CH02_pp157-250.qxd

1/14/10

1:13 PM

Page 165

SECTION 2.1

y=

165

Because b = - 2ah and c = ah2 + k in the last line above, h = -b/12a2 and k = c - ah2. Using these formulas, any quadratic function ƒ1x2 = ax 2 + bx + c can be rewritten in the form

y ax2 +

Linear and Quadratic Functions and Modeling

bx + c

ƒ1x2 = a1x - h22 + k. x=– b,a>0 2a

This vertex form for a quadratic function makes it easy to identify the vertex and axis of the graph of the function, and to sketch the graph.

x

(a)

Vertex Form of a Quadratic Function

y

Any quadratic function ƒ(x) = ax 2 + bx + c, a Z 0, can be written in the vertex form

y = ax2 + bx + c x=– b,a0 1y - k22 Standard 1x - h22 + = 1 equation a2 b2

Basic Formulas Distance d between points P1x 1, y12 and Q1x 2, y22:

Focal axis y = k

d = 21x 1 - x 222 + 1y1 - y222 Midpoint: x 1 + x 2 y1 + y2 a , b 2 2 y2 - y1 Slope of a line: m = x2 - x1

1h ! a, k2

1x - h22 b2

= 1

1h, k ! a2

a 2 = b 2 + c2 y

y

-1 m1

a2

+

1h, k ! c2

Pythagorean a 2 = b 2 + c2 relation

Condition for parallel lines: m 1 = m 2

1y - k22 x = h

1h ! c, k2

Foci Vertices

Condition for perpendicular lines: m 2 =

799

Formulas from Analytic Geometry

(h, k + a) (h – c, k)

(h, k + c) (h, k)

(h – a, k)

Equations of a Line

(h + a, k) (h + c, k) x

(h, k)

The point-slope form, slope m and through 1x 1, y12:

x

y - y1 = m1x - x 12 The slope-intercept form, slope m and y-intercept b:

(h, k – c) (h, k – a)

y = mx + b

Hyperbolas with Center (h, k)

Equation of a Circle The circle with center 1h, k2 and radius r:

Standard 1x - h22 1y - k22 = 1 equation a2 b2

1y - k22

Parabolas with Vertex (h, k)

Focal axis

y = k

x = h

Foci

1h ! c, k2

1h, k ! c2

1x - h22 + 1y - k22 = r 2

Standard equation Opens

1x - h22 = 4p1y - k2

1y - k22 = 4p1x - h2

Upward or downward

To the right or to the left

Focus Directrix

1h, k + p2

y = k - p

1h + p, k2

x = h - p

Axis

x = h

y = k

Vertices Pythagorean relation Asymptotes y

y

y

b2

c2 = a 2 + b 2 b y = ! 1x - h2 + k a

a y = ! 1x - h2 + k b

c2 = a 2 + b 2

y

x=h y = a (x – h) + k

(h + a, k) (h + c, k)

(h, k)

y=k (h, k + p)

x (h + p, k) (h, k)

y = – b (x – h) + k x

a

= 1

1h, k ! a2

y = b (x – h) + k

(h – a, k) (h – c, k)

x

a2

1x - h22

1h ! a, k2

a

(h, k)

-

(h, k + c) (h, k + a) (h, k – a) (h, k – c)

b

(h, k)

y = – a (x – h) + k b

x

6965_App_pp779-813.qxd

800

1/20/10

3:20 PM

Page 800

APPENDIX B

B.5 Gallery of Basic Functions

[–4.7, 4.7] by [–3.1, 3.1]

[– 4.7, 4.7] by [–1, 5]

[– 4.7, 4.7] by [–3.1, 3.1]

Identity Function ƒ1x2 = x Domain = 1 - q , q 2 Range = 1 - q , q 2

Squaring Function ƒ1x2 = x 2 Domain = 1- q , q 2 Range = 30, q 2

Cubing Function ƒ1x2 = x 3 Domain = 1- q , q 2 Range = 1- q , q 2

[–6, 6] by [–1, 7]

[– 4.7, 4.7] by [–3.1, 3.1]

[– 4.7, 4.7] by [–3.1, 3.1]

Absolute Value Function ƒ1x2 = ƒ x ƒ = abs (x2 Domain = 1 - q , q 2 Range = 30, q 2

Reciprocal Function 1 ƒ1x2 = x Domain = 1- q , 02 ´ 10, q 2 Range = 1- q , 0) ´ 10, q 2

Square Root Function ƒ1x2 = 1x Domain = 30, q 2 Range = 30, q 2

[–4, 4] by [–1, 5]

[– 4.7, 4.7] by [–0.5, 1.5]

[–2, 6] by [–3, 3]

Exponential Function ƒ1x2 = ex Domain = 1 - q , q 2 Range = 10, q 2

Logistic Function 1 ƒ1x2 = 1 + e -x Domain = 1- q , q 2 Range = 10, 12

Natural Logarithmic Function ƒ1x2 = ln x Domain = 10, q 2 Range = 1- q , q 2

[–6, 6] by [– 4, 4]

[–2p, 2p] by [– 4, 4]

[–2p, 2p] by [–4, 4]

Greatest Integer Function ƒ1x2 = int (x2 Domain = 1- q , q 2 Range = all integers

Sine Function ƒ1x2 = sin (x2 Domain = 1- q , q 2 Range = 3-1, 14

Cosine Function ƒ1x2 = cos (x2 Domain = 1- q , q 2 Range = 3-1, 14

6965_App_pp779-813.qxd

1/20/10

3:20 PM

Page 801

APPENDIX C.1

Logic: An Introduction

801

C.1 Logic: An Introduction What you’ll learn about •

Statements

•

Compound Statements

... and why These topics are important in the study of logic.

Statements Logic is a tool used in mathematical thinking and problem solving. In logic, a statement is a sentence that is either true or false, but not both. The following expressions are not statements because their truth values cannot be determined without more information. 1. She has blue eyes. 2. x + 7 = 18 3. 2y + 7 7 1

The expressions above become statements if, for (1), “she” is identified, and for (2) and (3), values are assigned to x and y, respectively. However, an expression involving he or she or x or y may already be a statement. For example, “If he is over 210 cm tall, then he is over 2 m tall,” and “2(x + y2 = 2x + 2y” are both statements because they are true no matter who he is or what the numerical values of x and y are. From a given statement, it is possible to create a new statement by forming a negation. The negation of a statement is a statement with the opposite truth value of the given statement. If a statement is true, its negation is false, and if a statement is false, its negation is true. Consider the statement “It is snowing.” The negation of this statement may be stated simply as “It is not snowing.”

EXAMPLE 1 Negation of Statements Negate each of the following statements: (a) 2 + 3 = 5 (b) A hexagon has six sides. (c) Today is not Monday. SOLUTION

(a) 2 + 3 Z 5 (b) A hexagon does not have six sides. (c) Today is Monday. Now try Exercise 5, parts (a), (b), and (c).

The statements “The shirt is blue” and “The shirt is green” are not negations of each other. A statement and its negation must have opposite truth values. If the shirt is actually red, then both of the above statements are false and, hence, cannot be negations of each other. However, the statements “The shirt is blue” and “The shirt is not blue” are negations of each other because they have opposite truth values no matter what color the shirt really is. Some statements involve quantifiers and are more complicated to negate. Quantifiers include words such as all, some, every, and there exists. The quantifiers all, every, and no refer to each and every element in a set and are universal quantifiers. The quantifiers some and there exists at least one refer to one or

6965_App_pp779-813.qxd

802

1/20/10

3:20 PM

Page 802

APPENDIX C

more, or possibly all, of the elements in a set. Some and there exists are called existential quantifiers. Examples with universal and existential quantifiers follow: 1. All roses are red. [universal] 2. Every student is important. [universal] 3. For each counting number x, x + 0 = x. [universal] 4. Some roses are red. [existential] 5. There exists at least one even counting number less than 3. [existential] 6. There are women who are taller than 200 cm. [existential]

Venn diagrams can be used to picture statements involving quantifiers. For example, Figures C.1a and C.1b picture statements (1) and (4). The x in Figure C.1b is used to show that there must be at least one element of the set of roses that is red. U

U

Red Objects Roses

x

Red Objects

Roses

(a)

(b)

FIGURE C.1 (a) All roses are red. (b) Some roses are red. Consider the following statement involving the existential quantifier some. “Some professors at Paxson University have blue eyes.” This means that at least one professor at Paxson University has blue eyes. It does not rule out the possibilities that all the Paxson professors have blue eyes or that some of the Paxson professors do not have blue eyes. Because the negation of a true statement is false, neither “Some professors at Paxson University do not have blue eyes” nor “All professors at Paxson have blue eyes” are negations of the original statement. One possible negation of the original statement is “No professors at Paxson University have blue eyes.” Statement Some a are b. Some a are not b. All a are b. No a is b.

Negation No a is b. All a are b. Some a are not b. Some a are b.

EXAMPLE 2 Negation with Quantifiers Negate each of the following statements: (a) All students like hamburgers. (b) Some people like mathematics. (c) There exists a counting number x such that 3x = 6. (d) For all counting numbers x, 3x = 3x.

6965_App_pp779-813.qxd

1/20/10

3:20 PM

Page 803

APPENDIX C.1

Logic: An Introduction

803

SOLUTION

(a) (b) (c) (d)

There is a symbolic system defined to help in the study of logic. If p represents a statement, the negation of the statement p is denoted by ' p. Truth tables are often used to show all possible true-false patterns for statements. Table C.1 summarizes the truth tables for p and ' p.

Table C.1 Negation p

'p

T F

F T

Some students do not like hamburgers. No people like mathematics. For all counting numbers x, 3x Z 6. There exists a counting number x such that 3x Z 3x. Now try Exercise 5, parts (e) and ( f ).

Observe that p and ' p are analogous to sets P and P. If x is an element of P, then x is not an element of P.

Compound Statements From two given statements, it is possible to create a new, compound statement by using a connective such as and. For example, “It is snowing” and “the ski run is open” together with and give “It is snowing and the ski run is open.” Other compound statements can be obtained by using the connective or. For example, “It is snowing or the ski run is open.”

Table C.2 Conjunction p

q

T T F F

T F T F

p ¿ q T F F F

Table C.3 Disjunction p

q

T T F F

T F T F

p ¡ q T T T F

The symbols ¿ and ¡ are used to represent the connectives and and or, respectively. For example, if p represents “It is snowing,” and if q represents “The ski run is open,” then “It is snowing and the ski run is open” is denoted by p ¿ q. Similarly, “It is snowing or the ski run is open” is denoted by p ¡ q.

The truth value of any compound statement, such as p ¿ q, is defined using the truth table of each of the simple statements. Because each of the statements p and q may be either true or false, there are four distinct possibilities for the truth values of p and q, as shown in Table C.2. The compound statement p ¿ q is the conjunction of p and q and is defined to be true if, and only if, both p and q are true. Otherwise, it is false.

The compound statement p ¡ q—that is, p or q—is a disjunction. In everyday language, or is not always interpreted in the same way. In logic, we use an inclusive or. The statement “I will go to a movie or I will read a book” means that I will either go to a movie, or read a book, or do both. Hence, in logic, p or q, symbolized as p ¡ q, is defined to be false if both p and q are false and true in all other cases. This is summarized in Table C.3.

EXAMPLE 3 Conjunction and Disjunction Given the following statements, classify each of the conjunctions and disjunctions as true or false: p: 2 + 3 = 5 q: 2 # 3 = 6 (a) p ¿ q (e) ' p ¿ q (i) s ¡ q

r: 5 + 3 = 9 s: 2 # 4 = 9 (b) p ¿ r (f) ' 1p ¿ q2 (j) r ¡ s

(c) s ¿ q (g) p ¡ q (k) ' p ¡ q

(d) r ¿ s (h) p ¡ r (l) ' 1p ¡ q2

(continued)

6965_App_pp779-813.qxd

804

1/20/10

3:20 PM

Page 804

APPENDIX C

SOLUTION

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)

p is true and q is true, so p ¿ q is true. p is true and r is false, so p ¿ r is false. s is false and q is true, so s ¿ q is false. r is false and s is false, so r ¿ s is false. ' p is false and q is true, so ' p ¿ q is false. p ¿ q is true [part (a)], so ' 1p ¿ q2 is false. p is true and q is true, so p ¡ q is true. p is true and r is false, so p ¡ r is true. s is false and q is true, so s ¡ q is true. r is false and s is false, so r ¡ s is false. ' p is false and q is true, so ' p ¡ q is true. p ¡ q is true [part (g)], so ' 1p ¡ q2 is false. Now try Exercise 7, parts (a) and ( f).

There is an analogy between the connectives ¿ and ¡ and the set operations of intersection 1¨2 and union 1´2. Just as the statement p ¿ q is true only when p and q are both true, so an element x belongs to the set P ¨ Q only when x belongs to both P and Q. Similarly, the statement p ¡ q is true when either p or q is true, and an element x belongs to the set P ´ Q when x belongs to either P or Q.

EXAMPLE 4 Statements and Sets Use set operations to construct a set that corresponds, by analogy, to each of the following statements: (a) p ¿ r (b) ' r ¡ q (c) ' 1p ¿ q2 (d) ' 1p ¡ ' r2 SOLUTION

(a) P ¨ R

Table C.4 T F T F

p ¿ q T F F F

q ¿ p T F F F

!

q

T T F F

!

p

(b) R ´ Q

(c) P ¨ Q

(d) P ´ R Now try Exercise 9.

Not only are truth tables used to summarize the truth values of compound statements, they also are used to determine if two statements are logically equivalent. Two statements are logically equivalent if, and only if, they have the same truth values. For example, we could show that p ¿ q is logically equivalent to q ¿ p by using a truth table as in Table C.4.

EXAMPLE 5 Logical Equivalence Use a truth table to determine if ' p ¡ ' q and ' 1p ¿ q2 are logically equivalent.

SOLUTION Table C.5 shows headings and the four distinct possibilities for p and q. In the column headed ' p, we write the negations of the p column. In the ' q column, we write the negations of the q column. Next, we use the values in the ' p and the ' q columns to construct the ' p ¡ ' q column. To find the truth values for ' 1p ¿ q2, we use the p and q columns to find the truth values for p ¿ q and then negate p ¿ q.

6965_App_pp779-813.qxd

1/20/10

3:20 PM

Page 805

APPENDIX C.1

Logic: An Introduction

805

Table C.5 q

'p

'q

T T F F

T F T F

F F T T

F T F T

'p ¡ 'q

T F F F

!

F T T T

p ¿ q

' 1p ¿ q2 F T T T

!

p

Since the values in the columns for ' p ¡ ' q and ' 1p ¿ q2 are identical, the statements are equivalent. Now try Exercise 4, parts (b) and (d).

APPENDIX C.1 EXERCISES 1. Determine which of the following are statements, and then classify each statement as true or false: (a) 2 + 4 = 8

(b) Shut the window.

(c) Los Angeles is a state.

(d) He is in town.

(e) What time is it?

(f) 5x = 15

(g) 3 # 2 = 6

(h) 2x 2 7 x

(i) This statement is false.

(j) Stay put!

2. Use quantifiers to make each of the following true where x is a natural number: (a) x + 8 = 11

(b) x + 0 = x

(c) x 2 = 4

(d) x + 1 = x + 2

(j) There exists a natural number x such that 3 # 1x + 22 = 12.

(k) Every counting number is divisible by itself and 1. (l) Not all natural numbers are divisible by 2. (m) For all natural numbers x, 5x + 4x = 9x. 6. If q stands for “This course is easy” and r stands for “Lazy students do not study,” write each of the following in symbolic form: (a) This course is easy and lazy students do not study. q ¿ r

(b) Lazy students do not study or this course is not easy.

3. Use quantifiers to make each equation in Exercise 2 false.

(c) It is false that both this course is easy and lazy students do not study.

4. Complete each of the following truth tables:

(d) This course is not easy.

(a)

p

'p

T F (b)

'p

p T F

' 1 ' p2 p ¡ 'p

7. If p is false and q is true, find the truth values for each of the following:

p ¿ 'p

(c) Based on part (a), is p logically equivalent to ' 1 ' p2? (d) Based on part (b), is p ¡ ' p logically equivalent to p ¿ ' p?

5. Write the negation for each of the following statements: (a) The book has 500 pages. (b) Six is less than eight. (c)

3#5 =

15

(d) Some people have blond hair. (e) All dogs have four legs. (f) Some cats do not have nine lives. (g) All squares are rectangles. (h) Not all rectangles are squares. (i) For all natural numbers x, x + 3 = 3 + x.

(a) p ¿ q (c) ' p

(b) p ¡ q (d) ' q

(i) ' 1 ' p ¿ q2

(j) ' q ¿ ' p

(e) ' 1 ' p2 (g) p ¿ ' q

(f ) ' p ¡ q (h) ' 1p ¡ q2

8. Find the truth value for each statement in Exercise 7 if p is false and q is false. 9. Use set operations to construct a set that corresponds, by analogy, to each of the following statements. (a) r ¡ s (b) q ¿ ' q (c) ' 1r ¡ q2

(d) p ¿ 1r ¡ s2

6965_App_pp779-813.qxd

806

1/20/10

3:20 PM

Page 806

APPENDIX C

10. For each of the following, is the pair of statements logically equivalent? (a) ' 1p ¡ q2 and ' p ¡ ' q (b) ' 1p ¡ q2 and ' p ¿ ' q (c) ' 1p ¿ q2 and ' p ¿ ' q (d) ' 1p ¿ q2 and ' p ¡ ' q

11. (a) Write two logical equivalences discovered in parts 10(a)–(d). These equivalences are called DeMorgan’s Laws for “and ” and “or.” (b) Write an explanation of the analogy between DeMorgan’s Laws for sets and those found in part (a).

12. Complete the following truth table: p

q

T T F F

T F T F

'p

'q

'p ¡ q

13. Restate the following in a logically equivalent form: (a) It is not true that both today is Wednesday and the month is June. (b) It is not true that yesterday I both ate breakfast and watched television. (c) It is not raining or it is not July.

6965_App_pp779-813.qxd

1/20/10

3:20 PM

Page 807

APPENDIX C.2

What you’ll learn about •

Forms of Statements

•

Valid Reasoning

... and why These topics are important in the study of logic.

Conditionals and Biconditionals

807

C.2 Conditionals and Biconditionals Forms of Statements Statements expressed in the form “if p, then q” are called conditionals, or implications, and are denoted by p : q. Such statements also can be read “p implies q.” The “if ” part of a conditional is called the hypotheses of the implication and the “then” part is called the conclusion. Many types of statements can be put in “if-then” form; an example follows: Statement: All first graders are 6 years old. If-then form: If a child is a first grader, then the child is 6 years old. An implication may also be thought of as a promise. Suppose Betty makes the promise, “If I get a raise, then I will take you to dinner.” If Betty keeps her promise, the implication is true; if Betty breaks her promise, the implication is false. Consider the following four possibilities:

(1) (2) (3) (4)

Table C.6 Implication p

q

p:q

T T F F

T F T F

T F T T

p

q

T T F F

T F T F

Betty gets the raise; she takes you to dinner. Betty gets the raise; she does not take you to dinner. Betty does not get the raise; she takes you to dinner. Betty does not get the raise; she does not take you to dinner.

The only case in which Betty breaks her promise is when she gets her raise and fails to take you to dinner, case (2). If she does not get the raise, she can either take you to dinner or not without breaking her promise. The definition of implication is summarized in Table C.6. Observe that the only cause for which the implication is false is when p is true and q is false. An implication may be worded in several equivalent ways, as follows: 1. If the sun shines, then the swimming pool is open. (If p, then q.) 2. If the sun shines, the swimming pool is open. (If p, q.) 3. The swimming pool is open if the sun shines. (q if p.) 4. The sun shines implies the swimming pool is open. ( p implies q.) 5. The sun is shining only if the pool is open. ( p only if q.) 6. The sun’s shining is a sufficient condition for the swimming pool to be open. ( p is

a sufficient condition for q.) 7. The swimming pool’s being open is a necessary condition for the sun to be shin-

ing. (q is a necessary condition for p.)

Any implication p : q has three related implication statements, as follows: Statement: Converse: Inverse: Contrapositive:

If p, then q. If q, then p. If not p, then not q. If not q, then not p.

p:q q:p 'p : 'q 'q : 'p

6965_App_pp779-813.qxd

3:20 PM

Page 808

APPENDIX C

EXAMPLE 1 Converse, Inverse, Contrapositive Write the converse, the inverse, and the contrapositive for each of the following statements: (a) If 2x = 6, then x = 3. (b) If I am in San Francisco, then I am in California. SOLUTION

(a) Converse: If x = 3, then 2x = 6. Inverse: If 2x Z 6, then x Z 3. Contrapositive: If x Z 3, then 2x Z 6. (b) Converse: If I am in California, then I am in San Francisco. Inverse: If I am not in San Francisco, then I am not in California. Contrapositive: If I am not in California, then I am not in San Francisco. Now try Exercise 3, parts (a) and (b).

Table C.7 shows that an implication and its converse do not always have the same truth value. However, an implication and its contrapositive always have the same truth value. Also, the converse and inverse of a conditional statement are logically equivalent.

Table C.7 Converse, Inverse, Contrapositive 'p

'q

T T F F

T F T F

F F T T

F T F T

T F T T

Inverse 'p : 'q

Contrapositive 'q : 'p

T T F T

T T F T

T F T T

!

!

q

Converse q:p

!

p

Implication p:q

!

808

1/20/10

Connecting a statement and its converse with the connective and gives 1p : q2 ¿ 1q : p2. This compound statement can be written as p 4 q and usually is read “p if and only if q.” The statement “p if and only if q” is a biconditional. A truth table for p 4 q is given in Table C.8. Observe that p 4 q is true if and only if both statements are true or both are false.

Table C.8 Biconditional

p

q

p:q

q:p

T T F F

T F T F

T F T T

T T F T

Biconditional 1p : q2 ¿ 1q : p2 or p 4 q T F F T

6965_App_pp779-813.qxd

1/20/10

3:20 PM

Page 809

APPENDIX C.2

Conditionals and Biconditionals

809

EXAMPLE 2 Biconditionals Given the following statements, classify each of the biconditionals as true or false: p: 2 = 2 q: 2 Z 1 (a) p 4 q (c) s 4 q

r: 2 = 1 s: 2 + 3 = 1 + 3 (b) p 4 r (d) r 4 s

SOLUTION

(a) (b) (c) (d)

p : q is true and q : p is true, so p 4 q is true. p : r is false and r : p is true, so p 4 r is false. s : q is true and q : s is false, so s 4 q is false. r : s is true and s : r is true, so r 4 s is true. Now try Exercise 5, parts (a) and (f ).

Now consider the following statement: It is raining or it is not raining. Table C.9 A Tautology p

'p

T F

F T

p ¡ 1 ' p2 T T

This statement, which can be modeled as p ¡ 1 ' p2, is always true, as shown in Table C.9. A statement that is always true is called a tautology. One way to make a tautology is to take two logically equivalent statements such as p : q and ' q : ' p (from Table C.7) and form them into a biconditional as follows: p : q 4 1 ' q : ' p2

Because p : q and ' q : ' p have the same truth values, 1p : q2 4 1 ' q : ' p2 is a tautology.

Valid Reasoning In problem solving, the reasoning used is said to be valid if the conclusion follows unavoidably from the hypotheses. Consider the following example: Hypotheses: Conclusion:

All roses are red. This flower is a rose. Therefore, this flower is red.

The statement “All roses are red” can be written as the implication “If a flower is a rose, then it is red” and pictured with the Venn diagram in Figure C.2a. U

U

Red Objects

Red Objects

Roses

Roses Flower

(a)

(b)

FIGURE C.2 (a) All roses are red. (b) This flower is a rose.

6965_App_pp779-813.qxd

1/20/10

3:20 PM

Page 810

APPENDIX C

810

The information “This flower is a rose” implies that this flower must belong to the circle containing roses, as pictured in Figure C.2b. This flower also must belong to the circle containing red objects. Thus the reasoning is valid because it is impossible to draw a picture satisfying the hypotheses and contradicting the conclusion. Consider the following argument: Hypotheses:

All elementary school teachers are mathematically literate. Some mathematically literate people are not children. Therefore, no elementary school teacher is a child.

Conclusion:

Let E be the set of elementary school teachers, M be the set of mathematically literate people, and C be the set of children. Then the statement “All elementary school teachers are mathematically literate” can be pictured as in Figure C.3a. The statement “Some mathematically literate people are not children” can be pictured in several ways. Three of these are illustrated in Figure C.3b–d.

M

M E

M E

M E

(b)

C

C

C (a)

E

(c)

(d)

FIGURE C.3 (a) All elementary school teachers are mathematically literate. (b)–(d) Some mathematically literate people are not children. According to Figure C.3d, it is possible that some elementary school teachers are children, and yet the given statements are satisfied. Therefore, the conclusion that “No elementary school teacher is a child” does not follow from the given hypotheses. Hence, the reasoning is not valid. If a single picture can be drawn to satisfy the hypotheses of an argument and contradict the conclusion, the argument is not valid. However, to show that an argument is valid, all possible pictures must be considered to show that there are no contradictions. There must be no way to satisfy the hypotheses and contradict the conclusion if the argument is valid.

EXAMPLE 3 Argument Validity Determine if the following argument is valid: Hypotheses: P

P S

S

Conclusion: T

(a)

(b)

FIGURE C.4 (a) In Washington, D.C., all senators wear power ties. (b) No one in Washington, D.C., over 6 ft tall wears a power tie.

In Washington, D.C., all senators wear power ties. No one in Washington, D.C., over 6 ft tall wears a power tie. Persons over 6 ft tall are not senators in Washington, D.C.

SOLUTION

If S represents the set of senators and P represents the set of people who wear power ties, the first hypothesis is pictured as shown in Figure C.4a. If T represents the set of people in Washington, D.C., over 6 ft tall, the second hypothesis is pictured in Figure C.4b. Because people over 6 ft tall are outside the circle representing power tie wearers and senators are inside the circle P, the conclusion is valid and no person over 6 ft tall can be a senator in Washington, D.C. Now try Exercise 14(a).

6965_App_pp779-813.qxd

1/20/10

3:20 PM

Page 811

APPENDIX C.2

Conditionals and Biconditionals

811

A different method for determining if an argument is valid uses direct reasoning and a form of argument called the Law of Detachment (or Modus Ponens). For example, consider the following true statements: If the sun is shining, then we shall take a trip. The sun is shining. Using these two statements, we can conclude that we shall take a trip. In general, the Law of Detachment is stated as follows: If a statement in the form “if p, then q” is true, and p is true, then q must also be true. The Law of Detachment is sometimes described schematically as follows, where all statements above the horizontal line are true and the statement below the horizontal line is the conclusion. p:q p q

The Law of Detachment follows from the truth table for p : q given in Table C.6. The only case in which both p and p : q are true is when q is true (line 1 in the table).

EXAMPLE 4 Applications of the Law of Detachment Determine if each of the following arguments is valid: Hypotheses: Conclusion: Hypotheses:

Conclusion:

If you eat spinach, then you will be strong. You eat spinach. Therefore, you will be strong. If Claude goes skiing, he will break his leg. If Claude breaks his leg, he cannot enter the dance contest. Claude goes skiing. Therefore, Claude cannot enter the dance contest.

SOLUTION

(a) Using the Law of Detachment, we see that the conclusion is valid. (b) By using the Law of Detachment twice, we see that the conclusion is valid. Now try Exercise 14(d). A different type of reasoning, indirect reasoning, uses a form of argument called Modus Tollens. For example, consider the following true statements: If Chicken Little had been hit by a jumping frog, he would have thought Earth was rising. Chicken Little did not think Earth was rising. What is the conclusion? The conclusion is that Chicken Little did not get hit by a jumping frog. This leads us to the general form of Modus Tollens: If we have a conditional accepted as true, and we know the conclusion is false, then the hypothesis must be false. Modus Tollens is sometimes schematically described as follows: p:q 'q 'p

The validity of Modus Tollens also follows from the truth table for p : q given in Table C.6. The only case in which both p : q is true and q is false is when p is false

6965_App_pp779-813.qxd

812

1/20/10

3:20 PM

Page 812

APPENDIX C

(line 4 in the table). The validity of Modus Tollens also can be established from the fact that an implication and its contrapositive are equivalent.

EXAMPLE 5 Applications of Modus Tollens Determine conclusions for each of the following sets of true statements: (a) If an old woman lives in a shoe, then she does not know what to do. Mrs. Pumpkin Eater, an old woman, knows what to do. (b) If Jack is nimble, he will not get burned. Jack was burned. SOLUTION

(a) Mrs. Pumpkin Eater does not live in a shoe. (b) Jack was not nimble. Now try Exercise 13(a). People often make invalid conclusions based on advertising or other information. Consider, for example, the statement “Healthy people eat Super-Bran cereal.” Are the following conclusions valid? If a person eats Super-Bran cereal, then the person is healthy. If a person is not healthy, the person does not eat Super-Bran cereal.

If the original statement is denoted by p : q, where p is “a person is healthy” and q is “a person eats Super-Bran cereal,” then the first conclusion is the converse of p : q— that is, q : p—and the second conclusion is the inverse of p : q—that is, ' p : ' q. Table C.7 points out that neither the converse nor the inverse is logically equivalent to the original statement, and consequently the conclusions are not necessarily true. The final reasoning argument to be considered here involves the Chain Rule. Consider the following statements: If my wife works, I will retire early. If I retire early, I will become lazy. What is the conclusion? The conclusion is that if my wife works, I will become lazy. In general, the Chain Rule can be stated as follows: If “if p, then q,” and “if q, then r” are true, then “if p, then r” is true. The Chain Rule is sometimes schematically described as follows: p:q q:r p:r

Notice that the Chain Rule shows that implication is a transitive relation.

EXAMPLE 6 Applications of the Chain Rule Remark Note that in Example 6, the Chain Rule can be extended to contain several implications.

Determine conclusions for each of the following sets of true statements: (a) If Alice follows the White Rabbit, she falls into a hole. If she falls into a hole, she goes to a tea party. (b) If Chicken Little is hit by an acorn, we think the sky is falling. If we think the sky is falling, we will go to a fallout shelter. If we go to a fallout shelter, we will stay there a month. SOLUTION

(a) If Alice follows the White Rabbit, she goes to a tea party. (b) If Chicken Little is hit by an acorn, we will stay in a fallout shelter for a month. Now try Exercise 13(c).

6965_App_pp779-813.qxd

1/20/10

3:20 PM

Page 813

APPENDIX C.2

Conditionals and Biconditionals

813

APPENDIX C.2 EXERCISES 1. Write each of the following in symbolic form if p is the statement “It is raining” and q is the statement “The grass is wet.” (a) If it is raining, then the grass is wet. (b) If it is not raining, then the grass is wet. (c) If it is raining, then the grass is not wet. (d) The grass is wet if it is raining. (e) The grass is not wet implies that it is not raining. (f) The grass is wet if, and only if, it is raining. 2. Construct a truth table for each of the following: (a) p : 1p ¡ q2 (c) p 4 ' 1 ' p2

(b) 1p ¿ q2 : q (d) ' 1p : q2

3. For each of the following implications, state the converse, inverse, and contrapositive. (a) If you eat Meaties, then you are good in sports. (b) If you do not like this book, then you do not like mathematics. (c) If you do not use Ultra Brush toothpaste, then you have cavities. (d) If you are good at logic, then your grades are high. 4. Can an implication and its converse both be false? Explain your answer. 5. If p is true and q is false, find the truth values for each of the following: (a) ' p : ' q (b) ' 1 p : q2 (c) 1 p ¡ q2 : 1 p ¿ q2 (e) 1 p ¡ ' p2 : p

(d) p : ' p

(f) 1 p ¡ q2 4 1 p ¿ q2

6. If p is false and q is false, find the truth values for each of the statements in Exercise 5. 7. Iris makes the true statement, “If it rains, then I am going to the movies.” Does it follow logically that if it does not rain, then Iris does not go to the movies? 8. Consider the statement “If every digit of a number is 6, then the number is divisible by 3.” Determine whether each of the following is logically equivalent to the statement. (a) If every digit of a number is not 6, then the number is not divisible by 3. (b) If a number is not divisible by 3, then some digit of the number is not 6. (c) If a number is divisible by 3, then every digit of the number is 6. 9. Write a statement logically equivalent to the statement “If a number is a multiple of 8, then it is a multiple of 4.” 10. Use truth tables to prove that the following are tautologies: (a) 1p : q2 : 31p ¿ r2 : q4 Law of Added Hypothesis (b) 31p : q2 ¿ p4 : q Law of Detachment

(c) 31p : q2 ¿ ' q4 : ' p Modus Tollens

(d) 31p : q2 ¿ 1q : r24 : 1p : r2 Chain Rule

11. (a) Suppose that p : q, q : r, and r : s are all true, but s is false. What can you conclude about the truth value of p? (b) Suppose that 1p ¿ q2 : r is true, r is false, and q is true. What can you conclude about the truth value of p? (c) Suppose that p : q is true and q : p is false. Can q be true? Why or why not?

12. Translate the following statements into symbolic form. Give the meanings of the symbols that you use. (a) If Mary’s little lamb follows her to school, then its appearance there will break the rules and Mary will be sent home. (b) If it is not the case that Jack is nimble and quick, then Jack will not make it over the candlestick. (c) If the apple had not hit Isaac Newton on the head, then the laws of gravity would not have been discovered. 13. For each of the following, form a conclusion that follows logically from the given statements: (a) All college students are poor. Helen is a college student. (b) Some freshmen like mathematics. All people who like mathematics are intelligent. (c) If I study for the final, then I will pass the final. If I pass the final, then I will pass the course. If I pass the course, then I will look for a teaching job. (d) Every equilateral triangle is isosceles. There exist triangles that are equilateral. 14. Investigate the validity of each of the following arguments: (a) All women are mortal. Hypatia was a woman. Therefore, Hypatia was mortal. (b) All squares are quadrilaterals. All quadrilaterals are polygons. Therefore, all squares are polygons. (c) All teachers are intelligent. Some teachers are rich. Therefore, some intelligent people are rich. (d) If a student is a freshman, then she takes mathematics. Jane is a sophomore. Therefore, Jane does not take mathematics. 15. Write the following in if-then form: (a) Every figure that is a square is a rectangle. (b) All integers are rational numbers. (c) Figures with exactly three sides may be triangles. (d) It rains only if it is cloudy.

6965_Bibl_pp814-815.qxd

1/20/10

3:26 PM

Page 814

Bibliography Recent national reports and position statements Achieve. (2009, February). Closing the expectations gap 2009: Fourth annual 50-state progress report on the alignment of high school policies with the demands of college and careers. Washington, DC: Author. Retrieved August 13, 2009, from http://www.achieve.org/ files/50-state-2009.pdf ACT. (2007). Aligning postsecondary expectations and high school practice: The gap defined. Policy implications of the ACT National Curriculum Survey results 2005–2006. Iowa City, IA: Author. American Diploma Project. (2004). Ready or not: Creating a high school diploma that counts. Washington, DC: Achieve. Retrieved February 5, 2007, from http://www.achieve.org/files/ADPreport_7.pdf American Mathematical Association of Two-Year Colleges. (2006). Beyond crossroads: Implementing mathematics standards in the first two years of college. Blair, R. (Ed.). Memphis, TN: Author. Baxter Hastings, N., Gordon, F. S., Gordon, S. P., and Narayan, J. (Eds.). (2006). A fresh start for collegiate mathematics: Rethinking the courses below calculus [MAA Notes No. 69]. Washington, DC: Mathematical Association of America. Bozick, R., and Ingels, S. J. (2008). Mathematics coursetaking and achievement at the end of high school: Evidence from the Education Longitudinal Study of 2002 (ELS:2002) (NCES 2008319). Washington, DC: National Center for Education Statistics, Institute of Education Science, U.S. Department of Education. College Board. (2006). College Board standards for college success: Mathematics and statistics. New York: Author. Franklin, C., Kader, G., Mewborn, D., Moreno, J., Peck, R., Perry, M., and Scheaffer, R. (2007). Guidelines for assessment and instruction in statistics education (GAISE) report: A pre-K–12 curriculum framework. Alexandria, VA: American Statistical Association. Lutzer, D. J., Rodi, S. B., Kirkman, E. E., and Maxwell, J. W. (2007). Statistical abstract of undergraduate programs in the mathematical sciences in the United States: Fall 2005 CBMS survey. Providence, RI: American Mathematical Society. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2006). Math takes time [Position statement]. Reston, VA: Author. Retrieved January 11, 2008, from http://nctm.org/about/content.aspx?=6348 National Council of Teachers of Mathematics. (2007). Teaching mathematics today: Improving practice, improving student

814

learning (2nd ed.). T. S. Martin (Ed.). Reston, VA: Author. (Originally published in 1991 as Professional standards for teaching mathematics) National Council of Teachers of Mathematics. (2009, June). Guiding principles for mathematics curriculum and assessment. Reston, VA: Author. Retrieved August 13, 2009, from http://www.nctm.org/standards/content.aspx?id=23273 Steen, L. A. (Executive Ed.). (2001). Mathematics and democracy: The case for quantitative literacy. Princeton, NJ: National Council on Education and the Disciplines and Woodrow Wilson National Fellowship Foundation. Other possible sources for the preface Groth, R. E. (2007). Toward a conceptualization of statistical knowledge for teaching. Journal for Research in Mathematics Education, 38, 427–437. Seeley, C. (2004, October). 21st century mathematics. Principal Leadership, 22–26. Steen, L. A. (2006). Twenty questions about precalculus. In N. Baxter Hastings, F. S. Gordon, S. P. Gordon, and J. Narayan (Eds.). A fresh start for collegiate mathematics: Rethinking the courses below calculus (pp. 8–12). Washington, DC: Mathematical Association of America. Waits, B. K., and Demana, F. (1988). Is three years enough? Mathematics Teacher, 81, 11–14. Zelkowski, J. S. (2008). Important secondary mathematics enrollment factors that influence the completion of a bachelor’s degree. Unpublished doctoral dissertation, Ohio University, Athens. Reference and resource for chapter 9 Peck, R., and Starnes, D., with Kranendonk, H., and Morita, J. (2009). Making sense of statistical studies [Teacher’s module]. Alexandria, VA: American Statistical Association. Other references and resources Bay-Williams, J. M., and Herrara, S. (2007). Is “just good teaching” enough to support the learning of English language learners? Insights from sociocultural learning theory. In W. G. Martin, M. E. Strutchens, and P. C. Elliott (Eds.). The learning of mathematics: 69th yearbook (pp. 43–63). Reston, VA: National Council of Teachers of Mathematics. Black, P., Harrison, C., Lee, C., Marshall, B., and Wiliam, D. (2003). Assessment for learning: Putting it into practice. Maidenhead, Berkshire, England: Open University Press.

6965_Bibl_pp814-815.qxd

1/20/10

3:26 PM

Page 815

BIBLIOGRAPHY

Marcus, R., Fukawa-Connelly, T., Conklin, M., and Fey, J. T. (2007/2008). New thinking about college mathematics: Implications for high school teaching. Mathematics Teacher, 101, 354–358. Marzano, R. J., and Pickering, D. J. (2005). Building academic vocabulary: Teacher’s manual. Alexandria, VA: Association for Supervision and Curriculum Development.

Murray, M. (2004). Teaching mathematics vocabulary in context: Windows, doors, and secret passageways. Portsmouth, NH: Heinemann. Winsor, M. S. (2007/2008). Bridging the language barrier in mathematics. Mathematics Teacher, 101, 372–378.

815

6965_GLOSS_pp816-832.qxd

1/22/10

2:42 PM

Page 816

Glossary The angle formed by two nonzero vectors sharing a common initial point, p. 468.

A value ƒ1c2 is an absolute maximum value of ƒ if ƒ1c2 Ú ƒ1x2 for all x in the domain of ƒ, p. 90.

Angle between vectors

A value ƒ1c2 is an absolute minimum value of ƒ if ƒ1c2 … ƒ1x2 for all x in the domain of ƒ, p. 90.

Angle of depression

The absolute value of the complex number z = a + bi is given by 2a 2 + b 2; also, the length of the segment from the origin to z in the complex plane, p. 504.

Angle of elevation

Absolute maximum Absolute minimum

Absolute value of a complex number

Denoted by ƒ a ƒ , represents the number a or the positive number -a if a 6 0, p. 13.

Absolute value of a real number Absolute value of a vector Acceleration due to gravity

p. 167. Acute angle

See Magnitude of a vector. g L 32 ft/sec2 L 9.8 m/sec,

An angle whose measure is between 0° and 90°,

p. 329. Acute triangle

A triangle in which all angles measure less than

90°, p. 434. P1A or B2 = P1A2 + P1B2 - P1A and B2. If A and B are mutually exclusive events, then P1A or B2 = P1A2 + P1B2, p. 662.

Addition principle of probability

Addition property of equality

u + w = v + z, p. 21.

If u = v and w = z, then

Addition property of inequality

u + w 6 v + w, p. 23.

If u 6 v, then

The acute angle formed by the line of sight (upward) and the horizontal, p. 388. Speed of rotation, typically measured in radians or revolutions per unit time, p. 323.

Angular speed

Annual percentage rate (APR)

complex number zero, p. 50.

0 + 0i is the

The opposite of b,

or -b, p. 6. Additive inverse of a complex number

The opposite of

a + bi, or -a - bi, p. 50. A combination of variables and constants involving addition, subtraction, multiplication, division, powers, and roots, p. 5.

Algebraic expression

The annual interest rate, p. 310.

The rate that would give the same return if interest were computed just once a year, p. 307.

Annual percentage yield (APY) Annuity

A sequence of equal periodic payments, p. 308.

Aphelion

The farthest point from the Sun in a planet’s orbit,

p. 595. The length of an arc in a circle of radius r intercepted by a central angle of u radians is s = r u, p. 322.

Arc length formula

Arccosecant function Arccosine function

See Inverse cosecant function.

See Inverse cosine function.

Arccotangent function Arcsecant function Arcsine function

See Inverse cotangent function.

See Inverse secant function.

See Inverse sine function.

Arctangent function

Additive identity for the complex numbers Additive inverse of a real number

The acute angle formed by the line of sight (downward) and the horizontal, p. 388.

See Inverse tangent function.

The argument of a + bi is the direction angle of the vector 8a, b9, p. 504.

Argument of a complex number

A sequence 5an6 in which an = an - 1 + d for every integer n Ú 2. The number d is the common difference, p. 672. ! Arrow The notation PQ denoting the directed line segment with initial point P and terminal point Q. Arithmetic sequence

Associative properties

a1bc2 = 1ab2c, p. 6.

a + 1b + c2 = 1a + b2 + c,

An equation that relates variable quantities associated with phenomena being studied, p. 65.

Augmented matrix

The case in which two sides and a nonincluded angle can determine two different triangles, p. 435.

Average rate of change of ƒ over [a, b]

Algebraic model Ambiguous case Amplitude Anchor

See Sinusoid.

See Mathematical induction.

Union of two rays with a common endpoint (the vertex). The beginning ray (the initial side) can be rotated about its endpoint to obtain the final position (the terminal side), p. 338.

Angle

816

A matrix that represents a system of

equations, p. 546. ƒ1b2 - ƒ1a2 b - a

The number

, provided a Z b, p. 160.

The change in position divided by the change in time, p. 736.

Average velocity

Axis of symmetry

See Line of symmetry.

A stemplot with leaves on either side used to compare two distributions, p. 696.

Back-to-back stemplot

6965_GLOSS_pp816-832.qxd

1/22/10

2:42 PM

Page 817

Glossary

Bar chart

A rectangular graphical display of categorical data,

p. 693. See Exponential function, Logarithmic function, nth power of a. 1 Basic logistic function The function ƒ1x2 = , p. 259. 1 + e -x Bearing Measure of the clockwise angle that the line of travel makes with due north, p. 321. A flaw in the design of a sampling process that systematically causes the sample to differ from the population with respect to the statistic being measured. Undercoverage bias results when the sample systematically excludes one or more segments of the population. Voluntary response bias results when a sample consists only of those who volunteer their responses. Response bias results when the sampling design intentionally or unintentionally influences the responses, pp. 720, 721.

Bias

A polynomial with exactly two terms, p. 652.

The numbers in Pascal’s triangle: n n! , p. 653. nCr = a b = r r!1n - r2! Binomial probability In an experiment with two possible outcomes, the probability of one outcome occurring k times in n! n independent trials is P1E2 = p k11 - p)n - k, k!1n - k2! where p is the probability of the outcome occurring once, Binomial coefficients

p. 665. A theorem that gives an expansion formula for 1a + b2n, p. 652.

Binomial theorem

An experiment in which subjects do not know if they have been given an active treatment or a placebo, p. 721.

Blind experiment

A feature of some experimental designs that controls for potential differences between subject groups by applying treatments randomly within homogeneous blocks of subjects, p. 722.

Blocking

Boundary

The two separate curves that make up a hyperbola,

p. 602.

Base

Binomial

Branches

817

The set of points on the “edge” of a region, p. 565.

A function ƒ is bounded if there are numbers b and B such that b … ƒ1x2 … B for all x in the domain of ƒ, p. 89.

Bounded

A limaçon whose polar equation is r = a ! a sin u, or r = a ! a cos u, where a 7 0, p. 498.

Cardioid

An association between the points in a plane and ordered pairs of real numbers; or an association between the points in three-dimensional space and ordered triples of real numbers, pp. 12, 629.

Cartesian coordinate system

In statistics, a nonnumerical variable such as gender or hair color. Numerical variables like zip codes, in which the numbers have no quantitative significance, are also considered to be categorical.

Categorical variable

A relationship between two variables in which the values of the response variable are directly affected by the values of the explanatory variable, p. 718.

Causation

An observational study that gathers data from an entire population, p. 720.

Census

The central point in a circle, ellipse, hyperbola, or sphere, pp. 15, 591, 602, 632.

Center

Central angle

An angle whose vertex is the center of a circle,

p. 320. Characteristic polynomial of a square matrix A

det1xIn - A2, where A is an n * n matrix, p. 556. A line segment with endpoints on the conic, pp. 583, 592, 603.

Chord of a conic

A set of points in a plane equally distant from a fixed point called the center, p. 15.

Circle

Circle graph

A circular graphical display of categorical data,

p. 693. Trigonometric functions when applied to real numbers are circular functions, p. 344.

Circular functions Closed interval

An interval that includes its endpoints, p. 4.

The real number multiplied by the variable(s) in a polynomial term, p. 185.

Coefficient

The number r 2 or R 2 that measures how well a regression curve fits the data, p. 146.

Coefficient of determination

A matrix whose elements are the coefficients in a system of linear equations, p. 546.

Coefficient matrix

An identity that relates the sine, secant, or tangent to the cosine, cosecant, or cotangent, respectively, p. 406.

A function ƒ is bounded above if there is a number B such that ƒ1x2 … B for all x in the domain of ƒ, p. 89.

Cofunction identity

A function ƒ is bounded below if there is a number b such that b … ƒ1x2 for all x in the domain of ƒ, p. 89.

Combination

Bounded above

Bounded below

An interval that has finite length (does not extend to q or - q ), p. 4.

Bounded interval

Boxplot (or box-and-whisker plot)

five-number summary, p. 709.

A graph that displays a

An arrangement of elements of a set, in which order is not important, p. 646.

There are n! such combinations, p. 646. nCr = r!1n - r2!

Combinations of n objects taken r at a time

A branch of mathematics related to determining the number of elements of a set or the number of ways objects can be arranged or combined, p. 642.

Combinatorics

6965_GLOSS_pp816-832.qxd

1/22/10

2:42 PM

Page 818

GLOSSARY

818

See Arithmetic sequence.

Common difference

A logarithm with base 10, p. 275.

Common logarithm Common ratio

a + b = b + a, ab = ba, p. 6.

Complements or complementary angles

p. 5. ƒ1x 12 = ƒ1x 22 for any x 1 and x 2 (in the interval), pp. 87, 93, 159.

Constant function (on an interval)

See Geometric sequence.

Commutative properties

A letter or symbol that stands for a specific number,

Constant

Two angles of posi-

Constant term

See Polynomial function, p. 161.

tive measure whose sum is 90°, p. 406.

Constant of variation

A method of adding a constant to an expression in order to form a perfect square, p. 41.

Constraints

Completing the square Complex conjugates

p. 51. Complex fraction

Complex numbers a + bi and a - bi,

See Compound fraction.

An expression a + bi, where a (the real part) and b (the imaginary part) are real numbers, p. 49.

Complex number

A coordinate plane used to represent the complex numbers. The x-axis of the complex plane is called the real axis and the y-axis is the imaginary axis, p. 503.

Complex plane

If a vector’s representative in standard position has a terminal point 1a, b2 1or 1a, b, c22, then 8a, b9 1or 8a, b, c92 is the component form of the vector, and a and b are the horizontal and vertical components of the vector (or a, b, and c are the x-, y-, and z-components of the vector, respectively), p. 456.

Component form of a vector

Components of a vector

See Component form of a vector.

Composition of functions

1ƒ ! g21x2 = ƒ1g1x22, p. 110.

A fractional expression in which the numerator or denominator may contain fractions, p. 793.

Compound fraction Compound interest

Interest that becomes part of the invest-

ment, p. 304. Compounded annually

See Compounded k times per year.

Interest compounded using the formula A = Pert, p. 306.

Compounded continuously

Compounded k times per year

Interest compounded using

kt

r b , where k = 1 is compounded k annually, k = 4 is compounded quarterly, k = 12 is compounded monthly, etc., pp. 304, 306. the formula A = P a1 + Compounded monthly

See Compounded k times per year.

The probability of an event A given that an event B has already occurred 1P1A ƒ B22, p. 663.

Conditional probability Cone

See Right circular cone.

A third variable that affects either of two variables being studied, making inferences about causation unreliable, p. 721.

Confounding variable

See Linear programming problem.

A function that is continuous on its entire domain, p. 102.

Continuous function

Continuous at x ! a

A sample that sacrifices randomness for convenience, p. 721.

Convenience sample

Convergence of a sequence

A sequence 5an6 converges

to a if lim an = a, p. 671. n: q

Convergence of a series

A series a ak converges to a q

k=1

sum S if lim a ak = S, p. 682. n: q n

k=1

Conversion factor

A ratio equal to 1, used for unit conversion,

p. 143. The number associated with a point on a number line, or the ordered pair associated with a point in the Cartesian coordinate plane, or the ordered triple associated with a point in the Cartesian three-dimensional space, pp. 3, 12, 629.

Coordinate(s) of a point

Coordinate plane

See Cartesian coordinate system.

A measure of the strength of the linear relationship between two variables, pp. 146, 162.

Correlation coefficient Cosecant

The function y = csc x, p. 364.

The function y = cos x, p. 351.

Cosine

Cotangent

The function y = cot x, p. 362.

Two angles having the same initial side and the same terminal side, p. 338.

Coterminal angles Course

See Bearing.

Cube root

p. 509. Cubic

nth root, where n = 3 (see Principal nth root),

A degree 3 polynomial function, p. 185.

The graph of the parametric equations x = t - sin t, y = 1 - cos t, p. 484.

Cycloid

The factor Ae -at in an equation such as cos bt, p. 373.

Damping factor

The line segment of length 2b that is perpendicular to the focal axis and has the center of the hyperbola as its midpoint, p. 603.

Data

Conjugate axis of a hyperbola

lim ƒ1x2 = ƒ1a2, p. 85.

x:a

The principle of experimental design that makes it possible to rule out other factors when making inferences about a particular explanatory variable, p. 721.

Control

A curve obtained by intersecting a double-napped right circular cone with a plane, p. 580.

Conic section (or conic)

See Power function.

y = Ae

-at

Facts collected for statistical purposes (singular form is datum), p. 693.

6965_GLOSS_pp816-832.qxd

1/22/10

2:42 PM

Page 819

GLOSSARY

De Moivre’s theorem

Directed distance n

n

1r1cos u + i sin u22 = r 1cos nu + i sin nu2, p. 507.

A function f is decreasing on an interval I if, for any two points in I, a positive change in x results in a negative change in ƒ1x2, pp. 87, 93.

Decreasing on an interval

The process of utilizing general information to prove a specific hypothesis, pp. 73, 80.

Deductive reasoning Definite integral b

The definite integral of the function ƒ over

ƒ1x2 dx = lim a ƒ1x i2 ¢x provided the n: q n

i=1 La limit of the Riemann sums exists, p. 750.

3a, b4 is

Unit of measurement (represented by the symbol °) for angles or arcs, equal to 1/360 of a complete revolution, p. 320.

Degree

See Polar coordinates.

Directed line segment

See Arrow.

The angle that the vector makes with the positive x-axis, p. 460.

Direction angle of a vector

A vector in the direction of a line in three-dimensional space, p. 634.

Direction vector for a line

The angle the arrow makes with the positive x-axis, p. 456.

Direction of an arrow

Directrix of a parabola, ellipse, or hyperbola

For the equation ax 2 + bx + c, the expression b - 4ac; for the equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0, the expression B 2 - 4AC, pp. 51, 617.

Discriminant 2

Distance (in a coordinate plane)

Distance (on a number line)

Distance (in Cartesian space)

An event whose probability depends on another event already occurring, p. 663.

Dependent event

Variable representing the range value of a function (usually y), p. 80.

Dependent variable

The function ƒ¿ defined by ƒ1x + h2 - ƒ1x2 for all of x where the ƒ¿1x2 = lim h:0 h limit exists, p. 740. ƒ1x2 - ƒ1a2 Derivative of ƒ at x ! a ƒ¿1a2 = lim x:a x - a provided the limit exists, p. 740. Derivative of ƒ

The gathering and processing of numerical information, p. 704.

Descriptive statistics

A number that is associated with a square matrix, p. 535.

Determinant

An identity involving a trigonometric function of u - v, pp. 422–424.

Difference identity

Difference of complex numbers

1a + bi2 - 1c + di2 = 1a - c2 + 1b - d2i, p. 49.

Difference of functions

p. 110.

1ƒ - g21x2 = ƒ1x2 - g1x2,

8u 1, u 29 - 8v1, v29 = 8u 1 - v1, u 2 - v29 or 8u 1, u 2, u 39 - 8v1, v2, v39 = 8u 1 - v1, u 2 - v2, u 3 - v39, p. 633.

Difference of two vectors

Differentiable at x ! a Dihedral angle

ƒ¿1a2 exists, p. 740.

An angle formed by two intersecting planes,

p. 446. Direct variation

See Power function.

Directed angle

See Polar coordinates.

The distance d1P, Q2

between P1x, y2 and Q1x, y2, d1P, Q2 = 21x 1 - x 222 + 1y1 - y222, p. 14.

p = g1x2, where x represents demand and p represents price, p. 525.

Demand curve

A line used to

determine the conic, pp. 581, 620.

The largest exponent on the variable in any of the terms of the polynomial (function), p. 158.

Degree of a polynomial (function)

819

The distance between real numbers a and b, or ƒ a - b ƒ , p. 13. The distance d1P, Q2 between and P1x, y, z2 and Q1x, y, z2, or d1P, Q221x 1 - x 222 + 1y1 - y222 + 1z 1 - z 222, p. 630.

Distributive property

properties, p. 6.

a1b + c2 = ab + ac and related

A sequence or series diverges if it does not converge, p. 682. 1 a Division = aa b, b Z 0, p. 6. b b Divergence

Given ƒ1x2, d1x2 Z 0 there are unique polynomials q1x2 (quotient) and r1x2 (remainder) with ƒ1x2 = d1x2q1x2 + r1x2 with either r1x2 = 0 or degree of r1x2 6 degree of d1x2, p. 197.

Division algorithm for polynomials

Divisor of a polynomial

See Division algorithm for

polynomials. The measure of an angle in degrees, minutes, and seconds, p. 320.

DMS measure

Domain of a function

The set of all input values for a

function, p. 80. The set of values of the variable for which both sides of the identity are defined, p. 404.

Domain of validity of an identity

The number found when the corresponding components of two vectors are multiplied and then summed, p. 467.

Dot product

An identity involving a trigonometric function of 2u, p. 428.

Double-angle identity

A blind experiment in which the researcher gathering data from the subjects is not told which subjects have received which treatment, p. 723.

Double-blind experiment

6965_GLOSS_pp816-832.qxd

1/22/10

2:42 PM

Page 820

GLOSSARY

820

A statement that describes a bounded interval, such as 3 … x 6 5, p. 25.

Expanded form

A nonnegative number that specifies how off-center the focus of a conic is, pp. 595–596, 606, 620.

Expanded form of a series

Double inequality Eccentricity

The following three row operations: Multiply all elements of a row by a nonzero constant; interchange two rows; and add a multiple of one row to another row, p. 547.

Elementary row operations

Elements of a matrix

See Matrix element.

A method of solving a system of linear

Elimination method

p. 6.

The right side of u1v + w2 = uv + uw,

A series written explicitly as a sum of terms (not in summation notation), p. 682.

In probability, a procedure that has one or more possible outcomes, p. 658. In statistics, a controlled study in which one or more treatments is imposed, p. 721.

Experiment

Explanatory variable

A variable that affects a response

variable, p. 642. A sequence in which the kth term is given as a function of k, p. 670.

Explicitly defined sequence

equations, p. 522.

Exponent

The set of all points in the plane such that the sum of the distances from a pair of fixed points (the foci) is a constant, p. 591.

Exponential decay function

Ellipse

A surface generated by rotating an ellipse about its major axis, p. 597.

Ellipsoid of revolution Empty set

A set with no elements, p. 659.

End behavior

The behavior of a graph of a function as

See nth power of a, p. 7.

Decay modeled by ƒ1x2 = a # b x, a 7 0 with 0 6 b 6 1, p. 254. Exponential form An equation written with exponents instead of logarithms, p. 274. A function of the form ƒ1x2 = a # b x, where a Z 0, b 7 0, b Z 1, p. 252.

Exponential function

Growth modeled by ƒ1x2 = a # b x, a 7 0, b 7 1, p. 254.

ƒ x ƒ : q , p. 187.

Exponential growth function

A polynomial q that the function approaches as ƒ x ƒ : , p. 221.

Exponential regression

A real number that represents one “end” of an interval, p. 5.

Extracting square roots

Complex numbers whose real parts are equal and whose imaginary parts are equal, p. 49.

Extraneous solution

Matrices that have the same order and equal corresponding elements, p. 530.

Factor

End behavior asymptote of a rational function Endpoint of an interval

Equal complex numbers Equal matrices

Outcomes of an experiment that have the same probability of occurring, p. 658.

Equally likely outcomes Equation

A statement of equality between two expressions,

A method for solving equations in the form x 2 = k, p. 41. Any solution of the resulting equation that is not a solution of the original equation, p. 228.

In algebra, a quantity being multiplied in a product. In statistics, a potential explanatory variable under study in an experiment, p. 721.

x - c is a factor of a polynomial if and only if c is a zero of the polynomial, p. 199.

Factor Theorem Factored form

p. 21. A point where the supply curve and demand curve intersect. The corresponding price is the equilibrium price, p. 525.

Equilibrium point

Equilibrium price

A procedure for fitting an exponential function to a set of data, p. 145.

See Equilibrium point.

Equivalent arrows

Arrows that have the same magnitude and

The left side of u1v + w2 = uv + uw,

Writing a polynomial as a product of two or more polynomial factors, p. 43.

Factoring (a polynomial)

Points that satisfy the constraints in a linear programming problem, p. 568.

Feasible points

Fibonacci numbers

direction, p. 457. Equations (inequalities) that have the same solutions, pp. 21, 24.

Equivalent equations (inequalities) Equivalent systems of equations

Systems of equations that

have the same solution, p. 544. Equivalent vectors

p. 6.

Vectors with the same magnitude and

The terms of the Fibonacci sequence,

p. 675. Fibonacci sequence

The sequence 1, 1, 2, 3, 5, 8, 13, . . . ,

p. 675. A function whose domain is the first n positive integers for some fixed integer n, p. 670.

Finite sequence

Sum of a finite number of terms, p. 678.

direction, p. 457.

Finite series

A function whose graph is symmetric about the y-axis 1ƒ1 -x2 = ƒ1x2 for all x in the domain of ƒ2, p. 90.

First-degree equation in x , y, and z

Even function

Event

A subset of a sample space, p. 658.

An equation that can be written in the form Ax + By + Cz + D = 0, p. 632. The points 1x, y, z2 in space with x 7 0, y 7 0, and z 7 0, p. 629.

First octant

6965_GLOSS_pp816-832.qxd

1/22/10

2:42 PM

Page 821

GLOSSARY

First quartile

See Quartile.

The set of all points in the coordinate plane corresponding to the solutions 1x, y2 of the inequality, p. 565.

Graph of an inequality in x and y

Finding a line or curve that comes close to passing through all the points in a scatter plot, p. 143.

Fitting a line or curve to data

The minimum, first quartile, median, third quartile, and maximum of a data set, p. 707.

Five-number summary

The set of all points in the coordinate plane corresponding to the ordered pairs determined by the parametric equations, p. 475.

Graph of parametric equations

Graphing calculator or a computer with graphing software, p. 31.

The line through the focus and perpendicular to the directrix of a conic, pp. 582, 591, 602, 620.

Grapher or graphing utility

The directed distance from the vertex to the focus, p. 583.

Graphical model

The length of the chord through the focus and perpendicular to the axis, p. 583.

Half-angle identity

Focal axis

Focal length of a parabola Focal width of a parabola Focus, foci

See Ellipse, Hyperbola, Parabola.

Frequency

Reciprocal of the period of a sinusoid, p. 353.

The number of individuals or observations with a certain characteristic, p. 697.

Frequency (in statistics) Frequency distribution

See Frequency table.

Frequency table (in statistics)

A table showing frequencies,

p. 697.

A visible representation of a numerical or algebraic model, p. 66. Identity involving a trigonometric function of u/2, p. 430.

The amount of time required for half of a radioactive substance to decay, p. 266.

Half-life

The graph of the linear inequality y Ú ax + b, y 7 ax + b, y … ax + b, or y 6 ax + b, p. 565.

Half-plane

An arrow with initial point (x 1, y1) and terminal point (x 2, y2) represents the vector 8x 2 - x 1, y2 - y19, p. 457.

Head minus tail (HMT) rule

The area of ¢ABC with semiperimeter s is given by 2s1s - a21s - b21s - c2, p. 445.

Heron’s formula

A relation that associates each value in the domain with exactly one value in the range, pp. 80, 633.

Function

A polynomial function of degree n 7 0 has n complex zeros (counting multiplicity), p. 210.

Fundamental Theorem of Algebra

The net amount of money returned from an annuity, p. 308.

Future value of an annuity Gaussian curve

821

See Normal curve.

A method of solving a system of n linear equations in n unknowns, p. 544.

Gaussian elimination

Ax + By + C = 0, where A and B are not both zero, p. 30.

General form (of a line)

A sequence 5an6 in which an = an-1 # r for every positive integer n Ú 2. The nonzero number r is called the common ratio, p. 673.

Geometric sequence

A series whose terms form a geometric sequence, p. 683.

Geometric series

The set of all points in the coordinate plane corresponding to the pairs 1x, ƒ1x22 for x in the domain of ƒ, p. 81.

Graph of a function ƒ

The set of all points in the polar coordinate system corresponding to the ordered pairs 1r, u2 that are solutions of the polar equation, p. 494.

Graph of a polar equation

The set of all points in the coordinate plane corresponding to the ordered pairs of the relation, p. 116.

Graph of a relation

The set of all points in the coordinate plane corresponding to the pairs 1x, y2 that are solutions of the equation, p. 31.

Graph of an equation in x and y

Higher-degree polynomial function

A polynomial function

whose degree is Ú 3, p. 185. A graph that visually represents the information in a frequency table using rectangular areas proportional to the frequencies, p. 697.

Histogram

The line y = b is a horizontal asymptote of the graph of a function ƒ if lim ƒ1x2 = b or

Horizontal asymptote

lim ƒ1x2 = b, p. 93.

x: - q

x: q

Horizontal component Horizontal line

See Component form of a vector.

y = b, p. 30.

A test for determining whether the inverse of a relation is a function, p. 122.

Horizontal Line Test

Horizontal shrink or stretch Horizontal translation

See Shrink, stretch.

A shift of a graph to the left or right,

p. 130. A set of points in a plane, the absolute value of the difference of whose distances from two fixed points (the foci) is a constant, p. 602.

Hyperbola

A surface generated by rotating a hyperbola about its transverse axis, p. 607.

Hyperboloid of revolution Hypotenuse

Side opposite the right angle in a right triangle,

p. 329. An equation that is always true throughout its domain, p. 404.

Identity

Identity function

The function ƒ1x2 = x, p. 99.

6965_GLOSS_pp816-832.qxd

1/22/10

2:42 PM

Page 822

GLOSSARY

822

A square matrix with 1’s in the main diagonal and 0’s elsewhere, p. 534.

Identity matrix

a + 0 = a, a # 1 = a, p. 6.

Identity properties Imaginary axis

See Complex plane.

Imaginary part of a complex number

See Complex number.

The complex number i = 2 -1, p. 49.

Imaginary unit

A function that is a subset of a relation defined by an equation in x and y, p. 115.

Implicitly defined function

The domain of a function’s algebraic expression, p. 82.

Implied domain

A function ƒ is increasing on an interval I if, for any two points in I, a positive change in x results in a positive change in ƒ1x2, p. 87.

Increasing on an interval

Events A and B such that P1A and B2 = P1A2P1B2, p. 661.

Independent events

Variable representing the domain value of a function (usually x), p. 80.

Independent variable Index

See Radical.

Index of summation Inductive step

See Summation notation.

See Mathematical induction.

A statement that compares two quantities using an inequality symbol, p. 3.

Inequality

Inequality symbol

6, 7 , …, or Ú , p. 3.

Using the science of statistics to make inferences about the parameters in a population from a sample, p. 704.

Inferential statistics

Infinite discontinuity at x ! a

lim ƒ1x2 = ! q , p. 85.

lim ƒ(x2 = ! q or

x:a +

The difference between the third quartile and the first quartile, p. 702.

Interquartile range

Connected subset of the real number line with at least two points, p. 4.

Interval

The composition of a one-toone function with its inverse results in the identity function, p. 124.

Inverse composition rule

Notation used to specify intervals, pp. 4, 5.

Interval notation

A special case of a limit that does not exist,

Inverse cosine function

p. 122. The inverse of a square matrix A, if it exists, is a matrix B, such that AB = BA = I, where I is an identity matrix, p. 534.

Inverse of a matrix

a + 1-a2 = 0, a #

Inverse properties

If the graph of a relation is reflected across the line y = x, the graph of the inverse relation results, p. 123. A relation that consists of all ordered pairs 1b, a2 for which 1a, b2 belongs to R, p. 121.

Inverse relation (of the relation R)

Inverse secant function

See Angle.

Initial value of a function

ƒ102, p. 161.

Instantaneous rate of change

See Derivative at x = a.

The instantaneous rate of change of a position function with respect to time, p. 737.

Instantaneous velocity Integers

The numbers . . . , -3, -2, -1, 0, 1, 2, . . . , p. 2. b

La Point where a curve crosses the x-, y-, or z-axis in a graph, pp. 31, 70, 629.

Integrable over [a, b]

ƒ1x2 dx exists, p. 750.

Intercept

Arc of a circle between the initial side and terminal side of a central angle, p. 320.

Intercepted arc

The function y = sec -1 x.

The function y = sin-1 x, p. 378.

Inverse tangent function

Invertible linear system

Initial side of an angle

1 = 11a Z 02, p. 6. a

Inverse reflection principle

Inverse variation

See Arrow.

The function y = cot -1 x.

The inverse relation of a one-to-one function,

Inverse function

A function whose domain is the set of all natural numbers, p. 670.

Initial point

The function y = cos -1 x, p. 380.

Inverse cotangent function

p. 760. Infinite sequence

The function y = csc -1 x.

Inverse cosecant function

Inverse sine function

x: a -

Infinite limit

If ƒ is a polynomial function and a 6 b, then ƒ assumes every value between ƒ1a2 and ƒ1b2, p. 190.

Intermediate Value Theorem

The function y = tan-1 x, p. 381.

See Power function.

A system of n linear equations in n variables whose coefficient matrix has a nonzero determinant, p. 550.

Irrational numbers Irrational zeros

Real numbers that are not rational, p. 2.

Zeros of a function that are irrational num-

bers, p. 201. A quadratic polynomial with real coefficients that cannot be factored using real coefficients, p. 214.

Irreducible quadratic over the reals

Jump discontinuity at x ! a

but are not equal, p. 85. kth term of a sequence

lim ƒ1x2 and lim+ ƒ1x2 exist

x:a -

x: a

The kth expression in the sequence,

p. 670. a 2 = b 2 + c2 - 2bc cos A, b = a + c - 2ac cos B, c2 = a 2 + b 2 - 2ab cos C, p. 442.

Law of cosines 2

2

2

6965_GLOSS_pp816-832.qxd

1/22/10

2:42 PM

Page 823

GLOSSARY

Law of sines

sin A sin B sin C = = , p. 434. a c b See Polynomial function in x.

Leading coefficient Leading term Leaf

See Polynomial function in x, p. 185.

The final digit of a number in a stemplot, p. 694.

Least-squares line Leibniz notation

See Linear regression line.

The notation dy/dx for the derivative of ƒ,

p. 742. Left-hand limit of f at x ! a

The limit of ƒ as x approaches a

from the left, p. 758. A graph of a polar equation of the form r 2 = a 2 sin 2u or r 2 = a 2 cos 2u, p. 499.

Lemniscate

Length of an arrow Length of a vector

See Magnitude of an arrow.

See Magnitude of a vector.

A graph of a polar equation r = a ! b sin u or r = a ! b cos u with a 7 0, b 7 0, p. 497.

Limaçon Limit

lim ƒ1x2 = L means that ƒ1x2 gets arbitrarily close

x :a

to L as x gets arbitrarily close (but not equal) to a, p. 755. Limit to growth Limit at infinity

See Logistic growth function. lim ƒ1x2 = L means that ƒ1x2 gets arbitrarily

x: q

close to L as x gets arbitrarily large; lim ƒ1x2 means x: - q

that ƒ1x2 gets arbitrarily close to L as -x gets arbitrarily large, pp. 748, 760. A graph of data in which consecutive data points are connected by line segments, p. 699.

Line graph

A line over which a graph is the mirror image of itself, p. 164.

Line of symmetry Line of travel

The path along which an object travels,

p. 321. An expression au + bv, where a and b are real numbers, p. 460.

Linear combination of vectors u and v

A scatter plot with points clustered along a line. Correlation is positive if the slope is positive and negative if the slope is negative, p. 161.

Linear correlation

An equation that can be written in the form ax + b = 0, where a and b are real numbers and a Z 0, p. 21.

Linear equation in x

A polynomial ƒ1x2 of degree n 7 0 has the factorization ƒ1x2 = a1x - z 12 1x - z 22 Á 1x - z n2 where the z i are the zeros of ƒ, p. 210.

Linear factorization theorem

A function that can be written in the form ƒ1x2 = mx + b, where m Z 0 and b are real numbers, p. 159.

Linear function

823

An inequality that can be written in one of the following forms: y 6 mx + b, y … mx + b, y 7 mx + b, or y Ú mx + b with m Z 0, p. 565.

Linear inequality in two variables x and y

An inequality that can be written in the form ax + b 6 0, ax + b … 0, ax + b 7 0, or ax + b Ú 0, where a and b are real numbers and a Z 0, p. 23.

Linear inequality in x

A method of solving certain problems involving maximizing or minimizing a function of two variables (called an objective function) subject to restrictions (called constraints), p. 567.

Linear programming problem

A procedure for finding the straight line that is the best fit for the data, p. 145.

Linear regression

Linear regression equation

Equation of a linear regression

line, p. 144. The line for which the sum of the squares of the residuals is the smallest possible, p. 144.

Linear regression line Linear system

A system of linear equations, p. 520.

Local extremum

A local maximum or a local minimum, p. 90.

A value ƒ1c2 is a local maximum of ƒ if there is an open interval I containing c such that ƒ1x2 … ƒ1c2 for all values of x in I, p. 90.

Local maximum

A value ƒ1c2 is a local minimum of ƒ if there is an open interval I containing c such that ƒ1x2 Ú ƒ1c2 for all values of x in I, p. 90.

Local minimum

An expression of the form logb x (see Logarithmic function), p. 274.

Logarithm

An equation written with logarithms instead of exponents, p. 274.

Logarithmic form

The inverse of the exponential function y = bx, denoted by y = logb x, pp. 274, 287.

Logarithmic function with base b

Transformation of a data set involving the natural logarithm: exponential regression, natural logarithmic regression, power regression, p. 298.

Logarithmic re-expression of data

Logarithmic regression

See Natural logarithmic regression.

A model of population growth: c c or ƒ1x2 = , where a, b, c, ƒ1x2 = 1 + a # bx 1 + ae -kx and k are positive with b 6 1. c is the limit to growth, p. 258.

Logistic growth function

A procedure for fitting a logistic curve to a set of data, p. 145.

Logistic regression

Any number b for which b … ƒ1x2 for all x in the domain of ƒ, p. 89.

Lower bound of f

A number c is a lower bound for the set of real zeros of ƒ if ƒ1x2 Z 0 whenever x 6 c, p. 202.

Lower bound for real zeros

6965_GLOSS_pp816-832.qxd

824

2/2/10

11:28 AM

Page 824

GLOSSARY

A test for finding a lower bound for the real zeros of a polynomial, p. 202.

Lower bound test for real zeros

A Riemann sum approximation of the area under a curve ƒ1x2 from x = a to x = b using x i as the left-hand endpoint of each subinterval, p. 750. ! Magnitude of an arrow The magnitude of PQ is the distance between P and Q, p. 456. LRAM

Magnitude of a real number

See Absolute value of a real

number. The magnitude of 8a, b9 is 2a 2 + b 2. The magnitude of 8a, b, c9 is 2a 2 + b 2 + c2, pp. 458, 633.

Magnitude of a vector

The diagonal from the top left to the bottom right of a square matrix, p. 534.

Main diagonal

The line segment through the foci of an ellipse with endpoints on the ellipse, p. 592.

Major axis

A function viewed as a mapping of the elements of the domain onto the elements of the range, p. 80.

Mapping

A mathematical structure that approximates phenomena for the purpose of studying or predicting their behavior, p. 64.

Mathematical model

A process for proving that a statement is true for all natural numbers n by showing that it is true for n = 1 (the anchor) and that, if it is true for n = k, then it must be true for n = k + 1 (the inductive step), p. 687.

Mathematical induction

A rectangular array of m rows and n columns of real numbers, p. 530.

Matrix, m : n

Matrix element

Any of the real numbers in a matrix, p. 530.

The value of ƒ r ƒ at the point on the graph of a polar equation that has the maximum distance from the pole, p. 496.

Maximum r-value

The sum of all the data divided by the total number of items, p. 704.

Mean (of a set of data)

The number of degrees or radians in an

Measure of an angle

angle, p. 320. A measure of the typical, middle, or average value for a data set, p. 705.

Measure of center

Measure of spread

A measure that tells how widely distributed

data are, p. 707.

Minute

Angle measure equal to 1/60 of a degree, p. 320.

The category or number that occurs most frequently in the set, p. 704.

Mode of a data set Modified boxplot

For the line segment with a + c b + d endpoints 1a, b2 and 1c, d2, a , b , p. 15. 2 2

Midpoint (in a coordinate plane)

Midpoint (on a number line)

For the line segment with

a + b , p. 14. 2

A boxplot with the outliers removed,

p. 710. Modulus

See Absolute value of a complex number.

Monomial function

A polynomial with exactly one term,

p. 175. A principle used to find the number of ways an event can occur, p. 643.

Multiplication principle of counting

If A and B are independent events, then P1A and B2 = P1A2 # P1B2. If A depends on B, then P1A and B2 = P1A|B2 # P1B2, p. 661.

Multiplication principle of probability

Multiplication property of equality

uw = vz, p. 21.

If u = v and w = z, then

If u 6 v and c 7 0, then uc 6 vc. If u 6 v and c 6 0, then uc 7 vc, p. 23.

Multiplication property of inequality Multiplicative identity for matrices

See Identity matrix.

Multiplicative inverse of a complex number

The reciprocal

1 a b = 2 - 2 i, p. 51. a + bi a + b2 a + b2 inverse of a matrix See Inverse of a matrix.

of a + bi, or Multiplicative

Multiplicative inverse of a real number

The reciprocal of b,

or 1/b, b Z 0, p. 6. The multiplicity of a zero c of a polynomial ƒ1x2 of degree n 7 0 is the number of times the factor 1x - c2 occurs in the linear factorization ƒ1x2 = a1x - z i2 1x - z 22 Á 1x - z n2, p. 189.

Multiplicity

Nappe

See Right circular cone.

Natural exponential function Natural logarithm

The middle number (or the mean of the two middle numbers) if the data are listed in order, p. 704.

Median (of a data set)

endpoints a and b,

For the line segment with endpoints 1x 1, y1, z 12 and 1x 2, y2, z 22, x 1 + x 2 y1 + y2 z 1 + z 2 a , , b , p. 630. 2 2 2 Minor axis The perpendicular bisector of the major axis of an ellipse with endpoints on the ellipse, p. 592. Midpoint (in Cartesian space)

The function ƒ1x2 = ex, p. 256.

A logarithm with base e, p. 277.

The inverse of the exponential function y = ex, denoted by y = ln x, p. 278.

Natural logarithmic function

A procedure for fitting a logarithmic curve to a set of data, p. 145.

Natural logarithmic regression Natural numbers

The numbers 1, 2, 3, . . . , p. 2.

Length of 1 minute of arc along the Earth’s equator, p. 324.

Nautical mile NDER ƒ(a)

See Numerical derivative of ƒ at x = a, p. 766.

6965_GLOSS_pp816-832.qxd

1/22/10

2:42 PM

Page 825

GLOSSARY

A relationship between two variables in which higher values of one variable are generally associated with lower values of the other variable, p. 717.

Negative association

Negative angle

Angle generated by clockwise rotation, p. 338.

Negative linear correlation

Real numbers shown to the left of the origin on a number line, p. 3.

Negative numbers

Newton’s law of cooling

T1t2 = Tm + 1T0 - Tm2e -kt, p. 296.

For any positive integer n, n factorial is n! = n # 1n - 12 # 1n - 22 # Á # 3 # 2 # 1; zero factorial is 0! = 1, p. 644.

n factorial

La

A calculator approximation to

NINT (ƒ(x), x, a, b)

p. 767. Nonsingular matrix

b

ƒ1x2dx,

A square matrix with nonzero determinant,

p. 534. Normal curve

2

The graph of ƒ1x2 = e -x /2, p. 711.

A distribution of data shaped like the normal curve, p. 711.

Normal distribution

n-set

A set of n objects, p. 644.

The number a n = a # a # Á # a 1with n factors of a2, where n is the exponent and a is the base, p. 7.

nth power of a nth root

See Principal nth root.

nth root of a complex number z n

A complex number v such

that v = z, p. 508. A complex number v such that v n = 1,

nth root of unity

p. 508.

The graph of the solutions of a linear inequality 1in x2 on a number line, p. 24.

Number line graph of a linear inequality Numerical derivative of ƒ at a

NDER ƒ1a2 =

ƒ1a + 0.0012 - ƒ1a - 0.0012 0.002

, p. 766.

A model determined by analyzing numbers or data in order to gain insight into a phenomenon, p. 64.

Numerical model

Objective function

See Linear programming problem.

A process for gathering data from a subset of a population through current or past observations. This differs from an experiment in that no treatment is imposed, p. 720.

Observational study

Obtuse triangle

A triangle in which one angle is greater than

90°, p. 434. The eight regions of space determined by the coordinate planes, p. 629.

Octants

For a basic trigonometric function f, an identity relating f (x) to f (-x) p. 407.

Odd-even identity

A function whose graph is symmetric about the origin 1ƒ1 -x2 = - ƒ1x2 for all x in the domain of ƒ2, p. 91.

Odd function

A function in which each element of the range corresponds to exactly one element in the domain, p. 122.

One-to-one function

One-to-one rule of exponents

p. 292.

See Linear correlation.

825

One-to-one rule of logarithms

logb x = logb y, p. 292. Open interval

x = y if and only if b x = b y, x = y if and only if

An interval that does not include its endpoints,

p. 5. A parabola y = ax 2 + bx + c opens upward if a 7 0 and opens downward if a 6 0, p. 584.

Opens upward or downward

See Additive inverse of a real number and Additive inverse of a complex number.

Opposite

Order of magnitude (of n)

log n, p. 294.

Order of an m : n matrix

The order of an m * n matrix is

m * n, p. 530.

Ordered pair

A pair of real numbers 1x, y2, p. 12.

A set is ordered if it is possible to compare any two elements and say that one element is “less than” or “greater than” the other, p. 3.

Ordered set

An annuity in which deposits are made at the same time interest is posted, p. 308.

Ordinary annuity

The number zero on a number line, or the point where the x- and y-axes cross in the Cartesian coordinate system, or the point where the x-, y-, and z-axes cross in Cartesian three-dimensional space, p. 3.

Origin

Orthogonal vectors

p. 469. Outcomes

Two vectors u and v with u # v = 0,

The various possible results of an experiment,

p. 658. Data items more than 1.5 times the IQR below the first quartile or above the third quartile, p. 709.

Outliers

The graph of a quadratic function, or the set of points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix), p. 581.

Parabola

A surface generated by rotating a parabola about its line of symmetry, p. 586.

Paraboloid of revolution

Two lines that are both vertical or have equal slopes, p. 32.

Parallel lines

Geometric representation of vector addition using the parallelogram determined by the position vectors.

Parallelogram representation of vector addition

Parameter

See Parametric equations.

Parameter interval Parametric curve

p. 475.

See Parametric equations.

The graph of parametric equations,

6965_GLOSS_pp816-832.qxd

1/22/10

2:42 PM

Page 826

GLOSSARY

826

Equations of the form x = ƒ1t2 and y = g1t2 for all t in an interval I. The variable t is the parameter and I is the parameter interval, pp. 119, 475.

Parametric equations

The line through P01x 0, y0, z 02 in the direction of the nonzero vector v = 8a, b, c9 has parametric equations x = x 0 + at, y = y0 + bt, z = z 0 + ct, p. 634.

Parametric equations for a line in space

Parametrization

A set of parametric equations for a curve,

p. 475. Partial fraction decomposition

See Partial fractions.

The process of expanding a fraction into a sum of fractions. The sum is called the partial fraction decomposition of the original fraction, p. 557.

Partial fractions

Partial sums

See Sequence of partial sums.

A number pattern in which row n 1beginning with n = 02 consists of the coefficients of the expanded form of 1a + b2n, p. 653. The closest point to the Sun in a planet’s orbit,

p. 595.

Period

See Periodic function.

A function ƒ for which there is a positive number c such that ƒ1t + c2 = ƒ1t2 for every value t in the domain of ƒ. The smallest such number c is the period of the function, p. 345.

Periodic function

An arrangement of elements of a set, in which order is important, p. 644.

Permutation

Permutations of n objects taken r at a time

n! such permutations, p. 645. nPr = 1n - r2!

Perpendicular lines

There are

Two lines that are at right angles to each

other, p. 31. PH

The measure of acidity, p. 295.

Phase shift

See Sinusoid.

A function whose domain is divided into several parts with a different function rule applied to each part, p. 104.

Piecewise-defined function

Pie chart

In an experimental study, an inactive treatment that is equivalent to the active treatment in every respect except for the factor about which an inference is to be made. Subjects in a blind experiment do not know if they have been given the active treatment or the placebo, p. 721.

Placebo

The graph of Ax + By + Cz + D = 0, where A, B, and C are not all zero, p. 632.

Plane in Cartesian space

Polar axis

The distance between the points with polar coordinates 1r1, u12 and 1r2, u22 = 2r 21 + r 22 - 2r1r2 cos 1u1 - u22, p. 493.

Polar distance formula

Polar equation

An equation in r and u, p. 490.

Polar form of a complex number

See Trigonometric form of

a complex number. See Polar coordinate system.

Polynomial function

A function in which ƒ1x2 is a polynomial

in x, p. 158. An expression that can be written in the form an x n + an - 1x n - 1 + Á + a1x + a0, where n is a nonnegative integer, the coefficients are real numbers, and an Z 0. The degree of the polynomial is n, the leading coefficient is an, the leading term is an x n, and the constant term is a0. (The number 0 is the zero polynomial), p. 158.

Polynomial in x

The process of fitting a polynomial of degree n to 1n + 12 points, p. 192.

Polynomial interpolation

Position vector of the point (a, b)

The vector 8a, b9, p. 456.

Angle generated by a counterclockwise rotation, p. 338.

Positive angle

A relationship between two variables in which higher values of one variable are generally associated with higher values of the other variable, p. 717.

Positive association

Positive linear correlation

See Linear correlation.

Real numbers shown to the right of the origin on a number line, p. 3.

Positive numbers

A function of the form ƒ1x2 = k # x a, where k and a are nonzero constants. k is the constant of variation and a is the power, p. 174.

Power function

A trigonometric identity that reduces the power to which the trigonometric functions are raised, p. 454.

Power-reducing identity

See Circle graph.

Point-slope form (of a line)

The numbers 1r, u2 that determine a point’s location in a polar coordinate system. The number r is the directed distance and u is the directed angle, p. 487.

Polar coordinates

Pole

Pascal’s triangle

Perihelion

A coordinate system whose ordered pair is based on the directed distance from a central point (the pole) and the angle measured from a ray from the pole (the polar axis), p. 487.

Polar coordinate system

y - y1 = m1x - x 12, p. 29.

See Polar coordinate system.

A procedure for fitting a curve y = a # x b to a set of data, p. 145.

Power regression

Power rule of logarithms

logb Rc = c logb R, R 7 0, p. 283.

Present value of an annuity

The net amount of your money

put into an annuity, p. 309. If b n = a, then b is an nth root of a. If b = a and a and b have the same sign, b is the principal nth root of a (see Radical), p. 508.

Principal nth root n

6965_GLOSS_pp816-832.qxd

1/22/10

2:42 PM

Page 827

GLOSSARY

Principle of mathematical induction

A principle related to

mathematical induction, p. 688. The collection of probabilities of outcomes in a sample space assigned by a probability function, p. 659.

Probability distribution

Probability of an event in a finite sample space of equally likely outcomes The number of outcomes in the event

divided by the number of outcomes in the sample space, p. 658. A function P that assigns a real number to each outcome O in a sample space satisfying: 0 … P1O2 … 1, P1"2 = 0, and the sum of the probabilities of all outcomes is 1, p. 659.

Probability function

-b ! 2b 2 - 4ac 2a used to solve ax 2 + bx + c = 0, p. 42.

Quadratic formula

A function that can be written in the form ƒ1x2 = ax 2 + bx + c, where a, b, and c are real numbers, and a Z 0, p. 164. A procedure for fitting a quadratic function to a set of data, p. 145.

Quadratic regression

The graph in three dimensions of a seconddegree equation in three variables, p. 633.

Quadric surface

A variable (in statistics) that takes on numerical values for a characteristic being measured, p. 693.

Quantitative variable

Quartic function

1a + bi21c + di2 = 1ac - bd2 + 1ad + bc2i, pp. 50, 505.

Quartile

Product of complex numbers

The product of scalar k and vector u = 8u 1, u 29 1or u = 8u 1, u 2, u 392 is k # u = 8ku 1, ku 291or k # u = 8ku 1, ku 2, ku 392, pp. 458, 633.

Product of a scalar and a vector

Product of functions

1ƒg21x2 = ƒ1x2g1x2, p. 110.

The matrix in which each entry is obtained by multiplying the entries of a row of A by the corresponding entries of a column of B and then adding, p. 532.

Product of matrices A and B

Product rule of logarithms

R 7 0, S 7 0, p. 283.

logb 1RS2 = logb R + logb S,

The movement of an object that is subject only to the force of gravity, p. 57. u#v 2 Projection of u onto v The vector projv u = a b v, ƒvƒ p. 470. Projectile motion

Proportional

See Power function.

Computer-generated numbers that can be used to approximate true randomness in scientific studies. Since they depend on iterative computer algorithms, they are not truly random, p. 723.

Pseudo-random numbers

sin2 u + cos2 u = 1, 1 + tan2 u = sec u, and 1 + cot u = csc2 u, p. 405.

Pythagorean identities 2

2

A degree 4 polynomial function, p. 185.

A procedure for fitting a quartic function to a set of data, p. 145.

Quartic regression

The first quartile is the median of the lower half of a set of data, the second quartile is the median, and the third quartile is the median of the upper half of the data, p. 707.

Quotient identities

Quotient polynomial

The measure of a central angle whose intercepted arc has a length equal to the circle’s radius, p. 321. The measure of an angle in radians, or, for a central angle, the ratio of the length of the intercepted arc to the radius of the circle, p. 322.

Radian measure

Radicand

See Radical.

The distance from a point on a circle (or a sphere) to the center of the circle (or the sphere), pp. 15, 632.

Radius

Random numbers

An angle in standard position whose terminal side lies on an axis, p. 342. An equation that can be written in the form ax 2 + bx + c = 01a Z 02, p. 41.

Quadratic equation in x

See Division algorithm for polynomials.

Radian

Random behavior

Quadrantal angle

sin u cos u and cot u = , p. 404. cos u sin u

a + bi ac + bd bc - ad = 2 + 2 i, pp. 51, 505. 2 c + di c + d c + d2 ƒ1x2 ƒ Quotient of functions a b1x2 = , g1x2 Z 0, p. 110. g g1x2 R Quotient rule of logarithms logb a b = logb R - logb S, S R 7 0, S 7 0, p. 283.

Any one of the four parts into which a plane is divided by the perpendicular coordinate axes, p. 12.

Quadrant

tan u =

Quotient of complex numbers

In a right triangle with sides a and b and hypotenuse c, c2 = a 2 + b 2, p. 14.

Pythagorean Theorem

The formula x =

Quadratic function

A numerical simulation of a probability experiment in which assigned numbers appear with the same probabilities as the outcomes of the experiment, p. 724.

Probability simulation

827

Behavior that is determined only by the laws of probability, p. 719.

Numbers that can be used by researchers to simulate randomness in scientific studies (they are usually obtained from lengthy tables of decimal digits that have been generated by verifiably random natural phenomena), p. 723.

A function that assigns real-number values to the outcomes in a sample space, p. 659.

Random variable

6965_GLOSS_pp816-832.qxd

828

1/22/10

2:42 PM

Page 828

GLOSSARY

The principle of experimental design that makes it possible to use the laws of probability when making inferences, p. 721.

Randomization

Two points that are symmetric with respect to a line or a point, p. 131.

Reflection

1x, y2 and 1x,-y2 are reflections of each other across the x-axis, p. 131.

Reflection across the x-axis

The set of all output values corresponding to elements in the domain, p. 80.

Reflection across the y-axis

The difference between the greatest and least values in a data set, p. 707.

Reflection through the origin

Range of a function Range (in statistics)

See Viewing window.

Range screen

An expression that can be written as a ratio of two polynomials, p. 791. ƒ1x2 Rational function Function of the form , where ƒ1x2 and g1x2 g1x2 are polynomials and g1x2 is not the zero polynomial, p. 218. Rational expression

Numbers that can be written as a/b, where a and b are integers, and b Z 0, p. 2.

Rational numbers Rational zeros

Zeros of a function that are rational numbers,

p. 201. A procedure for finding the possible rational zeros of a polynomial, p. 201.

Rational zeros theorem Real axis

See Complex plane.

Real number

Any number that can be written as a decimal,

p. 2. A horizontal line that represents the set of real numbers, p. 3.

Real number line

Real part of a complex number

See Complex number.

Zeros of a function that are real numbers, p. 189. 1 Reciprocal function The function ƒ1x2 = , p. 100. x Reciprocal identity An identity that equates a trigonometric function with the reciprocal of another trigonometric function, p. 404. Real zeros

Reciprocal of a real number

An equation found by regression and which can be used to predict unknown values, p. 144.

Relation

A set of ordered pairs of real numbers, p. 115.

The portion of the domain applicable to the situation being modeled, p. 82.

Relevant domain

A sequence defined by giving the first term (or the first few terms) along with a procedure for finding the subsequent terms, p. 670.

A matrix in row echelon form with every column that has a leading 1 having 0’s in all other positions, p. 548.

Reduced row echelon form

A transformation of a data set, p. 287.

See Reference triangle.

For an angle u in standard position, a reference triangle is a triangle formed by the terminal side of angle u, the x-axis, and a perpendicular dropped from a point on the terminal side to the x-axis. The angle in a reference triangle at the origin is the reference angle, p. 341.

See Division algorithm for

Remainder polynomial

polynomials. If a polynomial ƒ1x2 is divided by x - c, the remainder is ƒ1c2, p. 198.

Remainder theorem

Removable discontinuity at x ! a

lim ƒ1x2 = lim+ ƒ1x2

x: a -

x: a

but either the common limit is not equal to ƒ1a2 or ƒ1a2 is not defined, p. 84. Repeated zeros

p. 190.

Zeros of multiplicity Ú 2 (see Multiplicity),

The principle of experimental design that minimizes the effects of chance variation by repeating the experiment multiple times, p. 722.

Replication

The difference y1 - 1ax 1 + b2, where 1x 1, y12 is a point in a scatter plot and y = ax + b is a line that fits the set of data, p. 173.

Residual

A statistical measure that does not change much in response to outliers, p. 705.

Resistant measure

Finding the horizontal and vertical components of a vector, p. 460.

Response variable

Recursively defined sequence

a = a, p. 21.

Regression model

See Cartesian coordinate

system.

Reference triangle

Reflexive property of equality

Resolving a vector

Rectangular coordinate system

Reference angle

1x, y2 and 1 -x, -y2 are reflections of each other through the origin, p. 131.

See Multiplicative inverse of a

real number.

Re-expression of data

1x, y2 and 1 -x, y2 are reflections of each other across the y-axis, p. 131.

A variable that is affected by an explanatory variable, p. 642. A logarithmic scale used in measuring the intensity of an earthquake, pp. 290, 295.

Richter scale

A sum a ƒ1x i2¢x where the interval 3a, b4 is n

Riemann sum

i=1

divided into n subintervals of equal length ¢x and x i is in the ith subinterval, p. 750. Right angle

A 90° angle, p. 329.

The surface created when a line is rotated about a second line that intersects but is not perpendicular to the first line, p. 580.

Right circular cone

Right-hand limit of ƒ at x ! a

a from the right, p. 758.

The limit of ƒ as x approaches

6965_GLOSS_pp816-832.qxd

1/22/10

2:42 PM

Page 829

GLOSSARY

Right triangle

A triangle with a 90° angle, p. 14.

A transformation that leaves the basic shape of a graph unchanged, p. 129.

Rigid transformation Root of a number

See Principal nth root.

Root of an equation

A solution, p. 70.

A graph of a polar equation r = a cos nu or r = a sin nu, p. 496.

Rose curve

A matrix in which rows consisting of all 0’s occur only at the bottom of the matrix, the first nonzero entry in any row with nonzero entries is 1, and the leading 1’s move to the right as we move down the rows, p. 547.

Row echelon form

Row operations

See Elementary row operations.

A Riemann sum approximation of the area under a curve ƒ1x2 from x = a to x = b using x i as the right-hand end point of each subinterval, p. 750.

RRAM

Sample space

Set of all possible outcomes of an experiment,

p. 658. The standard deviation computed using only a sample of the entire population, p. 710.

Sample standard deviation

A process for gathering data from a subset of a population, usually through direct questioning, p. 720.

Sample survey Scalar

A real number, p. 458.

A plot of all the ordered pairs of a two-variable data set on a coordinate plane, p. 12.

Scatter plot

A positive number written as c * 10 m, where 1 … c 6 10 and m is an integer, p. 8.

Scientific notation Secant

The function y = sec x, p. 363.

Secant line of ƒ

A line joining two points of the graph of ƒ,

p. 740. Second

Angle measure equal to 1/60 of a minute, p. 320.

Second quartile

See Quartile.

Ax 2 + Bxy + Cy + Dx + Ey + F = 0, where A, B, and C are not all zero, p. 581.

Second-degree equation in two variables 2

A transformation of a graph obtained by multiplying all the x-coordinates (horizontal shrink) by the constant 1/c or all of the y-coordinates (vertical shrink) by the constant c, 0 6 c 6 1, p. 134.

Shrink of factor c

Motion described by d = a sin vt or d = a cos vt, p. 391.

Simple harmonic motion Sine

The function y = sin x, p. 350.

Singular matrix

A square matrix with zero determinant,

p. 534. A function that can be written in the form ƒ1x2 = a sin 1b1x - h22 + k or ƒ1x2 = a cos 1b1x - h22 + k. The number a is the amplitude, and the number h is the phase shift, p. 352.

Sinusoid

A procedure for fitting a curve y = a sin 1bx + c2 + d to a set of data, p. 145.

Sinusoidal regression

An end behavior asymptote that is a slant

Slant asymptote

line, p. 221.

A line that is neither horizontal nor vertical, p. 159. change in y Slope Ratio , p. 28. change in x Slope-intercept form (of a line) y = mx + b, p. 30. Slant line

Solution set of an inequality

The set of all solutions of an

inequality, p. 23. An ordered pair of real numbers that satisfies all of the equations or inequalities in the system, p. 520.

Solution of a system in two variables

A value of the variable (or values of the variables) for which the equation or inequality is true, pp. 21, 23.

Solution of an equation or inequality

Use an algebraic method, including paper and pencil manipulation and obvious mental work, with no calculator or grapher use. When appropriate, the final exact solution may be approximated by a calculator, p. 69.

Solve algebraically

Use a graphical method, including use of a hand sketch or use of a grapher. When appropriate, the approximate solution should be confirmed algebraically, p. 69.

Solve graphically

The distance from the center to a vertex of an ellipse, p. 592.

Solve an equation or inequality

The distance from the center of an ellipse to a point on the ellipse along a line perpendicular to the major axis, p. 592.

Solve a triangle

Semimajor axis Semiminor axis

One-half of the sum of the lengths of the sides of a triangle, p. 445.

Semiperimeter of a triangle Sequence

See Finite sequence, Infinite sequence.

The sequence 5Sn6, where Sn is the nth partial sum of the series, that is, the sum of the first n terms of the series, p. 682.

Sequence of partial sums

Series

A finite or infinite sum of terms, p. 678.

829

To find all solutions of the equation or inequality, pp. 21, 23. To find one or more unknown sides or angles of a triangle, p. 333.

Solve a system

To find all solutions of a system, p. 520.

Solve by elimination or substitution

Methods for solving sys-

tems of linear equations, p. 522. Solve by substitution

Method for solving systems of linear

equations, p. 520. The magnitude of the velocity vector, given by distance/time, p. 461.

Speed

A set of points in Cartesian space equally distant from a fixed point called the center, p. 632.

Sphere

6965_GLOSS_pp816-832.qxd

1/22/10

2:42 PM

Page 830

GLOSSARY

830

Spiral of Archimedes

p. 499.

The graph of the polar curve r = u,

A matrix whose number of rows equals the number of columns, p. 530.

Square matrix

Standard deviation

A measure of how a data set is spread,

p. 710. Standard form: equation of a circle

1x - h22 + 1y - k22 = r 2, p. 15.

equation of an ellipse

1y - k22 a2

+

1x - h22 b2

1x - h22 a2

a2

-

1x - h22 b2

b2

= 1 or

1x - h22 a2

-

1y - k22 b2

= 1 or

= 1, p. 603.

equation of a parabola

1x - h22 = 4p1y - k2 or

1y - k22 = 4p1x - h2, p. 583.

equation of a quadratic function

ƒ1x2 = ax 2 + bx + c1a Z 02, p. 164.

Standard form of a complex number

are real numbers, p. 49.

Sum of a finite arithmetic series

a1 + a2 n b = 32a1 + 1n - 12d4, p. 679. 2 2 a111 - r n2 Sum of a finite geometric series Sn = , p. 680. 1 - r a Sum of an infinite geometric series Sn = , ƒ r ƒ 6 1, 1 - r p. 683. Sn = na

Sum of an infinite series

See Convergence of a series.

1a + bi2 + 1c + di2 = 1a + c2 + 1b + d2i, p. 49.

Sum of complex numbers

= 1, p. 591.

equation of a hyperbola

1y - k22

+

1y - k22

An identity involving a trigonometric function of u + v, p. 421.

Sum identity

a + bi, where a and b

Standard form of a polar equation of a conic

ke ke r = or r = , pp. 621–622. 1 ! e cos u 1 ! e sin u Standard form of a polynomial function

ƒ1x2 = an x n + an - 1x n - 1 + Á + a1x + a0, p. 185.

An angle positioned on a rectangular coordinate system with its vertex at the origin and its initial side on the positive x-axis, pp. 329, 338.

Standard position (angle)

Sum of functions

1ƒ + g21x2 = ƒ1x2 + g1x2, p. 110.

8u 1, u 29 + 8v1, v29 = 8u 1 + v1, u 2 + v29 or 8u 1, u 2, u 39 + 8v1, v2, v39 = 8u 1 + v1, u 2 + v2, u 3 + v39, pp. 458, 633.

Sum of two vectors

The series a ak, where n is a natural k=1 number 1or q 2 is in summation notation and is read “the sum of ak from k = 1 to n (or infinity).” k is the index of summation, and ak is the kth term of the series, p. 678. n

Summation notation

p = ƒ1x2, where x represents production and p represents price, p. 525.

Supply curve

Symmetric difference quotient of ƒ at a

ƒ1x + h2 - ƒ1x - h2 2h

, p. 766.

A matrix A = 3aij4 with the property aij = aji for all i and j, p. 541.

Symmetric matrix

A graph in which 1- x, -y2 is on the graph whenever 1x, y2 is; or a graph in which 1-r, u2 or 1r, u + p2 is on the graph whenever 1r, u2 is, p. 91.

Symmetric about the origin

A representative arrow with its initial point at the origin, p. 456.

Symmetric about the x-axis

In the plane i = 81, 09 and j = 80, 19; in space i = 81, 0, 09, j = 80, 1, 09, and k = 80, 0, 19, pp. 460, 633.

Symmetric about the y-axis

Standard representation of a vector Standard unit vectors

A number that measures a quantitative variable for a sample from a population, p. 704.

Statistic

Statute mile

5280 feet, p. 324.

The initial digit or digits of a number in a stemplot, p. 694.

Stem

An arrangement of a numerical data set into a specific tabular format, p. 694.

Stemplot (or stem-and-leaf plot)

A transformation of a graph obtained by multiplying all the x-coordinates (horizontal stretch) by the constant 1/c, or all of the y-coordinates (vertical stretch) of the points by a constant c, c 7 1, p. 134.

Stretch of factor c

Subtraction

a - b = a + 1-b2, p. 6.

A graph in which 1x, -y2 is on the graph whenever 1x, y2 is; or a graph in which 1r, -u2 or 1-r, p - u2 is on the graph whenever 1r, u2 is, p. 91.

A graph in which 1- x, y2 is on the graph whenever 1x, y2 is; or a graph in which 1-r, - u2 or 1r, p - u2 is on the graph whenever 1r, u2 is, p. 90.

Symmetric property of equality

If a = b, then b = a, p. 21.

A procedure used to divide a polynomial by a linear factor, x - a, p. 199.

Synthetic division System Tangent

A set of equations or inequalities, p. 520. The function y = tan x, p. 361.

The line through 1a, ƒ1a22 with slope ƒ¿1a2 provided ƒ¿1a2 exists, p. 740.

Tangent line of ƒ at x ! a Terminal point

See Arrow.

Terminal side of an angle

See Angle.

6965_GLOSS_pp816-832.qxd

1/22/10

2:42 PM

Page 831

GLOSSARY

An expression of the form anx n in a polynomial (function), p. 185.

Term of a polynomial (function)

The range elements of a sequence,

Terms of a sequence

p. 670. Third quartile

See Quartile.

A line graph in which time is measured on the horizontal axis, p. 699.

Time plot

A function that maps real numbers to real numbers, p. 129.

Transformation

If a = b and b = c, then a = c. Similar properties hold for the inequality symbols 6 , …, 7 , Ú, pp. 21, 23.

Transitive property

Translation

See Horizontal translation, Vertical translation.

The matrix AT obtained by interchanging the rows and columns of A, p. 534.

Transpose of a matrix

The line segment whose endpoints are the vertices of a hyperbola, p. 603.

Transverse axis

Variance

The square of the standard deviation, p. 710.

Variation

See Power function.

An ordered pair 8a, b9 of real numbers in the plane, or an ordered triple 8a, b, c9 of real numbers in space. A vector has both magnitude and direction, pp. 456, 633.

Vector

The line through P01x 0, y0, z 02 in the direction of the nonzero vector V = 8a, b, c9 has vector equation r = r0 + tv, where r = 8x, y, z9, p. 634.

Vector equation for a line in space

A vector that specifies the motion of an object in terms of its speed and direction, p. 461.

Velocity

A visualization of the relationships among events within a sample space, p. 661.

Venn diagram

See Right circular cone.

Vertex of a cone

The point of intersection of a parabola and its line of symmetry, pp. 164, 582.

Vertex of a parabola Vertex of an angle

See Angle.

A visualization of the Multiplication Principle of Probability, p. 662.

Vertex form for a quadratic function

A special form for a system of linear equations that facilitates finding the solution, p. 544.

Vertical asymptote

Tree diagram

Triangular form

A number that is a sum of the arithmetic series 1 + 2 + 3 + Á + n for some natural number n, p. 656.

Triangular number

For real numbers a and b, exactly one of the following is true: a 6 b, a = b, or a 7 b, p. 4.

Trichotomy property

Trigonometric form of a complex number

p. 504. Unbounded interval

(or both), p. 5.

r1cos u + i sin u2,

An interval that extends to - q or q

The set of all elements that belong to A or B or both, p. 55.

Union of two sets A and B

A circle with radius 1 centered at the origin,

Unit circle

p. 344. Unit ratio

p. 165.

ƒ1x2 = a1x - h22 + k,

The line x = a is a vertical asymptote of the graph of the function ƒ if lim+ ƒ1x2 = ! q or x:a

lim- ƒ1x2 = ! q , pp. 93, 221.

x:a

Vertical component Vertical line

See Component form of a vector.

x = a, p. 30.

A test for determining whether a graph is a function, p. 81.

Vertical line test

Vertical stretch or shrink Vertical translation

See Stretch, Shrink.

A shift of a graph up or down, p. 131.

The points where the ellipse intersects its focal axis, p. 591.

Vertices of an ellipse

The points where a hyperbola intersects the line containing its foci, p. 602.

Vertices of a hyperbola

The rectangular portion of the coordinate plane specified by the dimensions [Xmin, Xmax] by [Ymin, Ymax], p. 31.

Viewing window

See Conversion factor.

Unit vector

Vector of length 1, pp. 459, 633.

A unit vector that has the same direction as the given vector, pp. 460, 633.

Unit vector in the direction of a vector

Any number B for which ƒ1x2 … B for all x in the domain of ƒ, p. 89.

Upper bound for ƒ

A number d is an upper bound for the set of real zeros of ƒ if ƒ1x2 Z 0 whenever x 7 d, p. 202.

Upper bound for real zeros

A test for finding an upper bound for the real zeros of a polynomial, p. 202.

Upper bound test for real zeros Variable

831

A letter that represents an unspecified number, p. 5.

A characteristic of individuals that is being identified or measured, p. 693.

Variable (in statistics)

A mean calculated in such a way that some elements of the data set have higher weights (that is, are counted more strongly in determining the mean) than others, p. 707.

Weighted mean

Weights

See Weighted mean.

Whole numbers

The numbers 0, 1, 2, 3, Á , p. 2.

The restrictions on x and y that specify a viewing window. See Viewing window.

Window dimensions

The product of a! force applied to an object over a given distance W = ƒ F ƒ ƒ AB ƒ , p. 471.

Work

The function that associates points on the unit circle with points on the real number line, p. 344.

Wrapping function

6965_GLOSS_pp816-832.qxd

832

1/22/10

2:42 PM

Page 832

GLOSSARY

Usually the horizontal coordinate line in a Cartesian coordinate system with positive direction to the right, pp. 12, 629.

x-axis

The directed distance from the y-axis 1yz-plane2 to a point in a plane (space), or the first number in an ordered pair (triple), pp. 12, 629.

x-coordinate

A point that lies on both the graph and the x-axis, pp. 31, 70.

x-intercept

The x-value of the right side of the viewing window, p. 31.

Xmax

The x-value of the left side of the viewing window, p. 31.

Xmin

The scale of the tick marks on the x-axis in a viewing window, p. 31.

Xscl

xy-plane xz-plane

The points 1x, y, 02 in Cartesian space, p. 629.

The points 1x, 0, z2 in Cartesian space, p. 629.

Usually the vertical coordinate line in a Cartesian coordinate system with positive direction up, pp. 12, 629.

y-axis

The directed distance from the x-axis 1xz-plane2 to a point in a plane (space), or the second number in an ordered pair (triple), pp. 12, 629.

y-coordinate

y-intercept

A point that lies on both the graph and the y-axis,

p. 29. Ymax Ymin

The y-value of the top of the viewing window, p. 31. The y-value of the bottom of the viewing window, p. 31.

The scale of the tick marks on the y-axis in a viewing window, p. 31.

Yscl

yz-plane z-axis

The points 10, y, z2 in Cartesian space, p. 629.

Usually the third dimension in Cartesian space, p. 629.

The directed distance from the xy-plane to a point in space, or the third number in an ordered triple, p. 629.

z-coordinate

Zero factor property

pp. 41, 69. Zero factorial

If ab = 0, then either a = 0 or b = 0,

See n factorial.

A value in the domain of a function that makes the function value zero, p. 70.

Zero of a function Zero matrix Zero vector

A matrix consisting entirely of zeros, p. 532. The vector 80, 09 or 80, 0, 09, pp. 456, 633.

A procedure of a graphing utility used to view more of the coordinate plane (used, for example, to find the end behavior of a function), p. 188.

Zoom out

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 833

SELECTED ANSWERS

833

Selected Answers Quick Review P.1 1. 51, 2, 3, 4, 5, 66

3. 5 - 3, - 2, -16

Exercises P.1

1. -4.625 (terminating)

5. (a) 1187.75 (b) - 4.72

3. -2.16 (repeating)

7. - 3; 1.375

5.

; All real numbers less than or equal to 2 0

!5 !4 !3 !2 !1

7. 0

1

2

3

4

5

6

7

8

2

3

4

5

!5 !4 !3 !2 !1

13. - q 6 x 6 5, or x 6 5

11. - 1 … x 6 1

1

9.

; All real numbers less than 7 !2 !1

9. 0, 1, 2, 3, 4, 5, 6

17. 1- 3, q 2

15. -1 6 x 6 2

0

1

2

3

4

5

19. 1- 2, -12

; All real numbers less than 0 21. 1- 3, 44

23. The real numbers greater than 4 and less than or equal to 9 25. The real numbers greater than or equal to -3, or the real numbers which are at least - 3 27. The real numbers greater than -1 29. -3 6 x … 4; endpoints -3 and 4; bounded; half-open 31. x 6 5; endpoint 5; unbounded; open 33. x Ú 29 or 329, q 2; x = Bill’s age 35. 1.099 … x … 1.399 or 31.099, 1.3994; x = dollars per gallon of gasoline 37. ax 2 + ab 39. 1a + d2x 2 41. p - 6 43. 5 45. (a) Associative property of multiplication

(b) Commutative property of multiplication (c) Addition inverse property (d) Addition identity property (e) Distributive property of x2 16 multiplication over addition 47. 2 49. 4 51. x 4y 4 53. 4.49595 * 10 11 55. 1.4347 * 10 10 y x 57. 4.839 * 10 8 59. 0.000 000 033 3 61. 5,870,000,000,000 63. 2.4 * 10 -8 65. (a) Because a m Z 0, a ma 0 = a m + 0 = a m implies that a 0 = 1. (b) Because a m Z 0, a ma -m = a m - m = a 0 = 1 implies that a -m = 67. False. For example, the additive inverse of -5 is 5, which is positive. 75. -6, - 5, - 4, -3, -2, - 1, 0, 1, 2, 3, 4, 5, 6

69. E

71. B

73. 0, 1, 2, 3, 4, 5, 6

SECTION P.2 Quick Review P.2 1.

0.5

1

1.5

2

2.5

Distance: 17 - 12 L 1.232

5.

7. 5.5

y

5

3

3.

!5 !4 !3 !2 !1

0

1

2

3

4

5

9. 10

A

B C 5

x

D

Exercises P.2 1. A11, 02, B12, 42, C1- 3, -22, D10, - 22

3. (a) First quadrant (b) On the y-axis, between quadrants I and II

(c) Second quadrant (d) Third quadrant 5. 6 7. 6 9. 4 - p 11. 19.9 13. 8 15. 5 19. Perimeter = 2 141 + 182 L 21.86; area = 20.5 21. Perimeter = 2120 + 16 L 24.94; area = 32 1 3 23. 0.65 25. 12, 62 27. a - , - b 3 4

17. 7

1 . am

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 834

SELECTED ANSWERS

834

29.

31.

U.S. Motor Vehicle Production

y

U.S. Imports from Mexico

y

Value (billions of dollars)

12,000

Number (in thousands)

10,000 8000 6000 4000 2000 2001

33.

2004 Year

100 50

2000

2004

x

2007

Year

35. (a) About $183,000 (b) About $277,000 37. The three sides have lengths 5, 5, and 5 12. Since two sides have the same length, the triangle is isosceles. 39. (a) 8; 5; 289 (b) 82 + 52 = 64 + 25 = 89 = 118922 41. 1x - 122 + 1y - 222 = 25

16

Surplus (billions of dollars)

150

x

2007

U.S. Agricultural Trade Surplus

y

220 200

14

43. 1x + 122 + 1y + 422 = 9 47. Center: 10, 02; radius: 15

12 10

45. Center: 13, 12; radius: 6 49. ƒ x - 4 ƒ = 3 51. ƒ x - c ƒ 6 d

53. 7; 6

5 7 55. Midpoint is a , b . Distances from this point to vertices are equal to 118.5. 2 2

8 6

57. x … - 8 or x Ú 2

4 2 2000

2004

2007

x

Year

59. True.

length of AM¿ length of AM length of AM 1 1 = because M is the midpoint of AB. By similar triangles, = = , so M¿ is the midpoint length of AB 2 length of AC length of AB 2

65. If the legs have lengths a and b, and the hypotenuse is c units long, then without loss of generality, we can a b a + 0 b + 0 assume the vertices are 10, 02, 1a, 02, and 10, b2. Then the midpoint of the hypotenuse is a , b = a , b. The distance to the other 2 2 2 2 of AC.

vertices is

61. C

63. E

a2 a 2 b 2 c b2 1 a b + a b = = = c. + B4 B 2 2 4 2 2

67. Q1a, -b2

69. Q1-a, - b2

SECTION P.3 Quick Review P.3 1. 4x + 5y + 9

3. 3x + 2y

5 y

5.

2x + 1 x

7.

9.

11x + 18 10

Exercises P.3 1. (a) and (c) 19. x =

3. (b)

7 = 1.75 4

5. Yes

21. x =

7. No

4 3

23. z =

is a solution of the equation 2x 2 + x - 6 = 0. 33. (b) and (c) 39. !5 !4 !3 !2 !1

19 5

8 19

11. x = 8

25. x =

0

45. -

1

0 2

1 3

2 4

1 17 … y … 2 2

3 5

4

13. t = 4

17 = 1.7 10

27. t =

15. x = 1

31 9

17. y = -

4 = - 0.8 5

29. (a) The figure shows that x = - 2

3 is a solution of the equation 2x 2 + x - 6 = 0. 2 37. x Ú -2

(b) The figure shows that x =

35. !1

43. x … -

9. No

5

6

7

8

-4 … x 6 3

x 6 6

9

!5 !4 !3 !2 !1

1

7

8

41. !2 !1

47. -

0

5 3 … z 6 2 2

0

49. x 7

1

21 5

2

3

4

5

51. y 6

6

7 6

2

3

4

5

x Ú 3

53. x …

34 7

55. x = 1

31. (a)

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 835

SELECTED ANSWERS

835

57. x = 3, 4, 5, 6 59. Multiply both sides of the first equation by 2. 61. (a) No (b) Yes 63. False. - 6 6 - 2 because - 6 lies to the left of -2 on the number line. 65. E 67. A 69. (a) (no answer) (b) (no answer) (c) 800/801 7 799/800 (d) -103/102 7 - 102/101 (e) If your calculator returns 0 when you enter 2x + 1 6 4, you can conclude that the value stored in x is not 9 2A a solution of the inequality 2x + 1 6 4. 71. b1 = 73. F = C + 32 - b2 h 5

SECTION P.4 Exploration 1 1. The graphs of y = mx + b and y = mx + c have the same slope but different y-intercepts. 3.

[–4.7, 4.7] by [–3.1, 3.1] m=1

[–4.7, 4.7] by [–3.1, 3.1] m=3

[–4.7, 4.7] by [–3.1, 3.1] m=4

[–4.7, 4.7] by [–3.1, 3.1] m=5

In each case, the two lines appear to be at right angles to one another.

Quick Review P.4 1. x = -

7 3

3. x = 12

5. y =

2 21 x 5 5

7. y =

17 5

9.

2 3

Exercises P.4 1. -2

3.

4 7

17. y + 3 = 0

5. 8

7. x = 2

19. x - y + 3 = 0

27.

9. y = 16

11. y - 4 = 21x - 12

21. y = - 3x + 5

31. 35. 39. 41. 43.

45. (a) 3187.5; 42,000

[–1, 5] by [–10, 80]

(b) 9.57 years (c) 3187.5t + 42,000 = 74,000; t = 10.04

4 3 7 , so asphalt shingles are acceptable. 8 12 51. (a) y = 0.24x - 473.8 (b) $7.64 trillion (c) $8.61 trillion

(d) 12 years

49. m =

(d)

[1985, 2010] by [0, 10]

53. (a)

y = 80.5x + 4453

[–5, 30] by [0, 7000]

15. x - y + 5 = 0

1 2 12 25. y = - x + x + 4 4 5 5 (a): the slope is 1.5, compared to 1 in (b). 33. x = 4; y = 21 37. Ymin = - 30, Ymax = 30, Yscl = 3 x = - 10; y = - 7 Ymin = - 20/3, Ymax = 20/3, Yscl = 2/3 1 7 (a) y = 3x - 1 (b) y = - x + 3 3 3 7 2 (a) y = - x + 3 (b) y = x 3 2 2

23. y = -

29.

[–5, 10] by [–10, 60]

13. y + 4 = - 21x - 52

(c) 6868 million

(b)

[–5, 30] by [0, 7000]

47. 32,000 ft

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 836

SELECTED ANSWERS

836

55. 9 57. b = 5; a = 6 59. (a) No; perpendicular lines have slopes with opposite signs. (b) No; perpendicular lines have slopes with opposite signs. 61. False. The slope of a vertical line is undefined. For example, the vertical line through 13, 12 and 13, 62 would have slope 16 - 12/13 - 32 = 5/0, which is undefined. 63. A 65. E

67. (a)

(b)

(c)

[–5, 5] by [–5, 5]

(d) a is the x-intercept and b is the y-intercept when c = 1.

[–5, 5] by [–5, 5]

[–5, 5] by [–5, 5]

(e)

(f) When c = - 1, a is the opposite of the x-intercept and b is the opposite of the y-intercept.

[–10, 10] by [–10, 10]

[–10, 10] by [–10, 10]

[–10, 10] by [–10, 10]

a is half the x-intercept and b is half the y-intercept when c = 2. 69. As in the diagram, we can choose one point to be the origin, and another to be on the x-axis. The midpoints of the sides, starting from the origin a a + b c b + d c + e d e , b , Ca , b , and D a , b. The opposite sides are and working around counterclockwise in the diagram, are then Aa , 0b , Ba 2 2 2 2 2 2 2 therefore parallel, since the slopes of the four lines connecting those points are: m AB =

c e c e = = ; m DA = . ;m ;m b BC d - a CD b d - a

c b c a + b c 71. A has coordinates a , b , while B is a , b , so the line containing A and B is the horizontal line y = , and the distance from 2 2 2 2 2 A to B is `

a + b b a - ` = . 2 2 2

SECTION P.5 Exploration 1 1.

3.

By this method, we have zeros at 0.79 and 2.21.

[–1, 4] by [–5, 10]

[–1, 4] by [–5, 10]

5. The answers in parts 2, 3, and 4 are the same.

[–1, 4] by [–5, 10]

7. 0.792893; 2.207107

Quick Review P.5 1. 9x 2 - 24x + 16

3. 6x 2 - 7x - 5

Exercises P.5 1. x = - 4 or x = 5 13. x = - 7 or x = 1

5. 15x - 222

3. x = 0.5 or x = 1.5 15. x =

5. x = -

7. 13x + 121x 2 - 52 2 or x = 3 3

7. x = "

7 7 - 211 L 0.18 or x = + 211 L 6.82 2 2

9.

5 2

1x - 221x + 12

12x + 121x + 32 9. x = - 4 "

17. x = 2 or x = 6

8 A3

11. y = "

7 A2

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 837

SELECTED ANSWERS

19. x = - 4 - 3 12 L - 8.24 or x = - 4 + 312 L 0.24

25. x-intercept: 3; y-intercept: -2 29.

21. x = - 1 or x = 4

23. -

5 273 5 273 + L 1.77 or - L - 6.77 2 2 2 2

27. x-intercepts: -2, 0, 2; y-intercept: 0 31.

[–5, 5] by [–5, 5]

[–5, 5] by [–5, 5]

35. x 2 + 2x - 1 = 0; x L 0.4

37. 1.62; -0.62

33.

[–5, 5] by [–5, 5]

[–5, 5] by [–5, 5]

39. t = 6 or t = 10

45. (a) y1 = 31x + 4 (the one that begins on the x-axis) and y2 = x 2 - 1

41. x = 1 or x = - 6

43. x = - 3 or x = 1

(b) y = 31x + 4 - x 2 + 1

(c) The x-coordinates of the 47. x = - 2 or x = 1

intersections in the first picture are the same as the x-coordinates where the second graph crosses the x-axis. 49. x = 3 or x = - 2

51. x L - 4.56 or x L - 0.44 or x = 1

837

53. x = - 2 " 213

55. x L - 2.41 or x L 2.91

57. (a) There must be 2 distinct real zeros, because b 2 - 4ac 7 0 implies that "2b 2 - 4ac are 2 distinct real numbers. (b) There must be b 1 real zero, because b 2 - 4ac = 0 implies that "2b 2 - 4ac = 0, so the root must be x = - . (c) There must be no real zeros, because 2a b 2 - 4ac 6 0 implies that "2b 2 - 4ac are not real numbers. 59. 80 yd wide; 110 yd long 61. L 11.98 ft 63. False. Notice that 21 - 322 = 18, so x could also be - 3.

65. B

67. E

69. (a) c = 2 (b) c = 4 (c) c = 5 (d) c = - 1 (e) There is no other

possible number of solutions of this equation. For any c, the solution involves solving two quadratic equations, each of which can have 0, 1, or 2 1 solutions. 71. 2.5 + 113, or approximately 0.697 and 4.303 2

SECTION P.6 Quick Review P.6 1. x + 9

5. x 2 - x - 6

3. a + 2d

7. x 2 - 2

9. x 2 - 2x - 1

Exercises P.6 1. 8 + 2i 17. 4i

3. 13 - 4i 19. 13i

5. 5 - 11 + 13 2i

21. x = 2, y = 3

7. -5 + i

11. -5 - 14i

25. 5 + 12i

13. -48 - 4i

15. 5 - 10i

29. 13 31. 25 7 115 i 33. 2/5 - 1/5i 35. 3/5 + 4/5i 37. 1/2 - 7/2i 39. 7/5 - 1/5i 41. x = - 1 " 2i 43. x = " 8 8 45. False. Any complex number bi has this property. 47. E 49. A 51. (a) i; -1; -i; 1; i; -1; - i; 1 (b) -i; - 1; i; 1; - i; -1; i; 1 (c) 1 (d) (no answer)

23. x = 1, y = 2

9. 7 + 4i

27. -1 + 0i

53. 1a + bi2 - 1a - bi2 = 2bi, real part is zero

55. 1a + bi2 # 1c + di2 = 1ac - bd2 - 1ad + bc2i = 1ac - bd2 - 1ad + bc2i and 1a + bi2 # 1c + di2 = 1a - bi2 # 1c - di2 = 1ac - bd2 - 1ad + bc2i are equal

57. 1 -i22 - i1 - i2 + 2 = 0 but 1i22 - i1i2 + 2 Z 0. Because the coefficient of x in x 2 - ix + 2 = 0 is not a real number, the complex conjugate, i, of -i, need not be a solution.

SECTION P.7 Quick Review P.7 1. - 2 6 x 6 5

3. x = 1 or x = - 5

5. x1x - 221x + 22

7.

z + 5 z

9.

Exercises P.7 1. !12!10 !8 !6 !4 !2

0

2

4

6

8

5. !5 !4 !3 !2 !1

0

1

2

3

4

5

1- q , - 94 ´ 31, q 2

2 10 a- , b 3 3

4x 2 - 4x - 1 1x - 1213x - 42

3. !2 !1

0

1

2

3

4

5

7.

!12!10 !8 !6 !4 !2

0

2

4

6

8

6

7

8

11, 52

1- q , -114 ´ 37, q 2

6965_SE_Ans_833-934.qxd

838

1/25/10

3:40 PM

Page 838

SELECTED ANSWERS

3 11. 1 - q , - 52 ´ a , q b 2

9. 3 -7, - 3/24

1 4 19. a - q , - b ´ a , q b 2 3 27. 3 -2.08, 0.174 ´ 31.19, q 2 (e) 1x + 121x - 42 7 0

21. 1- q , -1.414 ´ 30.08, q 2 29. 11.11, q 2

(f ) x1x - 42 Ú 0

43. D

15. 3- 1, 04 ´ 31, q 2

1 1 23. a - q , b ´ a , q b 2 2

31. (a) x 2 + 1 7 0

(b) x 2 + 1 6 0

17. 1 -0.24, 4.242

25. No solution

(c) x 2 … 0

(d) 1x + 221x - 52 … 0

33. (a) t = 4 sec up; t = 12 sec down (b) When t is in the interval 34, 124

(c) When t is in the interval 10, 44 or 312, 162

(b) When x is in the interval 11, 254.

than or equal to zero.

1 13. 1- q , - 22 ´ a , q b 3

35. Reveals the boundaries of the solution set

39. No more than $100,000

45. D

37. (a) 1 in. 6 x 6 34 in.

41. True. The absolute value of any real number is always greater

47. 1-5.69, - 4.112 ´ 10.61, 2.192

CHAPTER P REVIEW EXERCISES 1. Endpoints 0 and 5; bounded 11. (a) 1.45 * 10

10

3. 2x 2 - 2x

(b) 5.456 * 10

8

21. (a) 120 L 4.47, 180 L 8.94, 10 2 25. y + 1 = - 1x - 22 3 33. (a)

(c) 1.15 * 10

10

7. 3.68 * 10 9

(d) 9.7 * 10

7

17. 1x - 022 + 1y - 022 = 22, or x 2 + y 2 = 4

15. 5 15; 2 15; 1145

29. y = 4

5. v 4

27. y =

(e) 7.1 * 10 9

13. 19; 4.5

19. Center: 1- 5, -4); radius: 3

(b) 1 12022 + (18022 = 20 + 80 = 100 = 10 2

23. a = 7; b = 9

4 x - 4.4 5

2 11 31. y = - x 5 5 (b) y = 0.8x + 513.2

[–2, 10] by [500, 525]

9. 5,000,000,000

(c) 518.8, which is higher than 515. (d) 521.2

[–2, 10] by [500, 525]

3 1 or x = 3 2 7 5 117 3 117 3 45. x = or x = - 4 47. x = 49. x = 0, x = 3 51. 3 " 2i 53. x = L - 0.28 or x = + L 1.78 2 2 4 4 4 4 1 2 55. x = 0 or x = - or x = 7 57. x L 2.36 59. 1-6, 34 61. a - q , d 3 3 !10 !8 !6 !4 !2 0 2 4 6 8 10 35. 2.5

37. x = - 3

39. y = 3

41. x = 2 - 17 L - 0.65; x = 2 + 17 L 4.65

43. x =

2 63. 1 - q , -24 ´ c - , q b 65. 1 - q , - 0.372 ´ 11.37, q 2 67. 1- q , - 2.824 ´ 3-0.34, 3.154 3 69. 1 - q , -172 ´ 13, q 2 71. 1 - q , q 2 73. 1 + 3i 75. 7 + 4i 77. 25 + 0i 79. 4i 81. (a) t L 8 sec up; t L 12 sec down (b) When t is in the interval 10, 84 or 312, 204 (approximately) (c) When t is in the interval 38, 124 (approximately) 83. (a) When w is in the interval 10, 18.54 (b) When w is in the interval 122.19, q 2 (approximately)

SECTION 1.1 Exploration 1 1. 0.75

3. 0.8025

5. $124.61

Exploration 2 1. Percentages must be … 100. 3. A statistician might look for adverse economic factors in 1990, especially those that would affect people near or below the poverty line.

Quick Review 1.1 1. 1x + 421x - 42

3. 19y + 2219y - 22

5. 14h2 + 9212h + 3212h - 32

7. 1x + 421x - 12

9. 12x - 121x - 52

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 839

SELECTED ANSWERS

839

Exercises 1.1 1. (d)(q) 3. (a)(p) 5. (e)(l) 7. (g)(t) (b) 1974 to 1979 13. Women 1n2, Men 1+2

9. (i)(m)

11. (a) Increasing, except for a slight drop from 1999 to 2004 15. Women: y = 0.582x + 32.3; men: y = - 0.211x + 83.5 17. 2018, 69.9% 19. (a) and (b) L3 3.7975 4.375 5.5405 5.8986 6.657

[–5, 55] by [23, 92]

21. Square stones 23. y = 1.2t 2 25. The lower line shows the minimum salaries, since they are lower than the average salaries. 27. Year 15. There is a clear drop in the average salary right after the 1994 strike. 29. " 39.

13 A3

31. -1; 4

33. -1.5; 4

35. -

7 1105 " 2 2

41.

[–10, 10] by [–10, 10]

43.

[–10, 10] by [–10, 10]

x L - 1.47

x L 1.77

37. 5 45.

[–5, 5] by [–10, 10]

x L 3.91

[–4, 4] by [–10, 10]

x L 1.33 or x = 4

47. (a) $46.94 (b) 210 mi 49. (a) y = 1x 20021/200 = x 200/200 = x 1 = x for all x Ú 0. (b)

(c) Yes (d) For values of x close to 0, x 200 is so small that the calculator is unable to distinguish it from zero. It returns a value of 0 1/200 = 0 rather than x. [0, 1] by [0, 1]

51. (a) - 3 or 1.1 or 1.15 (b) - 3 53. Let n be any integer. n 2 + 2n = n1n + 22, which is either the product of two odd integers or the product of two even integers. The product of two odd integers is odd. The product of two even integers is a multiple of 4, since each even integer in the product contributes a factor of 2 to the product. Therefore, n 2 + 2n is either odd or a multiple of 4. 55. False; a product is zero if at least one factor is zero. 57. C 59. B 61. (a) March (b) $120 (c) June, after three months of poor performance (d) About $2000 (e) After reaching a low in June, the stock climbed back to a price near $140 by December. LaToya’s shares had gained $2000 by that point. (f) Any graph that decreases steadily from March to December would favor Ahmad’s strategy over LaToya’s. 63. (a)

Subscribers

Monthly Bills

[7, 18] by [35, 280]

[7, 18] by [35, 53]

(c) The fit is very good:

(b) The linear model for subscribers as a function of years after 1990 is y = 20.475x - 94.6.

(d) The monthly bill scatter plot has an obviously curved shape that could be modeled more effectively by a function with a curved graph. Some possibilities: quadratic (parabola), logarithm, sine, power (e.g., square root), logistic. (We will learn about these curves later in the book.)

6965_SE_Ans_833-934.qxd

840

(e)

1/25/10

3:40 PM

Page 840

SELECTED ANSWERS

Subscribers

Monthly Bills

[4, 18] by [10, 280]

[4, 18] by [35, 53]

(f ) Cellular phone technology was still emerging in 1995, so the growth rate was not as fast, explaining the lower slope on the subscriber scatter plot. The new technology was also more expensive before competition drove prices down, explaining the anomaly on the monthly bill scatter plot.

SECTION 1.2 Exploration 1 1. From left to right, the tables are (c) constant, (b) decreasing, and (a) increasing. 3. Positive; negative; 0

Quick Review 1.2 1. x = "4

3. x 6 10

5. x = 16

7. x 6 16

9. x 6 - 2, x Ú 3

Exercises 1.2 1. Function 3. Not a function; y has two values for each positive value of x. 5. Yes 7. No 9. 11. 1- q , q 2

[–5, 5] by [–5, 15]

[–10, 10] by [–10, 10]

17. 1 - q , 104

15.

19. 1- q , q 2 ´ 30, q 2

13.

[–10, 10] by [–5, 5]

21. Yes, non-removable

[–5, 5] by [–5, 5] [–10, 10] by [–10, 10]

23. Yes, non-removable

25. Local maxima at 1 -1, 42 and 15, 52, local minimum at 12, 22. The function increases on 1- q , -14, decreases on 3- 1, 24, increases on 32, 54, and decreases on 35, q 2. 27. 1- 1, 32 and 13, 32 are neither, 11, 52 is a local maximum, and 15, 12 is a local minimum. The function increases on 1 - q , 14, decreases on 31, 54, and increases on 35, q 2.

[–10, 10] by [–2, 2]

29. Decreasing on 1 - q , - 24; increasing on 3 -2, q 2

31. Decreasing on 1- q , -2]; constant on 3- 2, 14; increasing on 31, q 2

33. Increasing on 1- q , 14; decreasing on 31, q 2

[–4, 6] by [–25, 25]

[–10, 10] by [–2, 18] [–10, 10] by [0, 20]

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 841

SELECTED ANSWERS

35. Bounded

37. Bounded below

41. ƒ has a local minimum of y = 3.75 at x = 0.5. It has no maximum.

841

39. Bounded 43. Local minimum: y L - 4.09 at x L - 0.82. Local maximum: y L - 1.91 at x L 0.82.

45. Local maximum: y L 9.16 at x L - 3.20. Local minimum: y = 0 at x = 0 and y = 0 at x = - 4.

[–5, 5] by [–50, 50]

[–5, 5] by [0, 36]

[–5, 5] by [0, 80]

47. Even

49. Even

51. Neither

55.

53. Odd 57.

y = 1; x = 1

[–10, 10] by [–10, 10]

59.

[–8, 12] by [–10, 10]

61.

y = 1; x = 1; x = - 1

[–10, 10] by [–10, 10]

63. (b)

y = - 1; x = 3

y = 0; x = 2

[–4, 6] by [–5, 5]

65. (a)

67. (a) ƒ1x2 crosses the horizontal asymptote at 10, 02.

(b) g1x2 crosses the horizontal asymptote at 10, 02.

[–10, 10] by [–10, 10]

69. (a) The vertical asymptote is x = 0, and this function is undefined at x = 0 (because a denominator can’t be zero). (b)

(c) h1x2 intersects the horizontal asymptote at 10, 02.

[–10, 10] by [–5, 5]

71. True; this is the definition of the graph of a function. 73. B 75. C k = 1 77. (a)

[–3, 3] by [–2, 2] [–10, 10] by [–10, 10]

Add the point 10, 02. (c) Yes

(b)

x

6 1 3 x 6 1 + x 2 3 x 2 - x + 1 7 0; 1 + x2 but the discriminant of x 2 - x + 1 is negative 1-32, so the graph never crosses the x-axis on the interval 10, q 2.

[–5, 5] by [–5, 5]

(c) k = - 1 x 7 -1 3 x 7 (d) 1 + x2 -1 - x 2 3 x 2 + x + 1 7 0; but the discriminant of x 2 - x + 1 is negative 1- 32, so the graph never crosses the x-axis on the interval 1- q , 02.

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 842

SELECTED ANSWERS

842

79. (a) (b) (c) (d) (e) Answers vary. 81. (a) (b) (c) (d) Answers vary. 83. (a) 2. It is in the range. the range. (c) h1x2 is not bounded above. (d) 2. It is in the range. (e) 1. It is in the range.

(b) 3. It is not in

85. Since ƒ is odd, ƒ1- x2 = - ƒ1x2 for all x. In particular, ƒ1 -02 = - ƒ102. This is equivalent to saying that ƒ102 = - ƒ102, and the only number which equals its opposite is 0. Therefore ƒ102 = 0, which means the graph must pass through the origin. 87. (a) ƒ is continuous on 3 - 2, 44; the maximum value is 13, which occurs at x = 4, and the minimum value is - 3, which occurs at x = 0.

(b) ƒ is continuous on 31, 54; the maximum value is 1, which occurs at x = 1, and the minimum value is 0.2, which occurs at x = 5. (c) ƒ is continuous on 3 -4, 14; the maximum value is 5, which occurs at x = - 4, and the minimum value is 2, which occurs at x = - 1. (d) ƒ is continuous on 3 -4, 44; the maximum value is 5, which occurs at both x = - 4 and x = 4, and the minimum value is 3, which occurs at x = 0.

SECTION 1.3 Exploration 1 3. ƒ1x2 = 1/x, ƒ1x2 = e x, ƒ1x2 = 1/11 + e -x2

1. ƒ1x2 = 1/x, ƒ1x2 = ln x

Quick Review 1.3 1. 59.34

3. 7 - p

5. 0

7. 3

5. No. There is a removable discontinuity at x = 0.

9. - 4

Exercises 1.3 1. (e) 3. (j) 5. (i) 7. (k) 19. y = x, y = x 3, y = 1/x, y = sin x

9. (d) 11. (l) 13. Ex. 8 15. Ex. 7, 8 17. Ex. 2, 4, 6, 10, 11, 12 21. y = x 2, y = 1/x, y = ƒ x ƒ 23. y = 1/x, y = e x, y = 1/11 + e -x2

25. y = 1/x, y = sin x, y = cos x, y = 1/11 + e -x2 31. Domain: 1 -6, q 2; Range: all reals

27. y = x, y = x 3, y = 1/x, y = sin x

29. Domain: all reals; Range: 3- 5, q 2

35. (a) Increasing on 310, q 2 (b) Neither (c) Minimum value of 0 at x = 10 (d) Square root function, shifted 10 units right 37. (a) Increasing on 1 - q , q 2 (b) Neither (c) None (d) Logistic function, stretched vertically by a factor of 3 39. (a) Increasing on 30, q 2; decreasing on 1- q , 04 (b) Even (c) Minimum of - 10 at x = 0 (d) Absolute value function, shifted 10 units down 41. (a) Increasing on 32, q 2; decreasing on 1 - q , 24

(b) Neither

(c) Minimum of 0 at x = 2

45.

33. Domain: all reals; Range: all integers

(d) Absolute value function, shifted 2 units right 47.

y

43. y = 2, y = - 2

49.

y

y

5

5

5

x

5

x

5

No points of discontinuity

No points of discontinuity 51.

5

y

x

No points of discontinuity

53. (a)

55. (a)

5

5

x [–5, 5] by [–5, 5]

[–5, 5] by [–5, 5]

g1x2 = ƒ x ƒ

ƒ1x2 = x 2

x = 0

2

(b) ƒ1x2 = 2x = 2 ƒ x ƒ = ƒ x ƒ = g1x2

(b) The fact that ln1e x2 = x shows that the natural logarithm function takes on arbitrarily large values. In particular, it takes on the value L when x = eL.

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 843

SELECTED ANSWERS

57. Domain: all real numbers; Range: all integers; Continuity: There is a discontinuity at each integer value of x; Increasing/decreasing behavior: constant on intervals of the form 3k, k + 12, where k is an integer; Symmetry: none; Boundedness: not bounded; Local extrema: every non-integer is both a local minimum and local maximum; Horizontal asymptotes: none; Vertical asymptotes: none; End behavior: int1x2 : - q as x : - q and int1x2 : q as x : q . 59. True; the asymptotes are x = 0 and x = 1. 61. D 63. E 65. (a) Even (b) Even (c) Odd 67. (a) Pepperoni count ought to be proportional to the area of the pizza, which is proportional to the square of the radius (b) 0.75 (c) Yes, very well. (d) The fact that the pepperoni count fits the expected quadratic model so perfectly suggests that the pizzeria uses such a chart. If repeated observations produced the same results, there would be little doubt. 69. (a) ƒ1x2 = 1/x, ƒ1x2 = ex, ƒ1x2 = ln x, ƒ1x2 = cos x, ƒ1x2 = 1/(1 + e -x2 (b) ƒ1x2 = x (c) ƒ1x2 = ex (e) The odd functions: x, x 3, 1/x, sin x

(d) ƒ1x2 = ln x

SECTION 1.4 Exploration 1 ƒ

g

ƒ ! g

2x - 3

x + 3 2

x

ƒ 2x + 4 ƒ

1x - 221x + 22

x2

x2

ƒxƒ

x5

x 0.6

x3

x - 3

ln1e3x2

ln x

2 sin x cos x

x 2

sin x

2

1x

x sin a b 2

1 - 2x 2

cos x

Quick Review 1.4 1. 1 - q , - 32 ´ 1 - 3, q 2

Exercises 1.4

3. 1- q , 54

5. 31, q 2

7. 1- q , q 2

9. 1 -1, 12

1. 1ƒ + g21x2 = 2x - 1 + x 2; 1ƒ - g21x2 = 2x - 1 - x 2; 1ƒg21x2 = 12x - 121x 22 = 2x 3 - x 2. There are no restrictions on any of the domains, so all three domains are 1- q , q 2.

3. 1ƒ + g21x2 = 1x + sin x; 1ƒ - g21x2 = 1x - sin x; 1ƒg21x2 = 1x sin x. Domain in each case is 30, q 2. 1x + 3

; x + 3 Ú 0 and x Z 0, so the domain is 3- 3, 02 ´ 10, q 2. x2 x2 ; x + 3 7 0, so the domain is 1 -3, q 2. 1g/ƒ21x2 = 1x + 3

5. 1 ƒ/g21x2 =

7. 1 ƒ/g21x2 = x 2/ 21 - x 2; 1 - x 2 7 0, so x 2 6 1; the domain is 1 - 1, 12. 9.

1g/ƒ21x2 = 21 - x 2/x 2; 1 - x 2 Ú 0 and x Z 0; the domain is 3-1, 02 ´ 10, 14. 11. 5; - 6

13. 8; 3

15. ƒ1g1x22 = 3x - 1; 1 - q , q 2; g1ƒ1x22 = 3x + 1; 1- q , q 2

17. ƒ1g1x22 = x - 1; 3 -1, q 2; g1ƒ1x22 = 2x 2 - 1; 1 - q , -14 ´ 31, q 2 19. ƒ1g1x22 = 1 - x 2; 3 - 1, 14; g1ƒ1x22 = 21 - x 4; 3 -1, 14 [0, 5] by [0, 5]

843

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 844

SELECTED ANSWERS

844

21. ƒ1g1x22 =

3x 2x ; 1 - q , 02 ´ 10, q 2; g1ƒ1x22 = ; 1 - q , 02 ´ 10, q 2 2 3

23. One possibility: ƒ1x2 = 1x and g1x2 = x 2 - 5x 25. One possibility: ƒ1x2 = ƒ x ƒ and g1x2 = 3x - 2 29. One possibility: ƒ1x2 = cos x and g1x2 = 1x

27. One possibility: ƒ1x2 = x 5 + 2 and g1x2 = x - 3 4 4 31. V = pr 3 = p148 + 0.03t23; 775,734.6 in.3 33. t L 3.63 sec 3 3

37. y = 225 - x 2 and y = - 225 - x 2

35. 13, -12

39. y = 2x 2 - 25 and y = - 2x 2 - 25

43. y = x and y = - x or y = ƒ x ƒ and y = - ƒ x ƒ 45. False; x is not in the domain of 1ƒ/g21x2 if g1x2 = 0. 47. C 49. E 41. y = 1 - x and y = x - 1

51.

ƒ

g

D

ex

2 ln x

1x 2 + 222

1x - 2

10, q 2

1

x + 1 x

1x 2 - 222

12 - x

2

1x - 12

x 2 - 2x + 1 a

(b) g1x2 = 1

55.

(c) g1x2 = x

y =

- x 2 " 2x 4 + 20 2

32, q 2

1 - q , 24 x Z 0

[–9.4, 9.4] by [–6.2, 6.2]

1- q, q2

x + 1

x + 1 2 b x

53. (a) g1x2 = 0

1 x - 1

x Z 1

SECTION 1.5 Exploration 1 1. T starts at -4, at the point 1 -8, -32. It stops at T = 2, at the point 18, 32. 61 points are computed. 3. The graph is less smooth because the plotted points are further apart. 5. The grapher skips directly from the point 10, - 12 to the point 10, 12, corresponding to the T-values T = - 2 and T = 0. The two points are connected by a straight line, hidden by the Y-axis. 7. Leave everything else the same, but change Tmin back to -4 and Tmax to -1.

Quick Review 1.5 1. y =

1 x + 2 3

3. y = "1x - 4

5. y =

3x + 2 1 - x

7. y =

4x + 1 x - 2

9. y = x 2 - 3, y Ú - 3, and x Ú 0

Exercises 1.5 1. 16, 92

3. 115, 22

5. (a) 1 -6, - 102, 1 - 4, - 72, 1- 2, -42, 10, -12, 12, 22, 14, 52, 16, 82 (b) 1.5x - 1; It is a function. (c)

[–5, 5] by [–3, 3]

9. (a) No

(b) Yes

(b) x = 1y + 222; It is not a function. (c)

[–1, 5] by [–5, 1]

11. (a) Yes (b) Yes

17. ƒ -11x2 = x 2 + 3, x Ú 0

7. (a) 19, - 52, 14, - 42, 11, -32, 10, -22, 11, -12, 14, 02, 19, 12

13. ƒ-11x2 =

19. ƒ -11x2 = 2 3 x, 1- q , q 2

1 x + 2, 1- q , q 2 3

15. ƒ -11x2 =

21. ƒ -11x2 = x 3 - 5, 1- q , q 2

x + 3 , 1- q , 22 ´ (2, q 2 2 - x

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 845

SELECTED ANSWERS

23. One-to-one

845

25. One-to-one y

3

5

x

1 1 1 27. ƒ1g1x22 = 3 c 1x + 22 d - 2 = x + 2 - 2 = x; g1ƒ1x22 = 313x - 22 + 24 = 13x2 = x 3 3 3

29. ƒ1g1x22 = 31x - 121/343 + 1 = 1x - 121 + 1 = x - 1 + 1 = x; g1ƒ1x22 = 31x 3 + 12 - 141/3 = 1x 321/3 = x 1 = x

1 + 1 x - 1 1 = 1x - 12a + 1 b = 1 + x - 1 = x; g1ƒ1x22 = 31. ƒ1g1x22 = 1 x - 1 x + x x - 1 25 33. (a) 108 euros (b) y = x. This converts euros 1x2 to dollars 1y2. (c) $44.44 27 the domain of the function y = x 2 to the interval 30, q 2, then the restricted function and y

1 1

= § - 1

1 x x x ¥# = = = x x x + 1 x + 1 - x 1 - 1 x

35. y = ex and y = ln x are inverses. If we restrict = 1x are inverses.

37. y = ƒ x ƒ

39. True. All the ordered pairs swap domain and range values. 41. E 43. C 45. Answers may vary. (a) If the graph of ƒ is unbroken, its reflection in the line y = x will be also. (b) Both ƒ and its inverse must be one-to-one in order to be inverse functions. (c) Since f is odd, 1 - x, - y2 is on the graph whenever 1x, y2 is. This implies that 1-y, -x2 is on the graph of ƒ -1 whenever 1x, y2 is. That implies that ƒ -1 is odd. (d) Let y = ƒ1x2. Since the ratio of ¢ y to ¢ x is positive, the ratio of ¢ x to ¢ y is positive. Any ratio of ¢ y to ¢ x on the graph of ƒ -1 is the same as some ratio of ¢ x to ¢ y on the graph of ƒ, hence positive. This implies that ƒ -1 is increasing. 4 47. (a) y = 0.75x + 31 (b) y = 1x - 312. It converts scaled scores to raw scores. 49. (a) No (b) No (c) 45°; yes 3 51. When k = 1, the scaling function is linear. Opinions will vary as to which is the best value of k.

SECTION 1.6 Exploration 1 1. They raise or lower the parabola along the y-axis.

3. Yes

Exploration 2 1. Graph C. Points with positive y-coordinates remain unchanged, while points with negative y-coordinates are reflected across the x-axis. 3. Graph F. The graph will be a reflection across the y-axis of graph C.

Exploration 3 1.

The 1.5 and the 2 stretch the graph vertically; the 0.5 and the 0.25 shrink the graph vertically.

[–4.7, 4.7] by [–1.1, 5.1]

Quick Review 1.6 1. 1x + 122

3. 1x + 622

Exercises 1.6

5. 1x - 5/222

7. x 2 - x + 2

9. x 3 - 6x + 5

1. Vertical translation down 3 units 3. Horizontal translation left 4 units 5. Horizontal translation to the right 100 units 7. Horizontal translation to the right 1 unit, and Vertical translation up 3 units 9. Reflection across x-axis 11. Reflection across y-axis 1 3 13. Vertically stretch by 2 15. Horizontally stretch by 17. Translate right 6 units to get g = 5, or vertically shrink by 0.2 = 0.008 0.2

6965_SE_Ans_833-934.qxd

1/26/10

11:28 AM

Page 846

SELECTED ANSWERS

846

19. Translate left 4 units, and reflect across the x-axis to get g y y 21. 23. 3

h

h

–6

6

x

3

–6

6

f

x

f

g

g –6

–6

25. ƒ1x2 = 1x + 5

29. (a) - x 3 + 5x 2 + 3x - 2

27. ƒ1x2 = - 1x + 2 + 3 = 3 - 1x + 2

(b) - x 3 - 5x 2 + 3x + 2

31. (a) y = - ƒ1x2 = - 11 3 8x2 = - 21 3 x (b) y = ƒ1-x2 = 1 3x 3 81-x2 = - 2 1 33. Let ƒ be an odd function; that is, ƒ1-x2 = - ƒ1x2 for all x in the domain of ƒ. To reflect the graph of y = ƒ1x2 across the y-axis, we make the tranformation y = ƒ1- x2. But ƒ1- x2 = - ƒ1x2 for all x in the domain of ƒ, so this transformation results in y = - ƒ1x2. That is exactly the translation that reflects the graph of ƒ across the x-axis, so the two reflections yield the same graph. y y 35. 37. 39. (a) 2x 3 - 8x (b) 27x 3 - 12x 41. (a) 2x 2 + 2x - 4 (b) 9x 2 + 3x - 2 43. Starting with y = x 2, translate right 3 units, vertically stretch by 2, and translate down 4 units. x x 1 45. Starting with y = x 2, horizontally shrink by and translate 3 down 4 units. 47. y = 31x - 422 51.

53.

y

y 5

5 –5

5

x

5

–5

–5

x

49. y = 2 ƒ x + 2 ƒ - 4

55. Reflections have more effect on points that are farther away from the line of reflection. Translations affect the distance of points from the axes, and hence change the effect of the reflections. 9 57. First vertically stretch by , then translate up 32 units. 5 59. False; it is translated left. 61. C 63. A

–5

65. (a)

(b) Change the y-value by multiplying by the conversion rate from dollars to yen, a number that changes according to international market conditions. This results in a vertical stretch by the conversion rate.

y

Price (dollars)

30 25 20 15 10 5 0

x 1 2 3 4 5 6 7 8 9 10 11 12

Month

67. (a) The original graph is on the top; the graph of y = ƒ ƒ1x2 ƒ is on the bottom.

[–5, 5] by [–10, 10]

[–5, 5] by [–10, 10]

(b) The original graph is on the top; the graph of y = ƒ1 ƒ x ƒ 2 is on the bottom.

[–5, 5] by [–10, 10]

[–5, 5] by [–10, 10]

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 847

SELECTED ANSWERS

(c)

(d)

y

847

y

x

x

SECTION 1.7 Exploration 1 1.

n = 3; d = 0

n = 4; d = 2

n = 5; d = 5

n = 8; d = 20

n = 7; d = 14

n = 9; d = 27

n = 6; d = 9

n = 10; d = 35

2, 5, 9, 14, 20, 27 3. Linear: r 2 = 0.9758; power: r 2 = 0.9903; quadratic: R2 = 1; cubic: R2 = 1; quartic: R2 = 1 5. Since the quadratic curve fits the points perfectly, there is nothing to be gained by adding a cubic term or a quartic term. The coefficients of these terms in the regressions are zero.

Quick Review 1.7 1. h = 21A/b2

Exercises 1.7

3. h = V/1pr 22

3V 5. r = 3 A 4p

7. h =

A - 2pr 2 A = - r 2pr 2pr

9. P =

A = A11 + r/n2-nt 11 + r/n2nt

1. 3x + 5 3. 0.17x 5. 1x + 1221x2 7. 1.045x 9. 0.60x 11. Let C be the total cost and n be the number of items produced; C = 34,500 + 5.75n. 13. Let R be the revenue and n be the number of items sold: R = 3.75n. 15. V = (2/3)pr 3 2 2 17. A = a 115/4 19. A = 24r 21. x + 4x = 620; x = 124; 4x = 496 23. 1.035x = 36,432; x = 35,200 25. 182 = 52t, so t = 3.5 hr 27. 0.601332 = 19.8, 0.751272 = 20.25; The $33 shirt is a better bargain, because the sale price is cheaper. 29. 9%

31. (a) 0.10x + 0.451100 - x2 = 0.2511002

45% solution.

33. (a) V = x110 - 2x2118 - 2x2

(b) Use x L 57.14 gallons of the 10% solution and about 42.86 gal of the

(b) 10, 52

(c) Approx. 2.06 in. by 2.06 in.

35. 6 in.

37. Approx. 21.36 in.

39. Approx. 11.42 mph 41. True; the correlation coefficient is close to 1 if there is a good fit. 43. C 45. B 47. (a) C = 100,000 + 30x (b) R = 50x (c) x = 5000 pairs of shoes (d) The point of intersection corresponds to the break-even point, where C = R.

49. (a) y1 = u1x2 = 125,000 + 23x

(e)

(b) y2 = s1x2 = 125,000 + 31x

(c) y3 = Ru1x2 = 56x

(f) You should recommend stringing the rackets; fewer strung rackets need to be sold to begin making a profit (since the intersection of y2 and y4 occurs for smaller x than the intersection of y1 and y3).

[0, 10,000] by [0, 500,000]

(d) y4 = Rs1x2 = 79x

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 848

SELECTED ANSWERS

848

51. (a)

(b) List L3 = 5112.3, 106.5, 101.5, 96.6, 92.0, 87.2, 83.1, 79.8, 75.0, 71.7, 68, 64.1, 61.5, 58.5, 55.9, 53.0, 50.8, 47.9, 45.2, 43.26 (c) y = 118.07 * 0.951x. It fits the data extremely well.

[0, 22] by [100, 200]

CHAPTER 1 REVIEW EXERCISES 1. d

3. i

5. b

7. g q 15. (a) All reals (b) 38, 2

9. a

11. (a) All reals

(b) All reals

17. (a) All reals except 0 and 2

19. Continuous

21. (a) Vertical asymptotes at x = 0 and x = 5 23. (a) none (b) y = 7 and y = - 7 25. 1- q , q 2 27. 1 - q , - 12, 1 - 1, 12, 11, q 2 29. Not bounded 31. Bounded above

33. (a) None

35. (a) -1, at x = 0

(b) None

39. Neither

47.

[–5, 5] by [–5, 5]

(b) All reals except 0 (b) y = 0

(b) - 7, at x = - 1

37. Even

45.

13. (a) All reals (b) 30, q 2

41. 1x - 32/2

43. 2/x

[–5, 5] by [–5, 5]

49.

51.

[–5, 5] by [–5, 5]

[–5, 5] by [–5, 5]

2

53. 1ƒ ! g21x2 = 2x - 4; 1 - q , - 24 ´ 32, q 2 57. lim 1x = q 59. ps 2/2 61. 100ph x: q

65. (a)

55. 1ƒ ! g21x2 = 1x1x 2 - 42; 30, q 2 63. 40 - t/150p2

(b) The regression line is y = 79.58x + 1256.11.

[–1, 9] by [1160, 1970]

(c) 2450 (thousands of barrels)

[–1, 9] by [1160, 1970]

67. (a) h = 223 - r 2

(b) 2pr 2 23 - r 2

(c) 30, 134

(e) 12.57 in.3

(d)

3 h h 2

3 r

3 r

h = 2 3 – r2

[0, !"3 ] by [0, 20]

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 849

SELECTED ANSWERS

Chapter 1 Project 1.

3. 34.854; 49.785

5. The logistic model for the data is y =

16,098 1 + 432.23e -0.3678x

.

[–2, 20] by [–1720, 12000]

SECTION 2.1 Exploration 1 1. - $2000 per year

3. $50,000; $18,000

Quick Review 2.1 1. y = 8x + 3.6

3. y = - 0.6x + 2.8

5. x 2 + 6x + 9

7. 3x 2 - 36x + 108

9. 21x - 122

y 7

(!2, 4) (3, 1) 5

x

Exercises 2.1 1. Not a polynomial function because of the exponent -5 5. Not a polynomial function because of the radical 7. ƒ1x2 =

5 18 x + 7 7

3. Polynomial of degree 5 with leading coefficient 2

4 2 9. ƒ1x2 = - x + 3 3

y

11. ƒ1x2 = - x + 3 y

y

5

5

10 (2, 4)

3

(0, 3)

(–4, 6) (–1, 2)

x

10

(3, 0) x 5

x

(–5, –1)

13. (a)

15. (b)

17. (e)

19. Translate the graph of y = x 2 3 units right and the result 2 units down. y 10

21. Translate the graph of y = x 2 2 units left, vertically 1 shrink the resulting graph by a factor of , and translate 2 that graph 3 units down. y 10

10

x 10

x

849

6965_SE_Ans_833-934.qxd

850

1/26/10

11:31 AM

Page 850

SELECTED ANSWERS

23. Vertex: 11, 52; axis: x = 1

25. Vertex: 11, -72; axis: x = 1

5 73 5 5 2 73 27. Vertex: a - , - b; axis: x = - ; ƒ1x2 = 3a x + b 6 12 6 6 12 29. Vertex: 14, 192; axis: x = 4; ƒ1x2 = - 1x - 422 + 19

3 3 11 3 2 11 31. Vertex: a , b ; axis: x = ; g1x2 = 5 a x - b + 5 5 5 5 5 33. ƒ1x2 = 1x - 222 + 2; Vertex: 12, 22; axis: x = 2; opens upward; does not intersect x-axis

35. ƒ1x2 = - 1x + 822 + 74; Vertex: 1-8, 742; axis: x = - 8; opens downward; intersects x-axis at about -16.602 and 0.602, or 1- 8 ! 1742

37. ƒ1x2 = 2 ax +

3 2 5 b + ; 2 2

3 5 3 Vertex: a - , b ; axis: x = - ; 2 2 2 opens upward; does not intersect x-axis; vertically stretched by 2

[–4, 6] by [0, 20] [–20, 5] by [–100, 100] [–3.7, 1] by [2, 5.1]

39. y = 21x + 122 - 3 49. (a)

41. y = - 21x - 122 + 11 (b) Strong positive

43. y = 21x - 122 + 3

45. Strong positive

47. Weak positive

51. $940

[15, 45] by [20, 50]

53. (a) y L 0.371x + 2.891. The slope says that hourly compensation for production workers increases about 37¢/yr. 55. (a) 30, 1004 by 30, 10004 is one possibility. (b) either 107,335 units or 372,665 units 57. 3.5 ft

59. (a) R1x2 = 126,000 - 1000x210.50 + 0.05x2

(b)

(b) About $19.59

(c) 90 cents per can; $16,200

[0, 15] by [10,000, 17,000]

61. (a) About 215 ft above the field (b) About 6.54 sec (c) About 117 ft/sec downward 63. (a) h = - 16t 2 + 80t - 10 (b) 90 ft, 2.5 sec 65. 2006

[0, 5] by [–10, 100]

(b) y L 0.68x + 9.01

67. (a)

(d)

(e) L 29.41 lb

(c) On average, the children gained 0.68 pounds per month.

[15, 45] by [20, 40]

[15, 45] by [20, 40]

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 851

SELECTED ANSWERS

851

71. False. The initial value is ƒ102 = - 3

69. The Identity Function ƒ1x2 = x Domain: 1 - q , q 2; Range: 1 - q , q 2; Continuous; Increasing for all x; Symmetric about the origin; Not bounded; No local extrema; No horizontal or vertical asymptotes; End behavior: lim ƒ1x2 = - q , lim ƒ1x2 = q x: - q

73. E

75. B

x: q

[–4.7, 4.7] by [–3.1, 3.1]

77. (a) (i), (iii), and (v) because they are slant lines (b) (i), (iii), (iv), (v), and (vi) because they are not vertical (c) (ii) is not a function because it is vertical. 79. The line that minimizes the sum of the squares of vertical distances is nearly always different from the line that minimizes the sum of the squares of horizontal distances to the points in a scatter plot. For the data in Table 2.2, the regression line obtained from reversing the ordered pairs has a slope of - 1/15,974.90; whereas, the inverse of the function in Example 3 has a slope of - 1/15,358.93—close but not the same slope. 81. (a) The two solutions are

- b + 2b 2 - 4ac b b -b - 2b 2 - 4ac and ; their sum is 2 a - b = - . a 2a 2a 2a

(b) The product of the two solutions given above is

b 2 - (b 2 - 4ac)

=

c . a

83. a

1a - b22 a + b ,b 2 4

4a 2 85. Suppose ƒ1x2 = mx + b with m and b constants and m Z 0. Let x 1 and x 2 be real numbers with x 1 Z x 2. Then the average rate of change 1mx 2 + b2 - 1mx 1 + b2 m1x 2 - x 12 ƒ1x 22 - ƒ1x 12 of ƒ is = = = m, a nonzero constant. On the other hand, suppose m and x 1 are constants x2 - x1 x2 - x1 x2 - x1 ƒ1x2 - ƒ1x 12 and m Z 0. Let x be a real number with x Z x 1 and let ƒ be a function defined on all real numbers such that = m. Then x - x1 ƒ1x2 - ƒ1x 12 = m1x - x 12 and ƒ1x2 = mx + 1ƒ1x 12 - mx 12. Notice that the expression ƒ1x 12 - mx 1 is a constant; call it b. Then

ƒ1x 12 - mx 1 = b; so, ƒ1x 12 = mx 1 + b and ƒ1x2 = mx + b for all x Z x 1. Thus ƒ is a linear function.

SECTION 2.2 Exploration 1 1.

The pairs 10, 02, 11, 12 and 1 -1, - 12 are common to all three graphs.

[–2.35, 2.35] by [–1.5, 1.5]

[–5, 5] by [–15, 15]

[–20, 20] by [–200, 200]

Quick Review 2.2 3 x2 1. 2

3. 1/d 2

5 q4 5. 1/2

9. L 1.71x -4/3

7. 3x 3/2

Exercises 2.2 1 2 9. Power = - 2, constant = k

1. Power = 5, constant = -

3. Not a power function

5. Power = 1, constant = c2

11. Degree = 0, coefficient = - 4

7. Power = 2, constant =

g 2

13. Degree = 7, coefficient = - 6

2

15. Degree = 2, coefficient = 4p 17. A = ks 19. I = V/R 21. E = mc2 23. The weight w of an object varies directly with its mass m, with the constant of variation g. 25. The refractive index n of a medium is inversely proportional to v, the velocity of light in the medium, with constant of variation c, the constant velocity of light in free space. 27. Power = 4, constant = 2; Domain: 1 - q , q 2; Range: 30, q 2; Continuous; Decreasing on 1- q , 02. Increasing on 10, q 2; Even. Symmetric with respect to y-axis; Bounded below, but not above; Local minimum at x = 0; Asymptotes: none; End Behavior: lim 2x 4 = q , lim 2x 4 = q x: - q

x: q

[–5, 5] by [–1, 49]

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 852

SELECTED ANSWERS

852

1 1 , constant = ; Domain: 30, q 2; 4 2 Range: 30, q 2; Continuous; Increasing on 30, q 2; Bounded below; Neither even nor odd; Local minimum at 10, 02; Asymptotes: none; 1 End Behavior: lim 1 4x = q x: q 2

29. Power =

2 31. Shrink vertically by ; ƒ is even. 3

[–1, 99] by [–1, 4]

33. Stretch vertically by 1.5 and reflect over the x-axis; ƒ is odd.

1 35. Shrink vertically by ; ƒ is even. 4

[–5, 5] by [–20, 20]

[–5, 5] by [–1, 19]

[–5, 5] by [–1, 49]

1 . ƒ is increasing in Quadrant I. ƒ is undefined for x 6 0. 4 4 1 45. k = - 2, a = . ƒ is decreasing in Quadrant IV. ƒ is even. 47. k = , a = - 3. ƒ is decreasing in Quadrant I. ƒ is odd. 3 2 8 49. y = 2 , power = - 2, constant = 8 51. 2.21 L 53. 1.24 * 10 8 m/sec x 55. (a) (b) r L 231.204 # w -0.297 (c) (d) Approximately 37.67 beats/min, which is very close to Clark’s observed value 37. (g)

39. (d)

41. (h)

[–2, 71] by [50, 450]

57. (a)

43. k = 3, a =

(b) y L 7.932 # x -1.987; yes

[–2, 71] by [50, 450]

(d) Approximately 2.76 W/m2 and 0.697 W/m2, respectively

(c)

[0.8, 3.2] by [!0.3, 9.2]

[0.8, 3.2] by [–0.3, 9.2]

59. False, because ƒ1-x2 = 1- x21/3 = - x 1/3 = - ƒ1x2. The graph of ƒ is symmetric about the origin. 65. (a)

[0, 1] by [0, 5]

[0, 3] by [0, 3]

[–2, 2] by [–2, 2]

61. E

63. B

The graphs of ƒ1x2 = x -1 and h1x2 = x -3 are similar and appear in the first and third quadrants only. The graphs of g1x2 = x -2 and k1x2 = x -4 are similar and appear in the first and second quadrants only. The pair 11, 12 is common to all four functions.

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 853

SELECTED ANSWERS

ƒ

g

h

k

Domain

x Z 0

x Z 0

x Z 0

x Z 0

Range

y Z 0

y 7 0

y Z 0

y 7 0

Continuous

yes

yes

yes

yes

Increasing 1- q , 02, 10, q 2

Decreasing

1- q , 02

10, q 2

1- q , 02, 10, q 2

1 - q , 02

10, q 2

Symmetry

w.r.t. origin

w.r.t. y-axis

w.r.t. origin

w.r.t. y-axis

Bounded

not

below

not

below

Extrema

none

none

none

none

Asymptotes

x-axis, y-axis

x-axis, y-axis

x-axis, y-axis

x-axis, y-axis

End Behavior

lim ƒ(x) = 0

lim g(x) = 0

x: " q

x: " q

lim h(x) = 0

x: " q

lim k(x) = 0

x: " q

The graphs of ƒ1x2 = x 1/2 and h1x2 = x 1/4 are similar and appear in the first quadrant only. The graphs of g1x2 = x 1/3 and k1x2 = x 1/5 are similar and appear in the first and third quadrants only. The pairs 10, 02, 11, 12 are common to all four functions.

(b)

[0, 1] by [0, 1]

[0, 3] by [0, 2]

f Domain

30, q 2

h

k

1- q , q 2

y Ú 0

yes

1- q , q 2

yes

yes

yes

30, q 2

1- q , q 2

30, q 2

1- q, q2

Range

y Ú 0

Continuous Increasing Decreasing

[–3, 3] by [–2, 2]

g

30, q 2

3- q , q 2

1- q, q2

Symmetry

none

w.r.t. origin

none

w.r.t. origin

Bounded

below

not

below

not

Extrema Asymptotes End behavior

min at 10, 02

none

lim ƒ1x2 = q

x: q

min at 10, 02

none none

none

lim g1x2 = q

x: q

lim g1x2 = - q

853

lim h1x2 = q

x: q

none none lim k1x2 = q

x: q

lim k1x2 = - q

x: -q

x: -q

67. T L a 1.5. Squaring both sides shows that approximately T 2 = a 3.

69. If ƒ1x2 is even, g1 -x2 = 1/ƒ1 -x2 = 1/ƒ1x2 = g1x2. If ƒ1x2 is

odd, g1 -x2 = 1/ƒ1 - x2 = 1/1- ƒ1x22 = - 1/ƒ1x2 = - g1x2. If g1x2 = 1/ƒ1x2, then ƒ1x2 # g1x2 = 1 and ƒ1x2 = 1/g1x2. So by the reasoning used above, if g1x2 is even, so is ƒ1x2, and if g1x2 is odd, so is ƒ1x2. object and the acceleration a of the object. the velocity v of the object.

71. (a) The force F acting on an object varies jointly as the mass m of the

(b) The kinetic energy KE of an object varies jointly as the mass m of the object and the square of

(c) The force of gravity F acting on two objects varies jointly as their masses m1 and m 2 and inversely as the square

of the distance r between their centers, with the constant of variation G, the universal gravitational constant.

SECTION 2.3 Exploration 1 1. (a) q ; - q

(b) - q ; q

(c) q ; - q

(d) - q ; q

Exploration 2 1. y = 0.0061x 3 + 0.0177x 2 - 0.5007x + 0.9769

3. (a) - q ; q

(b) - q ; - q

(c) q ; q

(d) q ; - q

6965_SE_Ans_833-934.qxd

854

1/25/10

3:40 PM

Page 854

SELECTED ANSWERS

Quick Review 2.3 1. 1x - 421x + 32

3. 13x - 221x - 32

Exercises 2.3

1. Shift y = x 3 to the right by 3 units, stretch vertically by 2. y-intercept: 10, - 542

5. x13x - 221x - 12

7. x = 0, x = 1

3. Shift y = x 3 to the left by 1 unit, 1 vertically shrink by , reflect over the 2 x-axis, and then vertically shift up 2 units. 3 y-intercept: a0, b 2

y

10

9. x = - 6, x = - 3, x = 1.5 5. Shift y = x 4 to the left 2 units, vertically stretch by 2, reflect over the x-axis, and vertically shift down 3 units. y-intercept: 10, -352 y

40

y

10

x

5 5 5

7. Local maximum: L 10.79, 1.192, zeros: x = 0 and x L 1.26. 13. One possibility:

9. (c)

x

11. (a)

15. One possibility:

[–100, 100] by [–1000, 1000]

[–50, 50] by [–1000, 1000]

17.

19.

[–8, 10] by [–120, 100]

[–5, 3] by [–8, 3]

lim ƒ1x2 = q ; lim ƒ1x2 = - q

x: q

lim ƒ1x2 = - q ; lim ƒ1x2 = q

x: - q

21.

x: q

x: -q

23.

[–3, 5] by [–50, 50]

[–5, 5] by [–14, 6]

lim ƒ1x2 = q ; lim ƒ1x2 = q

x: q

25. q , q

x: - q

lim ƒ1x2 = q ; lim ƒ1x2 = q

x: q

27. - q , q

29. (a) There are 3 zeros: they are - 2.5, 1, and 1.1. 31. (c) There are 3 zeros: approximately - 0.273 a actually 33. -4 and 2

35.

2 1 and 3 3

2 37. 0, - , and 1 3

3 b , -0.25, and 1. 11

x: -q

x

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 855

SELECTED ANSWERS

39. Degree 3; zeros: x = 0 (mult. 1, graph crosses x-axis), x = 3 (mult. 2, graph is tangent)

855

41. Degree 5; zeros: x = 1 (mult. 3, graph crosses x-axis), x = - 2 (mult. 2, graph is tangent) y

y 10

10

6

x –5

x

5

–10

43.

45.

47.

[–3, 3] by [–10, 10]

[–3, 2] by [–10, 10]

-2.43, -0.74, 1.67 49. 0, -6, and 6

51. -5, 1, 11

[–6, 4] by [–100, 20]

-2.47, - 1.46, 1.94 3

2

53. ƒ1x2 = x - 5x - 18x + 72

-4.90, -0.45, 1, 1.35 3

2

55. ƒ1x2 = x - 4x - 3x + 12

57. y = 0.25x 3 - 1.25x 2 - 6.75x + 19.75 59. y = - 2.21x 4 + 45.75x 3 - 339.79x 2 + 1075.25x - 1231 61. It follows from the Intermediate Value Theorem. 63. (a) (c) 65. (a)

[0, 60] by [–10, 210] 2

(b) y = 0.051x + 0.97x + 0.26

[0, 60] by [–10, 210]

(d) L 56.39 ft

(e) 67.74 mph

[0, 0.8] by [0, 1.20]

(b) 0.3391 cm

67. 0 6 x … 0.929 or 3.644 … x 6 5 69. True. Because ƒ is continuous and ƒ112 = - 2 and ƒ122 = 2, the Intermediate Value Theorem assures us that the graph of f crosses the x-axis between x = 1 and x = 2. 71. C 73. B 75. The figure at left shows the end behavior and a zero of x L 9, but hides the other four zeros. The figure at right shows zeros near - 2, -1, 1, and 3, but hides the fifth zero and the end behavior. 77. The exact behavior near x = 1 is hard to see. A zoomed-in view around the point 11, 02 suggests that the graph just touches the x-axis at 0 without actually crossing it — that is, 11, 02 is alocal maximum. One possible window is 30.9999, 1.00014 by 3- 1 * 10 -7, 1 * 10 -74. 79. A maximum and minimum are not visible in the standard window, but can be seen on the window 30.2, 0.44 by 35.29, 5.34. 81. The graph of y = 31x3 - x2 increases, then decreases, then increases; the graph of y = x3 only increases. Therefore, this graph cannot be obtained from the graph of y = x3 by the transformations studied in Chapter 1 (translations, reflections, and stretching/shrinking). Since the right side includes only these transformations, there can be no solution. 83. (a) Substituting x = 2, y = 7, we find that 7 = 512 - 22 + 7, so Q is on line L, and also ƒ122 = - 8 + 8 + 18 - 11 = 7, so Q is on the graph of ƒ1x2.

(c) The line L also crosses the graph of ƒ1x2 at 1-2, -132.

(b)

[1.8, 2.2] by [6, 8]

y uy 8 8 x 8 # = y - 8 implies D - u = u1y - 82 . Combining these yields uy = u1y - 82 , which = and = imply D - u = u x u x D - u D D D - u 8 8 y - 8 8 8 8 8 8 x - 8 8x implies = . (b) Equation (a) says = 1 - . So, = 1 - = . Thus y = . (c) By the Pythagorean Theorem, x y x y y x x x - 8 8x 2 b - x 2. Thus, 500 1x - 822 = y 2 + D 2 = 900 and x 2 + D 2 = 400. Subtracting equal quantities yields y 2 - x 2 = 500. So, 500 = a x - 8

85. (a)

64x 2 - x 21x - 822, or 500x 2 - 8000x + 32000 = 64x 2 - x 4 + 16x 3 - 64x 2. This is equivalent to x 4 - 16x 3 + 500x 2 - 8000x + 32000 = 0.

(d) Notice that 8 6 x 6 20. So, the solution we seek is x L 11.71, which yields y L 25.24 and D L 16.21.

6965_SE_Ans_833-934.qxd

856

1/25/10

3:40 PM

Page 856

SELECTED ANSWERS

SECTION 2.4 Quick Review 2.4 1. x 2 - 4x + 7

3. 7x 3 + x 2 - 3

Exercises 2.4 1. ƒ1x2 = 1x - 122 + 2;

ƒ1x2

5. x1x + 221x - 22

= x - 1 +

x - 1 ƒ1x2 7 9/2 = 2x 2 - 5x + 2x + 1 2 2x + 1

2 x - 1

7. 41x + 521x - 32

9. 1x + 221x + 121x - 12

3. ƒ1x2 = 1x 2 + x + 421x + 32 - 21;

ƒ1x2 x + 3

= x2 + x + 4 -

21 x + 3

ƒ1x2

-32x + 18 = x 2 - 4x + 12 + 2 x + 2x - 1 x + 2x - 1 -11 9670 -1269 7. x 2 - 6x + 9 + 9. 9x 2 + 97x + 967 + 11. - 5x 3 - 20x 2 - 80x - 317 + 13. 3 15. - 43 x + 1 x - 10 4 - x 17. 5 19. Yes 21. No 23. Yes 25. ƒ1x2 = 1x + 321x - 12(5x - 172 27. 2x 3 - 6x 2 - 12x + 16 "1, "3, " 9 3 19 "1 29. 2x 3 - 8x 2 + 31. ƒ1x2 = 31x + 421x - 321x - 52 33. 35. x - 3 ;1 ; 2 "1, " 2, " 3, "6 " 1, "2 2 5. ƒ1x2 = 1x 2 - 4x + 1221x 2 + 2x - 12 - 32x + 18;

2

37. Numbers in last line—2, 2, 7, 19—are nonnegative, so 3 is an upper bound. 39. Numbers in last line—1, 1, 3, 7, 2—are nonnegative, so 2 is an upper bound. 41. Numbers in last line—3, -7, 8, -5—have alternating signs, so –1 is a lower bound. 43. Numbers in last line—1, - 4, 7, - 2—are nonnegative, so 0 is a lower bound. 45. No zeros outside window 47. There are zeros not shown 3 49. Rational zero: ; irrational zeros: "12 51. Rational: -3; irrational: 1 " 13 53. Rational: -1 1approx. -11.002 and 12.0032. 2 1 and 4; irrational: "12 55. Rational: - and 4; irrational: none 57. $36.27; 53.7 59. - 2 61. (b) 2 is a zero of ƒ1x2 2 3 2 (c) 1x - 221x + 4x - 3x - 192 (d) One irrational zero is x L 2.04. (e) ƒ1x2 L 1x - 221x - 2.0421x 2 + 6.04x + 9.31162 4 4 4 63. False. 1x + 22 is a factor if and only if ƒ1- 22 = 0. 65. A 67. B 69. (a) Volume of buoy = pr 3 = p # 1123 = p 3 3 3 4 d pd px # (b) p # = (c) V # d = 13r 2 + x 22 # d = pd # x13r 2 + x 22/6 (d) x L 0.6527 m 3 4 3 6 71. (a) Shown is one possible view, on the window 30, 6004 by 30, 5004. (b) The maximum population, after 300 days, is 460 turkeys. (c) P = 0 when t L 523.22 — about 523 days after release. (d) Answers will vary.

[0, 600] by [0, 500]

73. (a) 0 or 2 positive zeros, 1 negative zero (b) No positive zeros, 1 or 3 negative zeros (c) 1 positive zero, no negative zeros (d) 1 positive zero, 1 negative zero 75. Answers will vary, but should include a diagram of the synthetic division and a summary: 1 3 7 4x 3 - 5x 2 + 3x + 1 = a x - b a 4x 2 - 3x + b + 2 2 4 = 12x - 12a 2x 2 -

3 3 7 x + b + 2 4 4

7 1 77. (a) (b) - , , and 3 (c) There are no rational zeros. 79. (a) Approximate zeros: - 3.126, -1.075, 0.910, 2.291. 3 2 (b) ƒ1x2 L g1x2 = 1x + 3.12621x + 1.07521x - 0.91021x - 2.2912 (c) Graphically: graph the original function and the approximate factorization on a variety of windows and observe their similarity. Numerically: Compute ƒ1c2 and g1c2 for several values of c.

SECTION 2.5 Exploration 1 1. ƒ12i2 = (2i22 - i(2i2 + 2 = - 4 + 2 + 2 = 0; ƒ1- i2 = 1- i22 - i1-i2 + 2 = - 1 - 1 + 2 = 0; no. 3. The Complex Conjugate Zeros Theorem does not necessarily hold true for a polynomial function with complex coefficients.

Quick Review 2.5 1. 1 + 3i

3. 7 + 4i

5. 12x - 321x + 12

7.

119 5 " i 2 2

9. " 1, "2, " 1/3, "2/3

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 857

SELECTED ANSWERS

857

Exercises 2.5 1. x 2 + 9; zeros: "3i; x-intercepts: none 3

2

4

3

3. x 4 - 2x 3 + 5x 2 - 8x + 4; zeros: 1 1mult. 22, "2i; x-intercept: x = 1

2

3

2

5

4

3

5. x 2 + 1 2

7. x - x + 9x - 9 9. x - 5x + 7x - 5x + 6 11. x - 11x + 43x - 65 13. x + 4x + x - 10x - 4x + 8 15. x 4 - 10x 3 + 38x 2 - 64x + 40 17. (b) 19. (d) 21. 2 complex zeros; none real 23. 3 complex zeros; 1 real 25. 4 complex zeros; 2 real 29. Zeros: x = "1, x = -

27. Zeros: x = 1, x = -

1 119 1 " i; ƒ1x2 = 1x - 1212x + 1 + 119i212x + 1 - 119i2 2 2 4

1 123 1 " i; ƒ1x2 = 1x - 121x + 1212x + 1 + 123i212x + 1 - 123i2 2 2 4

3 7 31. Zeros: x = - , x = , x = 1 " 2i; ƒ1x2 = 13x + 7212x - 321x - 1 + 2i21x - 1 - 2i2 3 2 33. Zeros: x = "13, x = 1 " i; ƒ1x2 = 1x - 1321x + 1321x - 1 + i21x - 1 - i2

35. Zeros: x = "12, x = 3 " 2i; ƒ1x2 = 1x - 1221x + 1221x - 3 + 2i21x - 3 - 2i2 37. 1x - 221x 2 + x + 12 2

39. 1x - 12(2x 2 + x + 32

3

2

45. Yes, ƒ1x2 = 1x + 22 = x + 6x + 12x + 8.

41. 1x - 121x + 421x 2 + 12

43. h L 3.776 ft

47. No, either the degree would be at least 5 or some of the coefficients would be nonreal.

49. ƒ1x2 = - 2x 4 + 12x 3 - 20x 2 - 4x + 30

51. (a) D L - 0.0820t 3 + 0.9162t 2 - 2.5126t + 3.3779 (b) Sally walks toward the detector, turns and walks away (or walks backward), then walks toward the detector again. (c) t L 1.81 sec (D L 1.35 m2 and t L 5.64 sec 1D L 3.65 m2. [–1, 9] by [0, 5]

53. False. If 1 - 2i is a zero, then 1 + 2i must also be a zero. 59. (a)

Power

Real Part

Imaginary Part

7 8 9 10

8 16 16 0

-8 0 16 32

55. E

57. C

7

(b) 11 + i2 = 8 - 8i 11 + i28 = 16

11 + i29 = 16 + 16i

11 + i210 = 32i (c) Reconcile as needed.

61. ƒ1i2 = i 3 - i(i22 + 2i1i2 + 2 = - i + i - 2 + 2 = 0

63. Synthetic division shows that ƒ1i2 = 0 (the remainder), and at the same

2

time gives ƒ1x2 , 1x - i2 = x + 3x - i = h1x2, so ƒ1x2 = 1x - i21x 2 + 3x - i2.

SECTION 2.6 Exploration 1 1. g1x2 =

1 x - 2

3. k1x2 =

3 - 2 x + 4

[–1, 9] by [–5, 5]

[–3, 7] by [–5, 5]

Quick Review 2.6 1. x = - 3, x =

1 2

3. x = "2

5. x = 1

7. 2; 7

9. 3; - 5

65. -4, 2 + 2 13 i, 2 - 213 i

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 858

SELECTED ANSWERS

858

Exercises 2.6 1. Domain: all x Z - 3; lim - ƒ1x2 = - q , lim + ƒ1x2 = q x : -3

x: -3

3. Domain: all x Z - 2, 2; lim - ƒ1x2 = - q , lim + ƒ1x2 = q , lim- ƒ1x2 = q , lim+ ƒ1x2 = - q x : -2

x :2

x: -2

5. Translate right 3 units. Asymptotes: x = 3, y = 0

x:2

7. Translate left 3 units, reflect across x-axis, vertically stretch by 7, translate up 2 units. Asymptotes: x = - 3, y = 2

y

9. Translate left 4 units, vertically stretch by 13, translate down 2 units. Asymptotes: x = - 4, y = - 2 y

y

5

8

10

5

6

x 6

11. q

13. 0

15. q

17. 5

x

x

19. Vertical asymptote: none; horizontal asymptote: y = 2; lim ƒ1x2 = lim ƒ1x2 = 2 x: -q

x: q

21. Vertical asymptotes: x = 0, x = 1; horizontal asymptote: y = 0; lim- ƒ1x2 = q , lim+ ƒ1x2 = - q , lim- ƒ1x2 = - q , x: -q

x:1

x:0

x:0

lim+ ƒ1x2 = q , lim ƒ1x2 = lim ƒ1x2 = 0

x :1

x: q

2 23. Intercepts: a0, b and 12, 02; 3 asymptotes: x = - 1, x = 3, and y = 0

25. No intercepts; 27. Intercepts: 10, 22, 1 -1.28, 02, asymptotes: x = - 1, x = 0, and (0.78, 02; asymptotes: x = 1, and y = 0 x = 1, x = - 1, and y = 2

[–4.7, 4.7] by [–10, 10] [–4, 6] by [–5, 5]

3 29. Intercept: a 0, b ; 2 asymptotes: x = - 2, y = x - 4

[–5, 5] by [–4, 6]

[–20, 20] by [–20, 20]

31. (d); Xmin = - 2, Xmax = 8, Xscl 33. (a); Xmin = - 3, Xmax = 5, Xscl 35. (e); Xmin = - 2, Xmax = 8, Xscl 2 37. Intercept: a0, - b; asymptotes: x 3 lim

x :(3/2) +

ƒ1x2 = q ;

= 1, and Ymin = - 3, Ymax = 3, Yscl = 1 = 1, and Ymin = - 5, Ymax = 10, Yscl = 1 = 1, and Ymin = - 3, Ymax = 3, Yscl = 1 3 = - 1, x = , y = 0; lim - ƒ1x2 = q , lim + ƒ1x2 = - q , lim - ƒ1x2 = - q , x : -1 x: - 1 x :(3/2) 2 3 16 3 Domain: x Z - 1, ; Range: a - q , - d ´ 10, q 2; Continuity: all x Z - 1, ; 2 25 2 1 3 1 3 Increasing: 1 - q , -12, a -1, d ; Decreasing: c , b , a , q b; Unbounded; 4 4 2 2 1 16 Local Maximum at a , - b ; Horizontal asymptote: y = 0; 4 25 [–4.7, 4.7] by [–3.1, 3.1] 3 Vertical asymptotes: x = - 1, x = ; End behavior: lim ƒ1x2 = lim ƒ1x2 = 0 x: - q x: q 2

1 b , 11, 02; asymptotes: x = - 3, x = 4, y = 0; lim - h1x2 = - q , lim + h1x2 = q , lim- h1x2 = - q , x: -3 x : -3 x:4 12 lim+ h1x2 = q Domain: x Z - 3, 4; Range: 1- q , q 2; x :4 Continuity: all x Z - 3, 4; Decreasing: 1- q , - 32, 1- 3, 42, 14, q 2;

39. Intercepts: a0,

[–5.875, 5.875] by [–3.1, 3.1]

No symmetry; Unbounded; No extrema; Horizontal asymptote: y = 0; Vertical asymptotes: x = - 3, x = 4; End behavior: lim h1x2 = lim h1x2 = 0 x: - q

x: q

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 859

SELECTED ANSWERS

2 41. Intercepts: 1 -2, 02, 11, 02, a0, b; asymptotes: x = - 3, x = 3, y = 1; lim - ƒ1x2 = q , lim + ƒ1x2 = - q , lim- ƒ1x2 = - q , x : -3 x : -3 x:3 9 Domain: x Z - 3, 3; Range: 1- q , 0.2604 ´ 11, q 2; Continuity: all x Z - 3, 3; lim ƒ1x2 = q x:3 + Increasing: 1- q , - 32, 1 -3, -0.6752; Decreasing: 1-0.675, 32, (3, q 2; No symmetry; Unbounded; Local maximum at 1-0.675, 0.2602; Horizontal asymptote: y = 1; Vertical asymptotes: x = - 3, x = 3; End behavior: lim ƒ1x2 = lim ƒ1x2 = 1 x: - q

x: q

[–9.4, 9.4] by [–3, 3]

3 43. Intercepts: 1 -3, 02, 11, 02, a0, - b ; asymptotes: x = - 2, y = x; lim - h1x2 = q , lim + h1x2 = - q x : -2 x : -2 2 Domain: x Z - 2, Range: 1- q , q 2; Continuity: all x Z - 2; Increasing: 1- q , - 22, 1-2, q 2; No symmetry; Unbounded; No extrema; Horizontal asymptote: none; Vertical asymptote: x = - 2; Slant asymptote: y = x; End behavior: lim h1x2 = - q , lim h1x2 = q [–9.4, 9.4] by [–10, 20] x: -q

x: q

49. y = x 3 + 2x 2 + 4x + 6 (a)

47. y = x 2 - 3x + 6 (a)

45. y = x + 3 (a)

[–5, 5] by [–100, 200]

[–10, 10] by [–30, 60]

[–10, 20] by [–10, 30]

(b)

(b)

(b)

4 51. Intercept: a0, b ; 5 Domain: 1 - q , q 2; Range: 30.773, 14.2274;

Continuity: 1- q , q 2; Increasing: 3 - 0.245, 2.4454; Decreasing: 1 - q , - 0.2454, 32.445, q 2; No symmetry; Bounded;

Local max at 12.445, 14.2272, local min at 1- 0.245, 0.7732; Horizontal asymptote: y = 3; Vertical asymptote: none; End behavior: lim ƒ1x2 = lim ƒ1x2 = 3 x: -q

[–20, 20] by [–5000, 5000]

[–50, 50] by [–1500, 2500]

[–500, 500] by [–500, 500]

x: q

1 53. Intercepts: 11, 02, a0, b ; 2 Domain: x Z 2; Range: 1- q , q 2; Continuity: x Z 2; Increasing: 3- 0.384, 0.4424, 32.942, q 2; Decreasing: 1 - q , -0.3844, 30.442, 22, 12, 2.9424; No symmetry; Not bounded;

Local max at 10.442, 0.5862, local min at 1 - 0.384, 0.4432 and 12.942, 25.9702; Horizontal asymptote: none; Vertical asymptote: x = 2; End behavior: lim h1x2 = lim h1x2 = q ; x: -q

x: q

End behavior asymptote: y = x 2 + 2x + 4

[–15, 15] by [–5, 15] [–10, 10] by [–20, 50]

859

6965_SE_Ans_833-934.qxd

860

1/25/10

3:40 PM

Page 860

SELECTED ANSWERS

57. Intercept: 10, 12; Vertical asymptote: x = - 1; End behavior asymptote: y = x3 - x2 + x - 1

55. Intercepts: 11.755, 02, (0, 12; 1 1 Domain: x Z ; Range: 1 - q , q 2; Continuity: x Z ; 2 2 1 1 Increasing: B -0.184, b , a , q b ; 2 2

1 59. Intercepts: 11, 02, a 0, - b ; 2 Vertical asymptote: x = - 2; End behavior asymptote: y = x 4 - 2x 3 + 4x 2 - 8x + 16

Decreasing: 1- q , - 0.1844; No symmetry; Not bounded;

Local min at 1- 0.184, 0.9202;

[–5, 5] by [–30, 30]

Horizontal asymptote: none; 1 Vertical asymptote: x = ; 2 End behavior: lim ƒ1x2 = lim ƒ1x2 = q ; x: -q

[–5, 5] by [–200, 400]

x: q

End behavior asymptote: y =

1 2 3 1 x - x + 2 4 8

[–5, 5] by [–10, 10]

61. Intercepts: 1 - 1.476, 02, 10, - 22; Vertical asymptote: x = 1; End behavior asymptote: y = 2 y = x4 + x3 + x2 + x + 1

63. False.

1

is a rational function and has no vertical asymptotes. 65. E 67. D x + 1 69. (a) No: the domain of f is 1 - q , 32 ´ (3, q 2; the domain of g is all real numbers. 2

(b) No: while it is not defined at 3, it does not tend toward " q on either side. (c) Most grapher viewing windows do not reveal that f is undefined at 3. (d) Almost — but not quite; they are equal for all x Z 3.

[–5, 5] by [–5, 5]

71. (a) The volume is ƒ1x2 = k/x, where x is pressure and k is a constant. ƒ1x2 is a quotient of polynomials and hence is rational, but ƒ1x2 = k # x -1, so is a power function with constant of variation k and power -1. (b) If ƒ1x2 = kx a, where a is a negative integer, then the power function f is also a rational function. (c) 4.22 L 73. Horizontal asymptotes: y = - 2 and y = 2; 75. Horizontal asymptotes: y = "3; 3 3 intercepts: a0, - b , a , 0 b ; 2 2 2x - 3 x + 2 h1x2 = d 2x - 3 -x + 2

[–5, 5] by [–5, 5]

x Ú 0 x 6 0

5 5 intercepts: a 0, b , a , 0b ; 4 3

5 - 3x x + 4 ƒ1x2 = d 5 - 3x -x + 4

x Ú 0 x 6 0

[–10, 10] by [–5, 5]

1 77. The graph of f is the graph m = shifted horizontally -d/c units, stretched vertically by a factor of ƒ bc - ad ƒ /c2, reflected across the x-axis x if and only if bc - ad 6 0, and then shifted vertically by a/c.

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 861

SELECTED ANSWERS

861

SECTION 2.7 Quick Review 2.7 1. 2x 2 + 8x

3. LCD: 36; -

1 36

5. LCD: 12x + 121x - 32;

Exercises 2.7

x 2 - 7x - 2 12x + 121x - 32

7.

3 " 117 4

9.

-1 " 17 3

1. x = - 1 3. x = 2 or x = - 7 5. x = - 4 or x = 3; the latter is extraneous. 7. x = 2 or x = 5 9. x = 3 or x = 4 1 1 11. x = or x = - 1; the latter is extraneous. 13. x = - or x = 2; the latter is extraneous. 15. x = 5 or x = 0; the latter 2 3 is extraneous. 17. x = - 2 or x = 0; both of these are extraneous (there are no real solutions). 19. x = - 2 21. Both 23. x = 3 + 12 L 4.414 or x = 3 - 12 L 1.586 25. x = 1 27. No real solutions 29. x L - 3.100 or x L 0.661 or x L 2.439 31. (a) The total amount of solution is 1125 + x2 mL; of this, the amount of acid is x plus 60% of the original amount, or x + 0.611252.

x + 75 = 0.83; x L 169.12 mL x + 125 3000 + 2.12x (b) 4762 hats per week (c) 6350 hats per week C1x2 = x 364 (b) x L 13.49 (a square); P L 53.96 P1x2 = 2x + x 2px 3 + 1000 (b) Either x L 1.12 cm and h L 126.88 cm or x L 11.37 cm and h L 1.23 cm. S = x 2.3x 4.75 + t (b) x L 6.52 ohms 41. (a) D1t2 = (b) t L 5.74 h R1x2 = x + 2.3 4.75t (b) About 10.6 years 45. False. An extraneous solution is a solution of the equation cleared of fractions that is not a solution of the original equation.

(b) y = 0.83 33. (a) 35. (a) 37. (a) 39. (a) 43. (a)

(c) C1x2 =

[60, 110] by [0, 50]

47. D

53. x =

49. E

y y - 1

51. (a) ƒ1x2 =

55. x =

x 2 + 2x x 2 + 2x

(b) x Z 0, -2

(c) ƒ1x2 = e

1, x Z - 2, 0 undefined, x = - 2 or x = 0

(d) The graph appears to be the horizontal line y = 1 with holes at x = - 2 and x = 0.

2y - 3 y - 2

SECTION 2.8

[–4.7, 4.7] by [–3.1, 3.1]

Exploration 1 1. (a)

(+)(–)(+) Negative

(+)(–)(+) Negative –3

3. (a) (+)(+) (–)(–) (+)(+)(+)(–) (+)(+)(+)(–)

(+)(+)(+) Positive x

Positive

Negative –4

2

Negative

(b)

(b)

[–5, 5] by [–3000, 2000]

[–5, 5] by [–250, 50]

Quick Review 2.8 1. lim ƒ1x2 = q ; lim ƒ1x2 = - q x: q

x: - q

3 1 9. (a) "1, " , " 3, " 2 2

x

2

3. lim g1x2 = q , lim g1x2 = q x: q

(b) 1x + 1212x - 321x + 12

x: - q

5. 1x 3 + 52/x

7.

x 2 - 7x - 2 2x 2 - 5x - 3

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 862

SELECTED ANSWERS

862

Exercises 2.8 1. (a) x = - 2, -1, 5 (b) 3. (a) x = - 7, - 4, 6 (b) 5. (a) x = 8, -1 (b) - 1 11. 3 -2, 1/24 ´ 33, q 2

-2 6 x 6 - 1 or x 7 5 (c) x 6 - 2 or - 1 6 x 6 - 7 or -4 6 x 6 6 or x 7 6 (c) -7 6 x 7. 1 -1, 32 ´ 13, q 2 6 x 6 8 or x 7 8 q 13. 3 -1, 04 ´ 32, 2 15. 1-1, 3/22 ´ 12,

x 6 5 6 -4 9. 1- q , -12 ´ 11, 22 q2 17. 3-1.15, q 2

19. 13/2, 22 4 21. (a) 1 - q , q 2 (b) 1- q , q 2 (c) There are no solutions. (d) There are no solutions. 23. (a) x Z (b) 1- q , q 2 3 4 3 3 3 (c) There are no solutions. (d) x = 25. (a) x = 1 (b) x = - , 4 (c) - 6 x 6 1 or x 7 4 (d) x 6 - , or 1 6 x 6 4 3 2 2 2 1 27. (a) x = 0, -3 (b) x 6 - 3 (c) x 7 0 (d) -3 6 x 6 0 29. (a) x = - 5 (b) x = - , x = 1, x 6 - 5 2 1 1 (c) -5 6 x 6 - or x 7 1 (d) - 6 x 6 1 31. (a) x = 3 (b) x = 4, x 6 3 (c) 3 6 x 6 4 or x 7 4 2 2 (d) ƒ1x2 is never negative. 33. 1- q , - 22 ´ 11, 22 35. 3-1, 14 37. 1- q , - 42 ´ 13, q 2 39. 3-1, 04 ´ 31, q 2 1 43. a - 4, b 45. 10, 22 47. 1- q , 02 ´ ( 1 49. 1 - q , -12 ´ 31, 32 3 2, q 2 2 53. 35, q 2 55. Answers will vary; can be solved graphically, numerically, and algebraically: 3- 3.5, q 2.

41. 10, 22 ´ 12, q 2 51. 3 -3, q 2

57. 1 in. 6 x 6 34 in.

59. 0 in. … x … 0.69 in. or 4.20 in. … x … 6 in.

(b) 1.12 cm … x … 11.37 cm, 1.23 cm … h … 126.88 cm

(c) About 348.73 cm2

65. False, because the factor x 4 does not change sign at x = 0. 4 71. Vertical asymptotes: x = - 1, x = 3; x-intercepts: 1- 2, 02, 11, 02; y-intercept: a 0, b 3 (–)(+)2 0 (+)(+)2 (–)(+) (–)(+)

(b) About July 22, 2012

undefined

(–)(–)2 0 (–)(+)2 (–)(–) (–)(–)

undefined

63. (a) y L 2.726x + 282.132 69. D

61. (a) S = 2px 2 + 1000/x

67. C

(+)(+)2 (+)(+)

Negative Negative Positive Negative Positive x –2 –1 1 3

By hand:

Grapher

y 30

–10

5

10

x [0, 10] by [–40, 40]

[–5, 5] by [–5, 5]

-30

73. (a) ƒ x - 3 ƒ 6 1/3 Q ƒ 3x - 9 ƒ 6 1 Q ƒ 3x - 5 - 4 ƒ 6 1 Q ƒ ƒ1x2 - 4 ƒ 6 1. (b) If x stays within the dashed vertical lines, ƒ1x2 will stay within the dashed horizontal lines.

(c) ƒ x - 3 ƒ 6 0.01 Q ƒ 3x - 9 ƒ 6 0.03 Q ƒ 3x - 5 - 4 ƒ 6 0.03 Q ƒ ƒ1x2 - 4 ƒ 6 0.03. The dashed lines would be closer when x = 3 and y = 4.

75. 0 6 a 6 b Q a 2 6 ab and ab 6 b 2; so, a 2 6 b 2.

CHAPTER 2 REVIEW EXERCISES

3. Starting from y = x 2, translate right 2 units and vertically stretch by 3 (either order), then translate up 4 units.

1. y = - x - 5

y 10

[–15, 5] by [–15, 5] 6

5. Vertex: 1 - 3, 52; axis: x = - 3

7. Vertex: 1 -4, 12; axis: x = - 4

9. y = 15/921x + 222 - 3

x

11. y =

1 1x - 322 - 2 2

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 863

SELECTED ANSWERS

13.

863

17. S = kr 2

15.

[–4, 3] by [–30, 30]

[–10, 7] by [–50, 10]

19. The force F needed varies directly with the distance x from its resting position, with constant of variation k. 1 21. k = 4, a = , ƒ is increasing in the first quadrant, ƒ is odd. 23. k = - 2, a = - 3, ƒ is increasing in the fourth quadrant, ƒ is odd. 3 1 2 - 2x + 3 3 3 25. 2x 2 - x + 1 27. 2x 2 - 3x + 1 + 2 29. -39 31. Yes 37. "1, "2, "3, "6, " , " ; x - 3 2 2 2 x + 4 and 2 are zeros.

39. -2 + 2i

41. i

43. 3" 2i

45. (c)

47. (b)

51. Rational: none. Irrational: approximately - 2.34, 0.57, 3.77. No nonreal zeros.

49. Rational: 0. Irrational: 5 " 12. No nonreal zeros.

3 53. - , 3 "i; ƒ1x2 = 12x + 321x - 3 + i21x - 3 - i2 2

2 5 55. 1, -1, , and - ; ƒ1x2 = 13x - 2212x + 521x - 121x + 12 57. ƒ1x2 = 1x - 221x 2 + x + 12 3 2 59. ƒ1x2 = 12x - 321x - 121x 2 - 2x + 52 61. x 3 - 3x 2 - 5x + 15

63. 6x 4 - 5x 3 - 38x 2 - 5x + 6 65. x 4 - 4x 3 - 12x 2 + 32x + 64 67. Translate right 5 units and vertically stretch by 2 (either order), then translate down 1 unit. Horizontal asymptote: y = - 1; vertical asymptote: x = 5. 5 71. End behavior asymptote: y = x - 7; 73. y-intercept: a0, b , x-intercept: 1 - 2.55, 02; 69. Asymptotes: y = 1, x = - 1, and 2 x = 1; intercept: 10, -12 vertical asymptote: x = - 3; Domain: x Z - 2; Range: 1 - q , q 2; 5 intercept: a 0, b Continuity: all x Z - 2; 3 Decreasing: 1 - q , - 22, 1 - 2, 0.824; Increasing: 30.82, q 2; Unbounded; Local minimum: 10.82, 1.632; Vertical asymptote: x = - 2; End behavior asymptote: y = x 2 - x; lim ƒ1x2 = lim ƒ1x2 = q

[–5, 5] by [–5, 5] [–25, 25] by [–30, 20]

x: - q

3 or x = 4 77. 1 - q , -5/22 ´ 1 -2, 32 79. 3- 3, -22 ´ 12, q 2 2 1 81. x = - 3, x = 83. Yes; at approximately 10.0002 2 85. (a) V = x130 - 2x2170 - 2x2 in.3 (b) Either x L 4.57 or x L 8.63 in.

x: q

75. x =

87. (a) V =

4 3 px + px 21140 - 2x2 3

(b)

[–5, 5] by [–10, 20]

(c) The largest volume occurs when x = 70 (so it is actually a sphere). 4 This volume is p17023 L 1,436,755 ft 3. 3

[0, 70] by [0, 1,500,000]

89. (a) y = 1.401x + 4.331

[0, 15] by [0, 30]

(b) y = 0.188x 2 - 1.411x + 13.331

[0, 15] by [0, 30]

(c) Using linear regression: in 2008; using quadratic regression: in 2003

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 864

SELECTED ANSWERS

864

91. (a) P1152 = 325, P1702 = 600, P11002 = 648

(b) y =

640 0.8

(c) The deer population approaches (but never equals) 800.

50 100 L 33.33 (b) about 33.33 ounces of distilled water (c) x = 50 + x 3 95. (a) S = x 2 + 4000/x (b) 20 ft by 20 ft by 2.5 ft or x L 7.32, giving approximate dimensions 7.32 by 7.32 by 18.66. (c) 7.32 6 x 6 20 (lower bound approximate), so y must be between 2.5 and about 18.66. 93. (a) C1x2 =

Chapter 2 Project Answers are based on the sample date shown in the table. 1. 3. The sign of a affects the direction the parabola opens. The magnitude of a affects the vertical stretch of the graph. Changes to h cause horizontal shifts to the graph, while changes to k cause vertical shifts. 5. y L - 4.968x 2 - 10.913x - 5.160 [0, 1.6] by [–0.1, 1]

SECTION 3.1 Exploration 1 1. 10, 12 is in common; Domain: 1 - q , q 2; Range: 10, q 2; Continuous; Always increasing; Not symmetric; No local extrema; Bounded below by y = 0, which is also the only asymptote; lim ƒ1x2 = q . lim ƒ1x2 = 0 x: q

x: - q

Exploration 2 1.

3. k L 0.693

[–4, 4] by [–2, 8]

Quick Review 3.1 1. -6

3. 9

5. 1/212

7. 1/a 6

9. - 1.4

Exercises 3.1 1. Not exponential, a monomial function 3. Exponential function, initial value of 1 and base of 5 5. Not exponential, variable base 7. 3 9. -2 2 11. 3/2 # 11/22x 13. 3 # 2x/2 15. Translate ƒ1x2 = 2x by 3 units to the right. 17. Reflect ƒ1x2 = 4x over 33 x the y-axis. 19. Vertically stretch ƒ1x2 = 0.5 by a factor of 3 and then shift 4 units up. 21. Reflect ƒ1x2 = ex across the y-axis and x horizontally shrink by a factor of 2. 23. Reflect ƒ1x2 = e across the y-axis, horizontally shrink by a factor of 3, translate 1 unit to the right, and vertically stretch by a factor of 2. 25. Graph (a) is the only graph shaped and positioned like the graph of y = b x, b 7 1. 27. Graph (c) is the reflection of y = 2x across the x-axis. 29. Graph (b) is the graph of y = 3-x translated down 2 units. 31. Exponential decay; lim ƒ1x2 = 0, lim ƒ1x2 = q 33. Exponential decay; lim ƒ1x2 = 0, lim ƒ1x2 = q x: q

35. x 6 0

37. x 6 0

41.

x: -q

39. y1 = y3 since 3

x: q

2x + 4

21x + 22

= 3

43.

[–10, 20] by [–5, 15]

y-intercept: 10, 42 Horizontal asymptotes: y = 0, y = 12

2 x+2

= 13 2

= 9

x: - q

x+2

45.

[–5, 10] by [–5, 20]

y-intercept: 10, 42 Horizontal asymptotes: y = 0, y = 16

[–3, 3] by [–2, 8]

Domain: 1- q , q 2; Range: 10, q 2; Continuous; Always increasing; Not symmetric; Bounded below by y = 0, which is also the only asymptote; No local extrema; lim ƒ1x2 = q ; lim ƒ1x2 = 0 x: q

x: -q

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 865

SELECTED ANSWERS

47.

865

49.

[–3, 4] by [–1, 7]

[–2, 2] by [–1, 9]

Domain: 1- q , q 2; Range: 10, 52; Continuous; Always increasing; Symmetric about 10.69, 2.52;

Domain: 1 - q , q 2; Range: 10, q 2; Continuous;

Always increasing; Not symmetric; Bounded below by y = 0, which is the only asymptote; No local extrema; lim ƒ1x2 = q ; lim ƒ1x2 = 0

Bounded below by y = 0 and above by y = 5, both of which are asymptotes; No local extrema; lim ƒ1x2 = 5; lim ƒ1x2 = 0

x: -q

x: q

x: q

x: -q

51. In 2006 53. Near the end of 2003 55. In 1970 57. (a) 100 (b) L 6394 59. False. If a 7 0 and 0 6 b 6 1, then ƒ1x2 = a # b x is decreasing. 61. E 63. A 65. (a)

(b)

[–5, 5] by [–2, 5]

[–3, 3] by [–7, 5]

1 Domain: 1 - q , q 2; Range: c - , q b ; e

Domain: 1- q , 02 ´ 10, q 2; Range: 1 - q , -e4 ´ 10, q 2;

Decreasing on 1- q , - 14; Increasing on 3- 1, q 2;

1 1 Bounded below by y = - ; Local minimum at a - 1, - b ; e e Asymptote: y = 0; lim ƒ1x2 = q ; lim ƒ1x2 = 0 x: q

x: - q

Increasing on 1- q , -14; Decreasing on 3-1, 02 ´ 10, q 2; Not bounded; Local maximum at 1 -1, - e2; Asymptotes: x = 0, y = 0;

lim g1x2 = 0; lim g1x2 = - q x: -q

x: q

67. (a) y1 — ƒ1x2 decreases less rapidly as x increases. (b) y3 — as x increases, g1x2 decreases ever more rapidly. 71. a 7 0 and b 7 1, or a 6 0 and 0 6 b 6 1

69. a Z 0, c = 2

73. As x : - q , b x : q , so 1 + a # b x : q and

c : 0; 1 + a # bx c :c As x : q , b x : 0, so 1 + a # b x : 1 and 1 + a # bx

SECTION 3.2 Quick Review 3.2 1. 0.15

3. 23 # 1.07

5. "2

7. 1.01

9. 0.61

Exercises 3.2 1. exponential growth, 9% 11. 28,900 # 0.974x

23. 40/31 + 3 # 11/32x4

3. exponential decay, 3.2%

13. 18 # 1.052x

15. 0.6 # 2x/3

25. L 128/11 + 7 # 0.844x2

5. exponential growth, 100%

17. 592 # 2-x/6 20 27. 1 + 3 # 0.58x

19. 2.3 # 1.25x

7. 5 # 1.17x

9. 16 # 0.5x

21. L 4 # 1.15x

29. In 2021

1 t/14 33. (a) y = 6.6 a b , where t is time in days (b) After 38.11 days 2 35. One possible answer: Exponential and linear functions are similar in that they are always increasing or always decreasing. However, the two functions vary in how quickly they increase or decrease. While a linear function will increase or decrease at a steady rate over a given interval, the rate at which exponential functions increase or decrease over a given interval will vary. 37. One possible answer: From the graph, we see that doubling time for this model is 4 yr. This is the time required to grow from 50,000 to 100,000, from 100,000 to 200,000, or from any population size to twice that size. Regardless of the population size, it takes 4 yr for it to double. 39. When t = 1; every hour 41. 2.14 lb/in.2 43. About 4,178,000, overestimated by 344,000; 9% error 45. (a) 16 (b) About 14 days (c) In about 17 days 47. About 310.6 million 49. Model matches. 51. False. This holds true for logistic growth, not exponential. 53. C 55. D 57. (a) L277,900,000 (b) Underestimates actual population by 3.5 million. (c) Logistic model 31. (a) 12,315; 24,265

(b) 1966

6965_SE_Ans_833-934.qxd

866

1/25/10

3:40 PM

Page 866

SELECTED ANSWERS

e -x - e -1-x2 ex - e -x = = - sinh 1x2 2 2 x -x sinh 1x2 1e - e 2/2 ex - e -x = tanh 1x2 = x = x 61. (a) -x cosh 1x2 1e + e 2/2 e + e -x 59. sinh 1 - x2 =

(c) ƒ1x2 = 1 + tanh 1x2 = 1 +

(b) tanh 1-x2 =

sinh 1 - x2

cosh 1 -x2

=

-sinh 1x2 cosh 1x2

= - tanh 1x2

ex - e -x ex + e -x ex - e -x 2ex # e -x 2 , which is logistic. x -x = x -x + x -x = x -x -x = e + e e + e e + e e + e e 1 + e -2x

SECTION 3.3 Exploration 1 1.

[–6, 6] by [–4, 4]

Quick Review 3.3 1. 1/25 = 0.04

3. 1/5 = 0.2

5. 32

7. 51/2

9. e -1/2

Exercises 3.3 1. 1 3. 5 25. L 0.975

5. 2/3 7. 3 27. Undefined

41. Starting with y = ln x: shift left 3 units.

9. 5 11. 1/3 13. 3 15. - 1 17. 1/4 19. 3 21. 0.5 23. 6 29. L 1.399 31. Undefined 33. 100 35. 0.1 37. (d) 39. (a) 45. Starting from y = ln x: 43. Starting from y = ln x: reflect across the y-axis reflect across the y-axis and and translate right 2 units. translate up 3 units.

[–5, 5] by [–3, 3] [–7, 3] by [–3, 3]

[–4, 1] by [–3, 5]

47. Starting with y = log x: shift down 1 unit.

49. Starting from y = log x: reflect across both axes and vertically stretch by 2

51. Starting from y = log x: reflect across the y-axis, translate right 3 units, vertically stretch by 2, translate down 1 unit.

[–5, 15] by [–3, 3] [–8, 1] by [–2, 3]

53. Domain: 12, q 2; Range: 1 - q , q 2; Continuous; Always increasing; Not symmetric; Not bounded; No local extrema; Asymptote at x = 2; lim ƒ1x2 = q

[–5, 5] by [–4, 2]

57.

55.

x: q

[–2, 8] by [–3, 3]

[–1, 9] by [–3, 3]

Domain: 11, q 2; Range: 1- q , q 2; Continuous; Always decreasing; Not symmetric; Not bounded; No local extrema; Asymptote: x = 1; lim ƒ1x2 = - q x: q

[–3, 7] by [–3, 3]

Domain: 10, q 2; Range: 1- q , q 2; Continuous; Always increasing on its domain; Not symmetric; Not bounded; No local extrema; Asymptote: x = 0; lim x = q x: q

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 867

SELECTED ANSWERS

59. (a) 10 dB 69.

(b) 70 dB

61. 2023

3x

log3 x

1- q, q2

10, q 2

ƒ1x2 Domain

(c) 150 dB

Intercepts

10, q 2

1- q , q 2

Asymptotes

y = 0

x = 0

Range

10, 12

e 1e, e2 71. b = 1e;

11, 02

63. True, by definition.

65. C

67. B

[–6, 6] by [–4, 4]

73. Reflect across the x-axis

SECTION 3.4 Exploration 1 1. 0.90309 = 0.30103 + 0.60206

3. 0.90309 = 3 * 0.30103

5. 1.20412; 1.50515; 1.80618

Exploration 2 1. False

3. True

5. False

7. False

Quick Review 3.4 1. 2

3. -2

5. x 3y 2

7. ƒ x ƒ 3/ ƒ y ƒ

Exercises 3.4 1. 3 ln 2 + ln x

3. log 3 - log x

9. 1/13 ƒ u ƒ 2

5. 5 log2 y

7. 3 log x + 2 log y

9. 2 ln x - 3 ln y

13. log xy 15. ln 1y/32 17. log 1 19. ln 1x 2 y 32 21. log 1x 4y/z 32 23. 2.8074 3x 27. -3.5850 29. ln x/ln 3 31. ln 1a + b2/ln 2 33. log x/log 2 35. - log 1x + y2/log 2 37. Let x = logb R and y = logb S. Then

1 1 log x - log y 4 4 25. 2.4837

11.

R bx R = y = b x - y. So logb a b = x - y = logb R - logb S. S b S

39. Starting with g1x2 = ln x: vertically shrink by a factor of 1 L 0.72. ln 4

41. Starting with g1x2 = ln x: reflect across the x-axis, then vertically shrink by a factor of

[–1, 10] by [–2, 2]

43. (b): 3 -5, 54 by 3 - 3, 34, with Xscl = 1 and Yscl = 1 47.

[–1, 9] by [–1, 7]

Domain: 10, q 2; Range: 1- q , q 2; Continuous; Always increasing; Asymptote: x = 0; lim ƒ1x2 = q

x: q

51. (a) 0 (b) 10 (c) 60 (d) 80 (e) 100 (f ) 120 57. True, by the product rule for logarithms 59. B

1 L 0.91. ln 3

[–1, 10] by [–2, 2]

45. (d): 3-2, 84 by 3-3, 34, with Xscl = 1 and Yscl = 1 49.

[–10, 10] by [–2, 3]

Domain: 1- q , 02 ´ 10, q 2; Range: 1 - q , q 2; Discontinuous at x = 0; Decreasing on interval 1 - q , 02; Increasing on interval 10, q 2; Asymptote: x = 0; lim ƒ1x2 = q ; lim ƒ1x2 = q x: q

53. L 9.6645 lumens 61. A

x: - q

55. Vertical stretch by a factor of L 0.9102

867

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 868

SELECTED ANSWERS

868

63. (a) 2.75x 5.0 (b) 49,616 (c) ln1x2 1.39 1.87 2.14 ln1y2

7.94

65. (a) log1w2

- 0.70

-0.52

0.30

0.70

1.48

1.70

1.85

log1r2

2.62

2.48

2.31

2.08

1.93

1.85

1.86

2.30

10.37 11.71 12.52

(b) log r = 1 -0.302 log w + 2.36 (c)

[–1, 2] by [1.6, 2.8] [0, 3] by [0, 15]

[–1, 2] by [1.6, 2.8]

(d) 1ln y2 = 5.00 1ln x2 + 1.01

1 5

5

5

(e) a L 5, b L 1 so ƒ1x2 = e x = ex L 2.72x : The two equations are almost the same.

(e) One possible answer: Consider the power function y = a # x b. Then log y = log (a # x b2 = log a + log x b = log a + b log x = b1log x2 + log a, which is clearly a linear function of the form ƒ1t2 = mt + c where m = b, c = log a, ƒ1t2 = log y and t = log x. As a result, there is a linear relationship between log y and log x.

69. (a) Domain of ƒ and g: 13, q 2

67. 16.41, 93.352

Domain of g: 1 -3, q 2; Answers will vary.

(b) Domain of ƒ and g: 15, q 2 (c) Domain of ƒ: 1 - q , -32 ´ 1-3, q 2; log x log x = = log e 71. If x 7 0 and x Z 1, then ln x log x/ log e

SECTION 3.5 Exploration 1 1. 1.60206, 2.60206, 3.60206, 4.60206, 5.60206, 6.60206, 7.60206, 8.60206, 9.60206, 10.60206 3. The decimal parts are exactly equal.

Quick Review 3.5 1/2

1. ƒ1g1x22 = e2 ln1x

2

= eln x = x and g1ƒ1x22 = ln1e2x21/2 = ln1ex2 = x

1 1 ln1e3x2 = 13x2 = x and g1ƒ1x22 = e311/3 ln x2 = eln x = x 3 3 5. 7.783 * 10 8 km 7. 602,000,000,000,000,000,000,000 9. 5.766 * 10 12 3. ƒ1g1x22 =

Exercises 3.5 1. 10 3. 12 5. -3 7. 10,000 9. 5.25 11. L 24.2151 13. L 39.6084 15. L -0.4055 17. L 4.3956 19. Domain: 1- q , -12 ´ 10, q 2; graph (e) 21. Domain: 1 - q , -12 ´ 10, q 2; graph (d) 23. Domain: 10, q 2; graph (a) 25. x = 1000 or x = - 1000 27. "110 29. x L 3.5949 31. x L "2.0634 33. x L - 9.3780 35. x L 2.3028 37. 4 39. 3 41. 1.5 43. 3 45. About 20 times greater 47. (a) 1.26 * 10 -4; 1.26 * 10 -12 (b) 10 8 (c) 8 49. L 28.41 min 51. (a) (b) (c) 89.47°C

[0, 40] by [0, 80]

53. (a)

[0, 40] by [0, 80]

T1x2 L 79.47 # 0.93x (b) The scatter plot is better because it accurately represents the times between the measurements. The equal spacing on the bar graph suggests that the measurements were taken at equally spaced intervals, which distorts our perception of how the consumption has changed over time.

[0, 20] by [0, 15]

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 869

SELECTED ANSWERS

869

57. Exponential — the scatterplot of 1x, y2 is exactly

55. Logarithmic seems best — the scatterplot of 1x, y2 looks most logarithmic. (The data can be modeled by y = 3 + 2 ln x.)

exponential. a The data can be modeled by y =

[0, 5] by [0, 7]

3# x 2 .b 2

[0, 5] by [0, 30]

59. False. The order of magnitude is its common logarithm. 61. B 63. E 65. Logistic regression 67. (a) As k increases, the bell curve stretches vertically. (b) As c increases, the bell curve compresses horizontally. 69. (a) (b) 30, 104 by 3- 5, 34; 2.3807 71. y = a ln x + b, a logarithmic regression

[–10, 10] by [–10, 30]

r cannot be negative since it is a distance. 73. x L 1.3066 75. 0 6 x 6 1.7115 1approx.2

77. x 7 9

SECTION 3.6 Exploration 1 1.

k

A 10

1104.6

20

1104.9

30

1105

40

1105

50

1105.1

60

1105.1

70

1105.1

80

1105.1

90

1105.1

100

1105.1

A approaches a limit of about 1105.1.

Quick Review 3.6 1. 7

3. 1.8125%

5. 65%

7. 150

9. $315

Exercises 3.6 1. $2251.10 3. $19,908.59 5. $2122.17 7. $86,496.26 9. $1728.31 11. $30,402.43 13. $14,755.51 15. $70,819.63 17. $43,523.31 19. $293.24 21. 6.63 years — round to 6 years 9 months 23. 13.78 years — round to 13 years 10 months 25. L 10.13% 27. 7.07% 29. 12.14 — round to 12 years 3 months 31. 7.7016 years; $48,217.82 33. 17.33%; $127,816.26 35. 17.42 — round to 17 years 6 months 37. 10.24 — round to 11 years 39. 9.93 — round to 10 years 41. L 6.14% 43. L 6.50% 45. 5.1% quarterly 47. $42,211.46 49. $239.42 51. $219.51 53. $676.57 55. (a) 172 months (14 years, 4 months) (b) $137,859.60 57. One possible answer: The APY is the percentage increase from the initial balance S102 to the end-of-year balance S112; specifically, it is S112/S102 - 1. Multiplying the initial balance by P results in the end-of-year balance being multiplied by the same amount, so that the ratio remains unchanged. 59. One possible answer: Some of these situations involve counting things (e.g., populations), so that they can only take on whole-number values — exponential models which predict, e.g., 439.72 fish, have to be interpreted in light of this fact. Technically, bacterial growth, radioactive decay, and compounding interest also are “counting problems” — for example, we cannot have fractional bacteria, or fractional molecules of radioactive material, or fractions of pennies. However, because these are generally very large numbers, it is easier to ignore the

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 870

SELECTED ANSWERS

870

fractional parts. (This might also apply when one is talking about, e.g., the population of the whole world.) Another distinction: while we often use an exponential model for all these situations, it generally fits better (over long periods of time) for radioactive decay than for most of the others. Rates of growth in populations (esp. human populations) tend to fluctuate more than exponential models suggest. Of course, an exponential model also fits well in compound interest situations where the interest rate is held constant, but there are many cases where interest rates change over time. 61. False. The limit is A = Pert = 100e0.05 L $105.13. 63. B 65. E 67. $364.38 69. (a) 8% (b) 12 (c) $100

CHAPTER 3 REVIEW EXERCISES 3. y = 3 # 2x/2

1 5. ƒ1x2 = 2-2x + 3 — starting from 2x, horizontally shrink by , reflect across y-axis, and translate up 3 units. 2 1 7. ƒ1x2 = - 2-3x - 3 — starting from 2x, horizontally shrink by , reflect across y-axis, reflect across x-axis, translate down 3 units. 3 1 3 1 x 9. Starting from e , horizontally shrink by , then translate right units — or translate right 3 units, then horizontally shrink by . 2 2 2 11. y-intercept: 10, 12.52; Asymptotes: y = 0, y = 20 13. exp. decay; lim ƒ1x2 = 2, lim ƒ1x2 = q 1. - 3 1 34

x: -q

x: q

17.

15.

[–1, 4] by [–10, 30]

[–5, 10] by [–2, 8]

Domain: 1 - q , q 2; Range: 11, q 2; Continuous; Always decreasing; Not symmetric; Bounded below by y = 1, which is also the only asymptote; No local extrema; lim ƒ1x2 = 1; lim ƒ1x2 = q x: q

19. ƒ1x2 = 24 # 1.053x 27. 5

29. 1/3

Domain: 1- q , q 2; Range: 10, 62; Continuous; Increasing; Symmetric about 11.20, 32;

Bounded by the asymptotes y = 0, y = 6;

x: -q

x: q

21. ƒ1x2 = 18 # 2x/21 31. 35 = x

No extrema; lim ƒ1x2 = 6; lim ƒ1x2 = 0

33. y = xe2

23. ƒ1x2 L 30/11 + 1.5e -0.55x2

25. ƒ1x2 L

35. Translate left 4 units.

x: -q

20 1 + 3e -0.37x

37. Translate right 1 unit, reflect across x-axis, and translate up 2 units. 41.

39.

[–4.7, 4.7] by [–3.1, 3.1]

43. 57. 71. 73.

[–5, 5] by [–5, 25]

1 Domain: 10, q 2; Range: c - , q b L 3 -0.37, q 2; e Continuous; Decreasing on 10, 0.374; Increasing on 30.37, q 2; Not symmetric; Bounded below; 1 1 Local minimum at a , - b ; lim ƒ1x2 = q e e x: q

45. L 22.5171 47. 0.0000001 49. 4 log 4 L 0.6021 log x/log 5 59. (c) 61. (b) 63. $515.00 65. Pert P1t2 L 2.0956 # 1.01218t, P11052 L 7.5 million (a) 90 units (b) 32.8722 units (c)

[0, 4] by [0, 90]

Domain: 1- q , 02 ´ 10, q 2; Range: 3- 0.18, q 2; Discontinuous at x = 0; Decreasing on 1- q , -0.614, 10, 0.614; Increasing on 3-0.61, 02, 30.61, q 2; Symmetric across y-axis; Bounded below; Local minima at 1- 0.61, -0.182 and 10.61, - 0.182; No asymptotes; lim ƒ1x2 = q ; lim ƒ1x2 = q x: q

51. L 2.1049 67. $28,794.06

x: - q

53. L 99.5112 69. - 0.3054

55. ln x/ln 2

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 871

SELECTED ANSWERS

75. (a) P1t2 = 89,00010.9822t

79. (a) S1t2 = S0 # 11/22t/1.5

(b) 31.74 years

77. (a) P1t2 = 20 # 2t

(b) 1,099,500 metric tons

85. 137.7940 — about 11 years 6 months

(c) S0/2; S0/4

87. L 8.57%

(b) 81,920; 2.3058 * 10 19 81. 6.31

89. L 5.84 lumens

(c) L 8.9658 months

83. 11.75 years 1 1 6 b 6 10; 0 6 b 6 91. or b 7 10 10 10

1 days (c) 8.7413 — about 8 or 9 days. 95. L 41.54 minutes 97. (a) 9% (b) 4 (c) $100 2 99. (a) Grace’s balance will always remain $1000 since interest is not added to it. Every year she receives 5% of that $1000 in interest; after t years, she has been paid 5t% of the $1000 investment, meaning that altogether she has 1000 + 1000 # 0.05t = 100011 + 0.05t2. (b) Not Years Compounded Compounded 93. (a) 16

(b) About 11

0

1000.00

1000.00

1

1050.00

1051.27

2

1100.00

1105.17

3

1150.00

1161.83

4

1200.00

1221.40

5

1250.00

1284.03

6

1300.00

1349.86

7

1350.00

1419.07

8

1400.00

1491.82

9

1450.00

1568.31

10

1500.00

1648.72

Chapter 3 Project Answers are based on the sample data shown in the table. 3. 5. y L 2.7188 # 0.788x 7. A different ball would change the rebound percentage P. 9. y = Heln1P2x so y = 2.7188e -0.238x

[–1, 6] by [0, 3]

11. The linear regression is y L - 0.253x + 1.005. Since ln y = 1ln P2x + ln H, the slope is ln P and the y-intercept is ln H.

[–1, 6] by [–1.25, 1.25]

SECTION 4.1 Exploration 1 1. 2pr

3. No, not quite, since the distance pr would require a piece of thread p times as long, and p 7 3.

Quick Review 4.1 1. 5p in.

3.

6 m p

5. (a) 47.52 ft

(b) 39.77 km

7. 88 ft/sec

9. 6 mph

Exercises 4.1 1. 23.2° 17. 30°

3. 118.7375° 5. 21°12¿ 7. 118°19¿ 12– 9. p/3 19. 18° 21. 140° 23. L 114.59° 25. 50 in.

11. 2p/3 27. 6/p ft

13. L 1.2518 rad 15. L 1.0716 rad 29. 3 (radians) 31. 360/p cm

871

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 872

SELECTED ANSWERS

872

9 rad and s2 = 36 cm 35. 24 inches 37. L 5.4 inches 39. (a) 45° (b) 22.5° (c) 247.5° 11 41. ESE is closest at 112.5°. 43. L 4.23 statute miles 45. L 387.85 rpm 47. L 12,566.37 49. 51. L 778 nautical miles 53. (a) 16p L 50.265 in. (b) 2p L 6.283 ft 55. (a) 4p rad/sec (b) 28p cm/sec (c) 7p rad/sec 57. True. Horse A travels 2p12r2 = 212pr2 units of distance in the same amount of time as horse B travels 2pr units of distance, and so is moving twice as fast. 47° 4 mi 59. C 61. B 63. 38°02¿ 65. 5°37¿ 67. 80 naut mi 69. 902 naut mi 38° u u # 2 1 2 2 mi 71. The whole circle’s area is pr ; the sector with central angle u makes up of that area, or pr = r 2u. 2p 2p 2 73. 33. u =

37° 340° B

60 mi

A

SECTION 4.2 Exploration 1 1. sin and csc, cos and sec, and tan and cot

3. sec u

5. sin u and cos u

Exploration 2 13 2 1 cos u = 2

csc u =

1. Let u = 60°. Then sin u =

2 2 23 or 3 13

sec u = 2

tan u = 13 L 1.732

cot u =

1 23 or 3 13

3. The value of a trig function at u is the same as the value of its cofunction at 90° - u.

Quick Review 4.2 1. 5 12

3. 6

5. 100.8 in.

7. 7.9152 km

9. L 1.0101 (no units)

Exercises 4.2

13 4 3 4 5 5 3 12 5 12 13 , cos u = , tan u = , csc u = , sec u = , cot u = 3. sin u = , cos u = , tan u = , csc u = , sec u = , 5 5 3 4 3 4 13 13 5 12 5 8 11 7 1170 1170 11 157 5 7 cot u = 5. sin u = , cos u = , tan u = , csc u = , sec u = , cot u = 7. sin u = , cos u = , 12 11 7 11 7 11 11 1170 1170 2 110 157 11 2 110 7 11 8 3 7 tan u = , csc u = , sec u = , cot u = 9. cos u = , tan u = , csc u = , sec u = , cot u = 8 8 7 3 3 157 157 2 110 2 110 1106 4 16 4 16 11 11 5 5 9 11. sin u = , tan u = , csc u = , sec u = , cot u = 13. sin u = , cos u = , csc u = , 11 5 5 5 4 16 4 16 1106 1106 11 1106 9 3 11 3 1130 1130 sec u = , cot u = 15. sin u = , cos u = , tan u = , csc u = , sec u = , cot u = 9 5 11 3 11 3 1130 1130 9 817 9 23 8 17 13 12 17. sin u = , cos u = , tan u = , sec u = , cot u = 19. 21. 13 23. 25. 12 23 23 9 2 2 8 17 8 17 1. sin u =

27. 14/3 = 2/13 = 2 13/3 29. 0.961 31. 0.943 33. 0.268 35. 1.524 37. 0.810 p p p p 15 32 L 26.82 L 20.78 39. 2.414 41. 30° = 43. 60° = 45. 60° = 47. 30° = 49. 51. 6 3 3 6 sin 34° tan 57° 6 53. 55. b L 33.79, c L 35.96, b = 70° 57. b L 22.25, c L 27.16, a = 35° 59. As u gets smaller and smaller, L 10.46 sin 35° the side opposite u gets smaller and smaller, so its ratio to the hypotenuse approaches 0 as a limit. 61. L 205.26 ft 63. L 74.16 ft 2 65. L 378.80 ft 67. False. This is true only if u is an acute angle in a right triangle. 69. E 71. D 73. Sine values should be increasing, cosine values should be decreasing, and only tangent values can be greater than 1. Therefore, the first column is tangent, the second column is sine, and the third column is cosine.

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 873

SELECTED ANSWERS

873

75. The distance dA from A to the mirror is 5 cos 30°; the distance from B to the mirror is dB = dA - 2. dB dA - 2 2 4 Then PB = L 2.69 m = = 5 = 5 cos b cos 30° cos 30° 13

a 2 b 2 a2 b2 a2 + b 2 c2 77. One possible proof: 1sin u22 + 1cos u22 = a b + a b = 2 + 2 = = 2 = 1 1Pythagorean Theorem: a 2 + b 2 = c2.2 2 c c c c c c

SECTION 4.3 Exploration 1

1. The side opposite u in the triangle has length y and the hypotenuse has length r. Therefore sin u =

y opp = . r hyp

3. tan u = y/x

Exploration 2 1. The x-coordinates on the unit circle lie between - 1 and 1, and cos t is always an x-coordinate on the unit circle. 3. The points corresponding to t and -t on the number line are wrapped to points above and below the x-axis with the same x-coordinates. Therefore cos t and cos 1 - t2 are equal. 5. Since 2p is the distance around the unit circle, both t and t + 2p get wrapped to the same point. -y y 7. By the observation in 162, tan t and tan 1t + p2 are ratios of the form and , which are either equal to each other or both undefined. x -x 9. Answers will vary. For example, similar statements can be made about the functions cot, sec, and csc.

Quick Review 4.3 1. -30°

3. 1125°

Exercises 4.3

5. 13/3

7. 12

9. cos u =

12 5 13 13 12 , tan u = , csc u = , sec u = , cot u = 13 12 5 12 5

2 1 15 1 , cos u = , tan u = - 2, csc u = , sec u = - 15, cot u = 2 2 15 15 3 5 5 4 1 1 tan u = - , csc u = - , sec u = , cot u = 5. sin u = , cos u = , tan u = 1, csc u = - 12, sec u = - 12, cot u = 1 4 3 4 3 12 12 4 3 4 5 5 3 7. sin u = , cos u = , tan u = , csc u = , sec u = , cot u = sec u = - 1, cot u undefined 5 5 3 4 3 4 2 5 2 9. sin u = 1, cos u = 0, tan u undefined, csc u = 1, sec u undefined, cot u = 0 11. sin u = , cos u = , tan u = - , 5 129 129 129 129 5 , sec u = , cot u = 13. + , +, + 15. - , - , + 17. 19. 21. (a) 23. (a) 25. - 1/2 csc u = 2 5 2 13 1 13 27. 2 29. 31. 1 33. 35. 37. (a) - 1 (b) 0 (c) Undefined 39. (a) 0 (b) - 1 (c) 0 2 2 2 15 15 2 5 5 5 41. (a) 1 (b) 0 (c) Undefined 43. sin u = 45. tan u = 47. sec u = - ; csc u = ; sec u = ; tan u = 3 2 4 3 121 121 49. 1/2 51. 0 53. The calculator’s value of the irrational number p is necessarily an approximation. When multiplied by a very large number, the slight error of the original approximation is magnified sufficiently to throw the trigonometric functions off. sin 83° 55. m = 57. (a) 0.4 in. (b) L 0.1852 in. 59. The difference in the elevations is 600 ft, so d = 600/sin u. L 1.69 sin 36° Then: (a) L 848.53 ft (b) 600 ft (c) L 933.43 ft 61. True. Acute angles determine reference triangles in QI, where cosine is positive, while obtuse angles determine reference triangles in QII, where cosine is negative. 63. E 65. A 67. 5p/6 69. 7p/4 71. The two triangles are congruent: Both have hypotenuse 1, and the corresponding angles are congruent — the smaller acute angle has measure t in both triangles, and the two acute angles in a right triangle add up to p/2. 73. One possible answer: Starting from the point 1a, b2 on the unit circle — at an angle of t, so that cos t = a — then measuring a quarter of the way around the circle (which corresponds to adding p/2 to the angle), we end at 1 -b, a2, so that sin 1t + p/22 = a. For 1a, b2 in Quadrant I, this is shown in the figure; similar illustrations can be drawn for the other quadrants. 75. Starting from the point 1a, b2 on the unit circle — at an angle of t, y so that cos t = a — then measuring a quarter of the way around the P(a, b) circle (which corresponds to adding p/2 to the angle), we end at 1 -b, a2, so that sin (t + p/22 = a. This holds true when 1a, b2 is in t Quadrant II, just as it did for Quadrant I. (1, 0) 1. 450°

3. sin u =

t+π

2

Q(–b, a)

x

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 874

SELECTED ANSWERS

874

77. ƒ u ƒ 6 0.2441 (approximately) 79. This Taylor polynomial is generally a very good approximation for sin u — in fact, the relative error is less than 1% for ƒ u ƒ 6 1 1approx.2. It is better for u close to 0; it is slightly larger than sin u when u 6 0 and slightly smaller when u 7 0.

SECTION 4.4 Exploration 1

1. p/2 1at the point 10, 122 3. Both graphs cross the x-axis when the y-coordinate on the unit circle is 0. 5. The sine function tracks the y-coordinate of the point as it moves around the unit circle. After the point has gone completely around the unit circle (a distance of 2p), the same pattern of y-coordinates starts over again.

Quick Review 4.4 1. In order: +, +, - , -

3. In order: + , -, + , -

5. -5p/6

7. Vertically stretch by 3

9. Vertically shrink by 0.5

Exercises 4.4 1. Amplitude 2; vertical stretch by a factor of 2

3. Amplitude 4; vertical stretch by a factor of 4, reflection across x-axis 2p 2p 1 5. Amplitude 0.73; vertical shrink by a factor of 0.73 7. Period ; horizontal shrink by a factor of 9. Period ; horizontal shrink 3 3 7 1 1 11. Period p; horizontal shrink by a factor of , vertical stretch by a factor of 3 by a factor of , reflection across y-axis 7 2 3 13. Amplitude 3, period 4p, 15. Amplitude , period p, 17. 19. y y 2 1 1 frequency frequency 4p 3 2 p y

y

4 2 –3!

–!

4 –3!

x

x

–!

–2

2 3!

!

3!

!

x

–3

x

–2 4

21.

23.

y 0.5 –!

!

x

41. 45. 47. 51. 59.

4

x

–5

–!

!

x

–0.5

31. Period p; amplitude 3; 3-2p, 2p4 by 3-4, 44

–12! 12!

x

–4

3p 3p p p and b ; minimum: -2 aat - and b ; zeros: 0, "p, "2p 2 2 2 2 7p p 3p p 3p 5p Maximum: 1 1at 0, " p, " 2p2; minimum: - 1 a at " and " b; zeros: " , " , " , " 2 2 4 4 4 4 7p p 3p p 3p 5p Maximum: 1 a at " and " b ; minimum: -1 1at 0, "p, "2p2; zeros: " , " , " , " 2 2 4 4 4 4 1 One possibility is y = sin 1x + p2. 43. Starting from y = sin x, horizontally shrink by and vertically shrink by 0.5. 3 2 Starting from y = cos x, horizontally stretch by 3, vertically shrink by , reflect across x-axis. 3 3 5 Starting from y = cos x, horizontally shrink by and vertically stretch by 3. 49. Starting with y1, vertically stretch by . 2p 3 1 Starting with y1, horizontally shrink by . 53. (a) and (b) 55. (a) and (b) 57. One possibility is y = 3 sin 2x. 2 p One possibility is y = 1.5 sin 121x - 12. 61. Amplitude 2, period 2p, phase shift , vertical translation 1 unit up 4

33. Period 6; amplitude 4; 3 - 3, 34 by 3 -5, 54

39.

y

0.5 1.5!

29. Period p; amplitude 1.5; 3 - 2p, 2p4 by 3 - 2, 24

27.

y

5 –1.5!

–0.5

37.

25.

y

35. Maximum: 2 a at -

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 875

SELECTED ANSWERS

2p p , phase shift , vertical translation 0.5 unit up 3 18 Amplitude 2, period 1, phase shift 0, vertical translation 1 unit up 7 5 Amplitude , period 2p, phase shift - , vertical translation 1 unit down 3 2 y = 2 sin 2x 1a = 2, b = 2, h = 0, k = 02 (a) two (b) 10, 12 and 12p, 1.3-2p2 L 16.28, 0.192 73. L 15.90 sec

875

63. Amplitude 5, period 65. 67. 69. 71.

77. (a) The maximum d is approximately 21.4 cm. The amplitude is 7.1 cm; scatter plot:

75. (a) 1:00 A.M.

(b) L 0.83 sec

2px b + 14.3 (c) d1t2 = - 7.1 cos a 0.83 (d)

(b) 8.90 ft; 10.52 ft

(c) 4:06 A.M.

79. One possible solution is p T = 21.5 cos a 1x - 72b + 57.5. 6

[0, 2.1] by [7, 22] [0, 2.1] by [7, 22]

[0, 13] by [10, 80]

81. False. y = sin 2x is a horizontal stretch of y = sin 4x by a factor of 2, so it has twice the period. 83. D 85. C 87. (a) 89. (a) 1/262 sec 91. (a) a - b must equal 1. 1 (b) a - b must equal 2 (b) ƒ = 262 (“cycles per sec”), or 262 Hertz (c) a - b must equal k. sec (c) a - b must equal k. [–∏, ∏] by [–1.1, 1.1]

(b) 0.0246x4 + 0x3 - 0.4410x2 + 0x + 0.9703 (c) The coefficients are fairly similar. [0, 0.025] by [–2, 2]

93. B = 10, 32; C = a

3p , 0b 4

3p p 95. B = a , 2 b; C = a , 0b 4 4

97. (a) If b is negative, then b = - B, where B is positive. Then y = a sin 3-B1x - H24 + k = - a sin 3B1x - H24 + k, since sine is an odd function. We will see in part (d) what to do if the number out front is negative. (b) A sine graph can be translated a quarter of a period to the left to become a cosine graph of the same sinusoid. Thus p 1 # 2p b d + k = a sin cb a x - a h b b d + k has the same graph as y = a cos 3b1x - h24 + k. 4 b 2b p We therefore choose H = h . 2b (c) The angles u + p and u determine diametrically opposite points on the unit circle, so they have point symmetry with respect to the origin. The y-coordinates are therefore opposites, so sin (u + p2 = - sin u. p (d) By the identity in (c), y = a sin 3b1x - h2 + p4 + k = - a sin 3b1x - h24 + k. We therefore choose H = h - . b (e) Part (b) shows how to convert y = a cos 3b1x - h24 + k to y = a sin 3b1x - H24 + k, and parts (a) and (d) show how to ensure that a and b are positive. y = a sin c b a 1x - h2 +

SECTION 4.5 Exploration 1 1. The graphs do not seem to intersect.

Quick Review 4.5 1. p

3. 6p

5. Zero: 3; asymptote: x = - 4

7. Zero: -1; asymptotes: x = 2 and x = - 2

9. Even

Exercises 4.5 1. The graph of y = 2 csc x must be vertically stretched by 2 compared to y = csc x, so y1 = 2 csc x and y2 = csc x. 3. y1 = 3 csc 2x, y2 = csc x

6965_SE_Ans_833-934.qxd

876

1/25/10

3:40 PM

Page 876

SELECTED ANSWERS

5.

7.

∏ ∏ [– , ] by [–6, 6] 2 2

Horizontal shrink of y = tan x by factor 1/2; asymptotes at multiples of p/4

9.

[–

2∏ 2∏ , ] by [–6, 6] 3 3

11.

[–

Horizontal shrink of y = sec x by factor 1/3; asymptotes at odd multiples of p/6

∏ ∏ , ] by [–6, 6] 2 2

[–4∏, 4∏] by [–6, 6]

Horizontal shrink of y = cot x by factor 1/2; vertical stretch by factor 2; asymptotes at multiples of p/2

Horizontal stretch of y = csc x by factor 2; asymptotes at multiples of 2p

13. Graph (a); Xmin = - p and Xmax = p 15. Graph (c); Xmin = - p and Xmax = p 17. Domain: All reals except integer multiples of p; Range: 1- q , q 2; Continuous on its domain; Decreasing on each interval in its domain; Symmetry with respect to the origin (odd); Not bounded above or below; No local extrema; No horizontal asymptotes; Vertical asymptotes: x = kp for all integers k; End behavior: lim cot x and lim cot x do not exist. x: -q

x: q

19. Domain: All reals except integer multiples of p; Range: 1- q , -14 ´ 31, q 2; Continuous on its domain; On each interval centered at 3p p x = + 2kp, where k is an integer, decreasing on the left half of the interval and increasing on the right; for x = + 2kp, increasing on the 2 2 first half of the interval and decreasing on the second half; Symmetric with respect to the origin (odd); Not bounded above or below; Local 3p p minimum 1 at each x = + 2kp and local maximum -1 at each x = + 2kp, where k is an integer; No horizontal asymptotes; Vertical 2 2 asymptotes: x = kp for all integers k; End behavior: lim csc x and lim csc x do not exist. x: q

21. Starting with y = tan x, vertically stretch by 3.

x: -q

23. Starting with y = csc x, vertically stretch by 3.

25. Starting with y = cot x, horizontally stretch by 2, vertically stretch by 3, and reflect across x-axis. 2 27. Starting with y = tan x, horizontally shrink by , reflect across x-axis, and shift up by 2 units. p 29. p/3 31. 5p/6 33. 5p/2 35. x L 0.92 37. x L 5.25 39. x L 0.52 or x L 2.62 -b b 41. (a) The reflection of 1a, b2 across the origin is 1 -a, - b2. (b) Definition of tangent (c) tan t = = = tan 1t - p2 a -a (d) Since points on opposite sides of the unit circle determine the same tangent ratio, tan 1t " p2 = tan t for all numbers t in the domain. Other points on the unit circle yield triangles with different tangent ratios, so no smaller period is possible. (e) The same argument that uses the a b ratio above can be repeated using the ratio , which is the cotangent ratio. a b 1 1 1 1 43. For any x, a b1x + p2 = = = a b1x2. This is not true for any smaller value of p, since this is the smallest value that ƒ ƒ1x + p2 ƒ1x2 ƒ

works for ƒ. 45. (a) d = 350 sec x (b) L 16,831 ft 47. L 0.905 49. L 1.107 or L 2.034 51. False. It is increasing only over intervals on which it is defined; i.e., intervals bounded by consecutive asymptotes. 53. A 55. D 57. About 1 -0.44, 02 ´ (0.44, p2

59. cot x is not defined at 0; the definition of “increasing on 1a, b2” requires that the function be defined everywhere in 1a, b2. Also, choosing a = - p/4 and b = p/4, we have a 6 b but ƒ1a2 = 1 7 ƒ1b2 = - 1.

61. csc x = sec ax 63. d =

p b 2

30 = 30 sec x cos x

[–∏, ∏] by [–10, 10]

[–0.5∏, 0.5∏] by [0, 100] [–∏, ∏] by [–10, 10]

65. L 0.8952 radians L 51.29°

6965_SE_Ans_833-934.qxd

1/26/10

11:35 AM

Page 877

SELECTED ANSWERS

877

SECTION 4.6 Exploration 1

[–2∏, 2∏] by [–6, 6]

[–2∏, 2∏] by [–6, 6]

[–2∏, 2∏] by [–6, 6]

Sinusoid

Sinusoid

Not a Sinusoid

[–2∏, 2∏] by [–6, 6]

[–2∏, 2∏] by [–6, 6]

Sinusoid

[–2∏, 2∏] by [–6, 6]

Sinusoid

Not a Sinusoid

Quick Review 4.6 1. Domain: 1- q , q 2; range: 3 -3, 34 5. Domain: 1- q , q 2; range: 3 -2, q 2

3. Domain: 31, q 2; range: 30, q 2 7. As x : - q , ƒ1x2 : q ; as x : q , ƒ1x2 : 0.

9. 1ƒ ! g21x2 = x - 4, domain: 30, q 2; 1g ! ƒ21x2 = 2x2 - 4, domain: 1 - q , -24 ´ 32, q 2

Exercises 4.6 1. Periodic

3. Not periodic

5. Not periodic

[–2!, 2!] by [–5, 20]

[–2!, 2!] by [–1.5, 1.5]

9. Since the period of cos x is 2p, we have cos21x + 2p2 = 1cos 1x + 2p222 = 1cos x22 = cos2 x. The period is therefore an exact divisor of 2p, and we see graphically that it is p. A graph for - p … x … p is shown:

[–2!, 2!] by [–6, 6]

The period is therefore an exact divisor of 2p, and we see graphically that it is p. A graph for -p … x … p is shown:

[–!, !] by [–1, 2]

y

2

15. Domain: all x Z np, n an integer; range: 30, q 2; y

2 1 1 –2!

2!

[–2!, 2!] by [–10, 10]

11. Since the period of cos x is 2p, we have 2cos21x + 2p2 = 21cos1x + 2p222 = 21cos x22 = 2cos2 x.

[–!, !] by [–1, 2]

13. Domain: 1- q , q 2; range: 30, 14;

7. Periodic

x –2!

2!

x

6965_SE_Ans_833-934.qxd

1/26/10

11:35 AM

Page 878

SELECTED ANSWERS

878

19. y = 2x - 1; y = 2x + 1

p + np, 2 n an integer; range: 1- q , 04;

17. Domain: all x Z

21. y = 1 - 0.3x; y = 3 - 0.3x

23. Yes

y

–2!

1

2!

x [–10, 10] by [–4, 8]

[–10, 10] by [–20, 20]

–4

25. Yes 35.

27. No

29. a L 3.61, b = 2, h L 0.49 37.

31. a L 2.24, b = p, h L 0.35 39. (a) 41. (c)

33. a L 2.24, b = 1, h L - 1.11

[–∏, ∏] by [–5, 5]

[–∏, ∏] by [–3.5, 3.5]

43. The damping factor is e -x, and the damping occurs as x : q . 47. The damping factor is x 3, and the damping occurs as x : 0. 49. ƒ oscillates between 1.2-x and - 1.2-x. As x : q , ƒ1x2 : 0.

45. No damping 1 1 and - . x x As x : q , ƒ1x2 : 0.

51. ƒ oscillates between

[0, 4∏] by [–1, 1] [0, 4∏] by [–1.5, 1.5]

53. 2p

55. 2p

[–2∏, 2∏] by [–3.4, 2.8]

57. Period 2p

[–2∏, 2∏] by [–3, 3]

[–4∏, 4∏] by [–1, 4]

59. Not periodic

[–4∏, 4∏] by [–13, 13]

63. Domain: 1 - q , q 2; range: 1- q , q 2 65. Domain: 1- q , q 2; range: 31, q 2

61. Not periodic

[–4∏, 4∏] by [–7, 7]

67. Domain: Á ´ 3 - 2p, - p4 ´ 30, p4 ´ 32p, 3p4 ´ Á ; that is, all x with 2np … x … 12n + 12p, n an integer; range: 30, 14 69. Domain: 1 - q , q 2; range: 30, 14 71. (a)

73. Not periodic 75. (a) 77. Graph (d), shown on 3-2p, 2p] by 3- 4, 44 79. Graph (b), shown on 3-2p, 2p4 by 3-4, 44 81. False. For example, the function has a relative minimum of 0 at x = 0 that is not repeated anywhere else. 83. B 85. D

[0, 12] by [–0.5, 0.5]

(b) For t 7 0.51 (approximately).

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 879

SELECTED ANSWERS

879

p p = 0.0661 Á L 0.07; on a TI-82: = 0.0668 Á L 0.07; on a 47.5 47 p p TI-85: = 0.0498 Á L 0.05; on a TI-92: = 0.0263 Á L 0.03. 63 119 (b) Period: p = p/125 = 0.0251Á . For any of the TI graphers, there are from 1 to 3 cycles between each pair of pixels; the graphs produced are therefore inaccurate, since so much detail is lost. 91. Domain: 30, q 2; range: 1- q , q 2; 89. Domain: 1 - q , q 2; range: 3 -1, 14; zeros at np, n a nonnegative integer horizontal asymptote: y = 1; p zeros at ln a + np b, 2 n a nonnegative integer

87. (a) Answers will vary — for example, on a TI-81:

[–0.5, 4∏] by [–4, 4]

[–3, 3] by [–1.2, 1.2]

93. Domain: 1- q , 02 ´ 10, q 2; range: approximately 3 -0.22, 12; horizontal asymptote: y = 0; zeros at np, n a nonzero integer

95. Domain: 1- q , 02 ´ (0, q 2;

range: approximately 3- 0.22, 12;

horizontal asymptote: y = 1; 1 zeros at , n a nonzero integer np

[–5∏, 5∏] by [–0.5, 1.2] [–∏, ∏] by [–0.3, 1.2]

SECTION 4.7 Exploration 1 1. x

3. 21 + x 2

Quick Review 4.7

5. 21 + x 2

1. sin x: positive; cos x: positive; tan x: positive

3. sin x: negative; cos x: negative; tan x: positive

5. 1/2

7. - 1/2

9. -1/2

Exercises 4.7 1. p/3

3. 0

5. p/3 7. - p/4 9. -p/4 11. p/2 13. 21.22° 15. - 85.43° 17. 1.172 19. - 0.478 p p -1 2 21. lim tan 1x 2 = and lim tan 1x 2 = 23. 13/2 25. p/4 27. 1/2 29. p/6 31. 1/2 x: q x: - q 2 2 p p p 33. Domain: 3 -1, 14; Range: c - , d ; Continuous; Increasing; Symmetric with respect to the origin (odd); Bounded; Absolute maximum of 2 2 2 , p absolute minimum of - ; No asymptotes; No end behavior (bounded domain) 2 p p 35. Domain: 1- q , q 2; Range: a - , b ; Continuous; Increasing; Symmetric with respect to the origin (odd); Bounded; No local extrema; 2 2 p p p p Horizontal asymptotes: y = and y = - ; End behavior: lim tan-1 x = and lim tan-1 x = x: q x: - q 2 2 2 2 1 1 1 p p 37. Domain: c - , d ; Range: c - , d ; Starting from y = sin-1 x, horizontally shrink by . 2 2 2 2 2 -1

2

39. Domain: 1- q , q 2; Range: a -

5p 5p , b ; Starting from y = tan-1 x, horizontally stretch by 2 and vertically stretch by 5 (either order). 2 2

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 880

SELECTED ANSWERS

880

41. 1 53. (b)

43.

p 5p + 2np and + 2np, for all integers n 6 6 (c) 2 or 15 ft.

45.

1 3

47. x/ 21 + x 2

49. x/ 21 - x 2

51.

1 21 + 4x2

[0, 25] by [0, 55]

s (b) As s changes from 10 to 20 ft, u changes from about 1.1458° to 2.2906° — it almost exactly doubles (a 99.92% 500 increase). As s changes from 200 to 210 ft, u changes from about 21.80° to 22.78° — an increase of less than 1°, and a very small relative change (only about 4.5%). (c) The x-axis represents the height and the y-axis represents the angle: The angle cannot grow past 90° (in fact, it approaches but never exactly equals 90°). 57. False. This is only true for -1 … x … 1, the domain of sin-1 x. 59. E 61. C 63. The cotangent function restricted to the interval 10, p2 is one-to-one and has an inverse. The unique angle y between 0 and p (noninclusive) such that cot y = x is called the inverse cotangent (or arccotangent) of x, denoted cot -1 x or arccot x. The domain of y = cot -1 x is 1- q , q 2 and the range is 10, p2.

55. (a) u = tan-1

65. (a) Domain all reals, range 3 -p/2, p/24, period 2p

[–2∏, 2∏] by [–0.5∏, 0.5∏]

(b) Domain all reals, range 30, p4, period 2p

(c) Domain all reals except p/2 + np, n an integer, range 1 -p/2, p/22, period p. Discontinuity is not removable.

[–2∏, 2∏] by [0, ∏] [–2∏, 2∏] by [–∏, ∏]

18 p - tan-1 x tan-1 x + 33 69. p 2 71. (a) y = p/2 (b) y = p/2, y = 3p/2 (c) The graph on the left (d) The graph on the left 67. y =

SECTION 4.8 Exploration 1 1. The unit circle 3. Since the grapher is plotting points along the unit circle, it covers the circle at a constant speed. Toward the extremes its motion is mostly vertical, so not much horizontal progress (which is all that we see) occurs. Toward the middle, the motion is mostly horizontal, so it moves faster.

Quick Review 4.8 1. b = 15 cot 31° L 24.964, c = 15 csc 31° L 29.124 3. b = 28 cot 28° - 28 cot 44° L 23.665, c = 28 csc 28° L 59.642, 5. Complement: 58°; supplement: 148° 7. 45° 9. Amplitude: 3; period: p a = 28 csc 44° L 40.308

Exercises 4.8 1. 300 13 L 519.62 ft 3. 120 cot 10° L 680.55 ft 5. wire length = 5 sec 80° L 28.79 ft; tower height = 5 tan 80° L 28.36 ft 7. 185 tan 80°1¿12– L 1051 ft 9. 100 tan 83°12¿ L 839 ft 11. 10 tan 55° L 14.3 ft 13. 4.25 tan 35° L 2.98 mi 15. 2001tan 40° - tan 30°2 L 52.35 ft 17. Distance: 6012 L 84.85 naut mi; bearing is 140°. 19. 1097 cot 19° L 3186 ft 550 21. 325 tan 63° L 638 ft 23. 36.5 tan 15° L 9.8 ft 25. L 2931 ft cot 70° - cot 80° 27. (a) 8 cycles/sec (b) d = 6 cos 16pt (c) About 4.1 in. left of the starting position 29. d = 3 cos 4pt cm 31. (a) 25 ft (b) 33 ft (c) p/10/radians sec 33. (a) p/6 (b) a = 182 - 482/2 = 17 and k = 82 - 17 = 65 (c) 13 + 1 = 42 (d) The fit is very good: (e) Setting 17 sin 1p/61t - 422 + 65 = 70, we get t = 4.57 or t = 9.43. These represent (approximately) days #139 and #287 of a 365-day year, namely May 19 and October 14. [0, 13] by [42, 88]

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 881

SELECTED ANSWERS

35. (a) March (b) November 37. True. Since the frequency and the period are reciprocals, the higher the frequency, the shorter the period. 43. (a) (b) The first is the best. 1232 2464 (c) About = L 392 oscillations/sec. p 2p

39. D

881

41. D

[0, 0.0062] by [–0.5, 1]

p L 5.2 cm 7

45. 2.5 cot

47. AC L 33.6 in.; BD L 12.9 in

51. (a)

49. tan-1 0.06 L 3.4°

(b) One pretty good match is y = 1.51971 sin324671t - 0.000224 (that is, a = 1.51971, b = 2467, h = 0.00022. Answers will vary but should be close to these values. 2467 (c) Frequency: about L 393 Hz; It appears to be a G. 2p (d) G [0, 0.0092] by [–1.6, 1.6]

CHAPTER 4 REVIEW EXERCISES 1. positive y-axis; 450° 3p 9. 270° or radians 2

17. 1/2

19. 1

3. QIII; - 3p/4 11. 30° = p/6 rad

21. 1/2

23. 2

5. QI; 13p/30

7. QI; 15°

13. 120° = 2p/3 rad

25. -1

15. 360° + tan-11- 22 L 296.565° L 5.176 radians

27. 0

p 1 p 13 p 1 p p 2 p , tan a- b = , csc a - b = - 2, sec a- b = , cot a - b = - 13 b = - , cos a - b = 6 2 6 2 6 6 6 6 13 13 1 1 , cos 1 -135°2 = , tan 1 -135°2 = 1, csc 1-135°2 = - 12, sec 1- 135°2 = - 12, cot 1 -135°2 = 1 sin 1 - 135°2 = 12 12 12 5 12 5 13 13 , cos a = , tan a = , csc a = , sec a = , cot a = sin a = 13 13 12 5 12 5 8 15 8 15 17 17 , cos u = , tan u = , csc u = , sec u = , cot u = sin u = 17 17 8 15 8 15 39. a = 15 sin 35° L 8.604, b = 15 cos 35° L 12.287, b = 55° L 4.075 radians 7 43. a = 216 L 4.90, a L 44.42°, b L 45.58° 45. QIII 47. QII b = 7 tan 48° L 7.774, c = L 10.461, a = 42° cos 48° 2 1 15 1 sin u = , cos u = , tan u = - 2, csc u = , sec u = - 15, cot u = 2 2 15 15 3 5 3 134 134 5 sin u = , cos u = , tan u = , csc u = , sec u = , cot u = 5 3 5 3 134 134 p Starting from y = sin x, 55. Starting from y = cos x, translate left units, 2 translate left p units. reflect across x-axis, and translate up 4 units.

29. sin a 31. 33. 35. 37. 41. 49. 51. 53.

[–2∏, 2∏] by [–1.2, 1.2]

[–2∏, 2∏] by [–1, 6]

6965_SE_Ans_833-934.qxd

882

1/25/10

3:40 PM

Page 882

SELECTED ANSWERS

57. Starting from y = tan x, 1 horizontally shrink by . 2

59. Starting from y = sec x, horizontally stretch by 2, vertically stretch by 2, and reflect across x-axis (in any order).

[–4∏, 4∏] by [–8, 8] [–0.5∏, 0.5∏] by [–5, 5]

2p ; phase shift: 0; domain: 1 - q , q 2; range: 3-2, 24 3 p 63. Amplitude: 1.5; period: p; phase shift: ; domain: 1- q , q 2; range: 3-1.5, 1.54 8 1 65. Amplitude: 4; period: p; phase shift: ; domain: 1 - q , q 2; range: 3- 4, 44 2 67. a L 4.47, b = 1, and h L 1.11 69. L 49.996° L 0.873 radians 71. 45° = p/4 rad 61. Amplitude: 2; period:

1 1 1 p p 73. Starting from y = sin-1 x, horizontally shrink by . Domain: c - , d ; range: c - , d 3 3 3 2 2

p 1 2 p 75. Starting from y = sin-1 x, translate right 1 unit, horizontally shrink by , translate up 2 units. Domain: c 0, d ; range: c2 - , 2 + d 3 3 2 2 77. 5p/6

79. 3p/4

81. 3p/2

89. Periodic; period: p; domain: x Z 91. Not periodic; domain: x Z 95. 100 tan 78° L 470 m 99. north tower

83. As ƒ x ƒ : q ,

sin x x2

: 0.

p + np, n an integer; range: 31, q 2 2

p + np, n an integer; range: 1 - q , q 2 2

85. 1

87. 3/4

93. 4p/3

97. 1501cot 18° - cot 42°2 L 295 ft 101. 62 tan 72°24¿ L 195.4 ft 103. 22p/15 L 4.6 in.

23° 128° south tower

105. (a) Day 123 (May 3)

(b) Day 287 (October 14)

Chapter 4 Project Answers are based on the sample data shown in the table. 1.

3. The constant a represents half the distance the pendulum bob swings as it moves from its highest point to its lowest point; k represents the distance from the detector to the pendulum bob when it is in midswing. 5. y L 0.22 sin 13.87x - 0.162 + 0.71; Most calculator/computer regression models are expressed in the form y = a sin 1bx + p2 + k, where -p/b = h in the equation y = a sin1b1x - h22 + k. The latter equation form differs from y = a cos1b1x - h22 + k only in h.

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 883

SELECTED ANSWERS

883

SECTION 5.1 Exploration 1 1 1 sin u , sec u = , and tan u = sec u cos u cos u cos u 1 1 3. csc u = , cot u = , and cot u = sin u tan u sin u 1. cos u =

Quick Review 5.1 1. 1.1760 rad = 67.380°

3. 2.4981 rad = 143.130°

Exercises 5.1

5. 1a - b22

7. (2x + y)(x - 2y)

9.

y - 2x xy

11. xy

1. sin u = 3/5 and cos u = 4/5 3. tan u = - 115 and cot u = - 1/115 = - 115/15 5. 0.45 7. -0.73 9. sin x 11. 1 13. tan2 x 15. cos x sin2 x 17. -1 19. -1 21. 1 23. cos x 25. 2 27. sec x 29. tan x 31. tan x 33. 2 csc2 x 35. - sin x 37. cot x 39. 1cos x + 122 41. 11 - sin x22 43. 12 cos x - 121cos x + 12 45. 12 tan x - 122 47. 1 - sin x 49. 1 - cos x 51. p/6, p/2, 5p/6, 3p/2 53. 0, p 55. p/3, 2p/3, 4p/3, 5p/3 p 57. " + 2np, n = 0, " 1, " 2, . . . 59. np, n = 0, "1, "2, . . . 61. np, n = 0, "1, " 2, . . . 3 63. 5 " 1.1918 + 2np|n = 0, " 1, "2, . . .6 65. 50.3047 + 2np or 2.8369 + 2np|n = 0, " 1, "2, . . .6 67. 5"0.8861 + np|n = 0, " 1, "2, . . .6 69. ƒ sin u ƒ 71. 3 ƒ tan u ƒ 73. 9 ƒ sec u ƒ 75. True. Since secant is an even function, sec a x 77. D

79. C

81. sin x, cos x = " 21 - sin2 x, tan x = "

p p b = sec a - xb , which equals csc x by one of the cofunction identities. 2 2

sin x

21 - sin2 x

83. The two functions are parallel to each other, separated by 1 unit for every x. At any x, the distance between the two graphs is sin2 x - 1-cos2 x2 = sin2 x + cos2 x = 1.

, csc x =

21 - sin2 x 1 1 , sec x = " , cot x = " sin x sin x 21 - sin2 x

[–2!, 2!] by [–4, 4]

85. (a)

(b) The equation is y = 13,111 sin 10.22997 x + 1.5712 + 238,855.

[–6, 70] by [220000, 260000] [–6, 70] by [220000, 260000]

(c) 12p2/0.22998 L 27.32 days. This is the number of days that it takes the moon to make one complete orbit of the Earth (known as the moon’s sidereal period). (d) 225,744 miles (e) y = 13,111 cos 1-0.22997x2 + 238855, or y = 13,111 cos 10.22997x2 + 238855.

87. Factor the left-hand side: sin4 u - cos4 u = 1sin2 u - cos2 u21sin2 u + cos2 u2 = 1sin2 u - cos2 u2 # 1 = sin2 u - cos2 u 89. Use the hint: sin 1p - x2 = sin 1p/2 - 1x - p/222 = cos 1x - p/22 Cofunction identity = cos 1p/2 - x2 Since cos is even = sin x Cofunction identity 91. Since A, B, and C are angles of a triangle, A + B = p - C. So: sin 1A + B2 = sin 1p - C2 = sin C.

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 884

SELECTED ANSWERS

884

SECTION 5.2 Exploration 1 1. The graphs lead us to conclude that this is not an identity.

3. Yes

[–2!, 2!] by [–4, 4]

5. No. The graph window cannot show the full graphs, so they could differ outside the viewing window. Also, the function values could be so close that the graphs appear to coincide.

Quick Review 5.2 1.

sin x + cos x sin x cos x

3.

1 sin x cos x

5. 1

7. No. Any negative x.

9. No. Any x for which sin x 6 0, e.g., x = - p/2.

11. Yes

Exercises 5.2

x1x 2 - x2 x3 - x2 - 1x - 121x + 12 = - 1x 2 - 12 = x 2 - x - 1x 2 - 12 = - x + 1 = 1 - x x x 1x + 221x - 22 1x + 321x - 32 x2 - 9 x2 - 4 3. One possible proof: 5. Yes 7. No = = x + 2 - 1x - 32 = 5 x - 2 x + 3 x - 2 x + 3 sin x cos x 11. 1cos x21tan x + sin x cot x2 = cos x # + cos x sin x # = sin x + cos2 x cos x sin x 1. One possible proof:

13. 11 - tan x22 = 1 - 2 tan x + tan2 x = 11 + tan2 x2 - 2 tan x = sec2 x - 2 tan x 15.

11 - cos u211 + cos u2

=

1 - cos2 u

=

sin2 u

cos2 u cos2 u cos2 u 19. Multiply out the expression on the left side.

= tan2 u

17.

cos2 x - 1 -sin2 x sin x = = sin x = - tan x sin x cos x cos x cos x

21. 1cos t - sin t22 + 1cos t + sin t22 = cos2 t - 2 cos t sin t + sin2 t + cos2 t + 2 cos t sin t + sin2 t = 2 cos2 t + 2 sin2 t = 2 23.

1 + tan2 x 2

2

sin x + cos x

=

sec2 x = sec2 x 1

25.

11 - sin b211 + sin b2 cos b cos2 b 1 - sin2 b 1 - sin b = = = = 1 + sin b cos b11 + sin b2 cos b11 + sin b2 cos b11 + sin b2 cos b

27.

tan2 x sec2 x - 1 1 1 - cos x = = sec x - 1 = - 1 = sec x + 1 sec x + 1 cos x cos x

29. cot 2 x - cos2 x = a

cos2 x11 - sin2 x2 cos x 2 cos2 x = cos2 x # = cos2 x cot 2 x b - cos2 x = 2 sin x sin x sin2 x

31. cos4 x - sin4 x = 1cos2 x + sin2 x21cos2 x - sin2 x2 = 11cos2 x - sin2 x2 = cos2 x - sin2 x

33. 1x sin a + y cos a22 + 1x cos a - y sin a22 = 1x 2 sin2 a + 2xy sin a cos a + y 2 cos2 a2 + 1x 2 cos2 a - 2xy cos a sin a + y 2 sin2 a2 = x 2 sin2 a + y 2 cos2 a + x 2 cos2 a + y 2sin2 a = 1x 2 + y 221sin2 a + cos2 a2 = x 2 + y 2

tan x1sec x + 12 tan x1sec x + 12 tan x sec x + 1 = = = . See also #26. 2 sec x - 1 tan x sec x + 1 tan2 x sin2 x - 11 - sin2 x2 1sin x - cos x21sin x + cos x2 sin x - cos x sin2 x - cos2 x 2 sin2 x - 1 = = = = 37. sin x + cos x 1 + 2 sin x cos x 1 + 2 sin x cos x 1sin x + cos x22 sin2 x + 2 sin x cos x + cos2 x 35.

39.

sin2 t + 11 + cos t211 - cos t2 211 - cos2 t2 211 + cos t2 sin t 1 + cos t 1 - cos2 t + 1 - cos2 t + = = = = 1 - cos t sin t 1sin t211 - cos t2 1sin t211 - cos t2 1sin t211 - cos t2 sin t

41. sin2 x cos3 x = sin2 x cos2 x cos x = sin2 x11 - sin2 x21cos x2 = 1sin2 x - sin4 x21cos x2

43. cos5 x = cos4 x cos x = 1cos2 x221cos x2 = 11 - sin2 x22 1cos x2 = 11 - 2 sin2 x + sin4 x21cos x2

9. Yes

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 885

SELECTED ANSWERS

45.

cot x tan x # sin x cot x # cos x sin2 x/cos x cos2 x/sin x sin x cos x tan x + = + = a + b 1 - cot x 1 - tan x 1 - cot x sin x 1 - tan x cos x sin x - cos x cos x - sin x sin x cos x =

47.

885

sin3 x - cos3 x sin2 x + sin x cos x + cos2 x 1 + sin x cos x 1 = = = + 1 = csc x sec x + 1 sin x cos x1sin x - cos x2 sin x cos x sin x cos x sin x cos x

2 tan x 1 - tan2 x

1

+

2 cos2 x - 1

=

2 tan x

2

# cos2 x

+

1

=

2 sin x cos x

1 - tan2 x cos x cos2 x - sin2 x cos2 x - sin2 x 2 2 1cos x + sin x2 2 sin x cos x + cos x + sin x cos x + sin x = = = 1cos x - sin x21cos x + sin x2 1cos x - sin x21cos x + sin x2 cos x - sin x

+

cos2 x + sin2 x cos2 x - sin2 x

2

49. cos3 x = 1cos2 x21cos x2 = 11 - sin2 x21cos x2

51. sin5 x = 1sin4 x21sin x2 = 1sin2 x221sin x2 = 11 - cos2 x221sin x2 = 11 - 2 cos2 x + cos4 x21sin x2

53. (d)

55. (c)

57. (b)

59. True. If x is in the domain of both sides of the equation, then x Ú 0. The equation holds for all x Ú 0, so it is an identity. 61. E 63. B 65. sin x 67. 1 69. 1 71. If A and B are complementary angles, then sin2 A + sin2 B = sin2 A + sin2 1p/2 - A2 = sin2 A + cos2 A = 1. 73. Multiply and divide by 1 - sin t under the radical:

11 - sin t22 11 - sin t22 ƒ 1 - sin t ƒ 1 - sin t # 1 - sin t = = = since A 1 + sin t 1 - sin t C 1 - sin2 t C cos2 t ƒ cos t ƒ

2a 2 = ƒ a ƒ . Now, since 1 - sin t Ú 0, we can dispense with the absolute value in the numerator, but it must stay in the denominator.

75. sin6 x + cos6 x = 1sin2 x23 + cos6 x = 11 - cos2 x23 + cos6 x = 11 - 3 cos2 x + 3 cos4 x - cos6 x2 + cos6 x = 1 - 3 cos2 x 11 - cos2 x2 = 1 - 3 cos2 x sin2 x

77. ln ƒ tan x ƒ = ln

ƒ sin x ƒ

ƒ cos x ƒ

= ln ƒ sin x ƒ - ln ƒ cos x ƒ

79. (a) They are not equal. Shown is the window 3- 2p, 2p4 by 3- 2, 24; graphing on nearly any viewing window does not show any apparent difference—but using TRACE, one finds that the y-coordinates are not identical. Likewise, a table of values will show slight differences; for example, when x = 1, y1 = 0.53988 while y2 = 0.54030.

(b) One choice for h is 0.001 (shown). The function y3 is a combination of three sinusoidal functions 11000 sin 1x + 0.0012, 1000 sin x, and cos x2, all with period 2p.

[–2!, 2!] by [–0.001, 0.001]

[–2!, 2!] by [–2, 2]

81. In the decimal window, the x-coordinates used to plot the graph on the calculator are (e.g.) 0, 0.1, 0.2, 0.3, etc.—that is, x = n/10, where n is an integer. Then 10px = pn, and the sine of integer multiples of p is 0; therefore, cos x + sin 10px = cos x + sin pn = cos x + 0 = cos x. 1 However, for other choices of x, such as x = , we have cos x + sin 10px = cos x + sin 10 Z cos x. p

SECTION 5.3 Exploration 1 1. No

3. tan a

p p p p + b = - 13, tan + tan = 2 13. (Many other answers are possible.) 3 3 3 3

Quick Review 5.3 1. 45° - 30°

3. 210° - 45°

5. 2p/3 - p/4

7. No

9. Yes

Exercises 5.3 1. 1 16 - 122/4 15. tan 66°

3. 1 16 + 122/4

17. cos a x -

p b 7

5. 112 + 162/4

19. sin 2x

7. 2 + 13

21. tan 12y + 3x2

9. 112 - 162/4

11. sin 25°

13. sin 7p/10

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 886

SELECTED ANSWERS

886

23. sin ax -

25. cos a x -

p p p b = sin x cos - cos x sin = sin x # 0 - cos x # 1 = - cos x 2 2 2 p p p b = cos x cos + sin x sin = cos x # 0 + sin x # 1 = sin x 2 2 2

p p b = sin x cos + cos x sin 6 6 tan u + tan 1p/42 p 29. tan au + b = = 4 1 - tan u tan 1p/42 27. sin ax +

31. Equations (b) and (f )

p 13 1 = sin x # + cos x # 6 2 2 tan u + 1 1 + tan u = 1 - tan u # 1 1 - tan u

33. Equations (d) and (h).

35. x = np, n = 0, "1, "2, . . .

p p p 37. sin a - u b = sin cos u - cos sin u = 1 # cos u - 0 # sin u = cos u 2 2 2 cos 1p/2 u2 sin u p 39. cot a - u b = = = tan u, using the first two cofunction identities. 2 sin 1p/2 - u2 cos u 41. csc a

p 1 1 - ub = = = sec u, using the second cofunction identity. 2 sin 1p/2 - u2 cos u

43. y L 5 sin 1x + 0.92732

45. y L 2.236 sin 13x + 0.46362

47. sin 1x - y2 + sin 1x + y2 = 1sin x cos y - cos x sin y2 + 1sin x cos y + cos x sin y2 = 2 sin x cos y

49. cos 3x = cos 31x + x2 + x4 = cos 1x + x2 cos x - sin 1x + x2 sin x = 1cos x cos x - sin x sin x2 cos x - 1sin x cos x + cos x sin x2 sin x = cos3 x - sin2 x cos x - 2 cos x sin2 x = cos3 x - 3 sin2 x cos x

51. cos 3x + cos x = cos 12x + x2 + cos 12x - x2; use Exercise 48 with x replaced with 2x and y replaced with x.

tan x - tan y tan2 x - tan2 y tan x + tan y since both the numerator and denominator are b#a b = 1 - tan x tan y 1 + tan x tan y 1 - tan2 x tan2 y factored forms for differences of squares. sin 1x + y2 sin x cos y + cos x sin y sin x cos y + cos x sin y 1/1cos x cos y2 # = = 55. sin 1x - y2 sin x cos y - cos x sin y sin x cos y - cos x sin y 1/1cos x cos y2 53. tan 1x + y2 tan 1x - y2 = a

=

1sin x cos y2/1cos x cos y2 + 1cos x sin y2/1cos x cos y2

1sin x cos y2/1cos x cos y2 - 1cos x sin y2/1cos x cos y2

1sin x/cos x2 + 1sin y/cos y2

=

1sin x/cos x2 - 1sin y/cos y2

=

tan x + tan y tan x - tan y

57. False. For example, cos 3p + cos 4p = 0 but 3p and 4p are not supplementary. 59. A 61. B cos u sin v sin v sin u cos v sin u sin 1u - v2 cos u cos v cos u cos v cos u cos v sin u cos v - cos u sin v tan u - tan v = = = = 63. tan 1u - v2 = sin u sin v cos 1u - v2 cos u cos v + sin u sin v cos u cos v sin u sin v 1 + tan u tan v 1 + + cos u cos v cos u cos v cos u cos v 3p 3p 3p b sin ax sin x cos - cos x sin 2 3p 2 2 3p = b = 65. The identity would involve tan a b , which does not exist. tan ax 2 2 3p 3p 3p cos x cos + sin x sin b cos ax 2 2 2 sin x # 0 - cos x # 1- 12 = = - cot x cos x # 0 + sin x # 1- 12

cos x 1cos h - 12 - sin x sin h cos x cos h - sin x sin h - cos x cos h - 1 sin h = = cos x a b - sin x h h h h h 69. sin 1A + B2 = sin 1p - C2 = sin p cos C - cos p sin C = 0 # cos C - 1 -12 sin C = sin C 67.

cos 1x + h2 - cos x

=

71. tan A + tan B + tan C = = =

sin A1cos B cos C2 + sin B1cos A cos C2 sin C1cos A cos B2 sin A sin B sin C + + = + cos A cos B cos C cos A cos B cos C cos A cos B cos C

cos C1sin A cos B + cos A sin B2 + sin C1cos A cos B2

=

cos A cos B cos C cos C sin 1p - C2 + sin C1cos 1p - C2 + sin A sin B2 cos A cos B cos C

sin A sin B sin C = tan A tan B tan C = cos A cos B cos C

cos C sin 1A + B2 + sin C1cos 1A + B2 + sin A sin B2

=

cos A cos B cos C cos C sin C + sin C1 -cos C2 + sin C sin A sin B cos A cos B cos C

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 887

SELECTED ANSWERS

887

73. This equation is easier to deal with after rewriting it as cos 5x cos 4x + sin 5x sin 4x = 0. The left side of this equation is the expanded form of cos 15x - 4x2, which of course equals cos x; the graph shown is simply y = cos x. The equation cos x = 0 is easily solved on the interval p 3p 3 - 2p, 2p4: x = " or x = " . The original graph is so crowded that one cannot see where crossings occur. The window shown is 2 2 3 -2p, 2p4 by 3 - 1.1, 1.14.

[–2!, 2!] by [–1.1, 1.1]

75. B = Bin + Bref = =

E0 E0 E0 vx vx vx vx cos a vt b + cos a vt + b = ccos a vt b + cos a vt + bd c c c c c c c

E0 E0 vx vx b = 2 cos vt cos a2 cos vt cos c c c c

The next-to-last step follows by the identity in Exercise 48.

SECTION 5.4 Exploration 1 1. sin2

1 - cos 1p/42 1 - 1 12/22 2 p # = 2 - 12 = = 8 2 2 2 4

3. sin2

1 - cos 19p/42 1 - 1 12/22 2 9p # = 2 - 12 = = 8 2 2 2 4

Quick Review 5.4 p p 3. x = + np, n = 0, " 1, "2, . . . + np, n = 0, " 1, "2, . . . 4 2 p 5p 2p 7. x = + 2np or x = + 2np or x = " + 2np, n = 0, "1, "2, . . . 6 6 3

1. x =

5. x = -

p + np, n = 0, "1, "2, . . . 4

9. 10 1/2 sq units

Exercises 5.4 1. cos 2u = cos 1u + u2 = cos u cos u - sin u sin u = cos2 u - sin2 u

3. Starting with the result of Exercise 1: cos 2u = cos2 u - sin2 u = 11 - sin2 u2 - sin2 u = 1 - 2 sin2 u 5p 7p p 3p 11. 2 sin u cos u + cos u or 1cos u212 sin u + 12 , , p, , 4 4 4 4 3 13. 2 sin u cos u + 4 cos u - 3 cos u or 2 sin u cos u + cos3 u - 3 sin2 u cos u 2 2 1 # sin x 17. 2 csc 2x = = = = csc2 x tan x sin 2x 2 sin x cos x sin2 x cos x

5. 0, p

7.

9. 0,

15. sin 4x = sin 212x2 = 2 sin 2x cos 2x

19. sin 3x = sin 2x cos x + cos 2x sin x = 2 sin x cos2 x + 12 cos2 x - 12 sin x = 1sin x214 cos2 x - 12 21. cos 4x = cos 212x2 = 1 - 2 sin2 2x = 1 - 212 sin x cos x22 = 1 - 8 sin2 x cos2 x 23.

p 5p , p, 3 3

25.

p p 3p 5p 3p 7p , , , , , 4 2 4 4 2 4

27. 0,

29.

4p 3p 5p p p 2p , , , p, , , 3 2 3 3 2 3

31. 11/22 32 - 23

[0, 2! ] by [–2, 2]

Solution: 50.1p, 0.5p, 0.9p, 1.3p, 1.5p, 1.7p6

33. 11/22 32 - 23

37. (a) Starting from the right side: 1 1 11 - cos 2u2 = 31 - 11 - 2 sin2 u24 = 2 2 (b) Starting from the right side: 1 1 11 + cos 2u2 = 31 + 12 cos2 u - 124 = 2 2

35. - 2 - 13 1 12 sin2 u2 = sin2 u. 2

1 12 cos2 u2 = cos2 u. 2

2 1 1 1 1 39. sin4 x = 1sin2 x22 = c 11 - cos 2x2 d = 11 - 2 cos 2x + cos2 2x2 = c1 - 2 cos 2x + 11 + cos 4x2 d 2 4 4 2 1 1 = 12 - 4 cos 2x + 1 + cos 4x2 = 13 - 4 cos 2x + cos 4x2 8 8

p 5p 3p , , 6 6 2

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 888

SELECTED ANSWERS

888

1 1 41. sin3 2x = sin 2x sin2 2x = sin 2x # 11 - cos 4x2 = 1sin 2x211 - cos 4x2 2 2 p p 5p 43. , p, ; general solution: " + 2np or p + 2np, n = 0, "1, "2, . . . 3 3 3 p p + 2np, n = 0, "1, "2, . . . 45. 0, ; general solution: 2np or 2 2 47. False. For example, ƒ1x2 = 2 sin x has period 2p and g1x2 = cos x has period 2p, but the product ƒ1x2 g1x2 = 2 sin x cos x = sin 2x has period p. 49. D 51. E 53. (a) In the figure, the triangle with side lengths x/2 and R is a right triangle, since R is given as the u x/2 perpendicular distance. Then the tangent of the angle u/2 is the ratio “opposite over adjacent”: tan = . Solving for x gives the desired 2 R equation. The central angle u is 2p/n since one full revolution of 2p radians is divided evenly into n sections. u 2p p (b) 5.87 L 2R tan , where u = , so R L 5.87/a2 tan b L 9.9957, R = 10. 2 11 11 p 1 1 1 # 1 1 1 55. u = ; the maximum value is about 12.99 ft 3. 57. csc 2u = = = # = csc u sec u 6 sin 2u 2 sin u cos u 2 sin u cos u 2 1 1 csc2 u csc2 u 1 59. sec 2u = = a b a b = = cos 2u 1 - 2 sin2 u 1 - 2 sin2 u csc2 u csc2 u - 2 61. sec 2u = 63. (a)

1 1 sec2 u csc2 u sec2 u csc2 u 1 = a b a b = = cos 2u cos2 u - sin2 u cos2 u - sin2 u sec2 u csc2 u csc2 u - sec2 u (b)

(c) The residual list: 53.73, 7.48, 3.05, -6.50, -9.77, - 3.67, 5.63, 9.79, 4.50, -3.66, -8.62, - 2.396.

[–30, 370] by [–60, 60]

[–30, 370] by [–60, 60]

Y1 = 44.52 sin 10.015x - 0.822 - 0.59

This is a fairly good fit, but not really as good as one might expect from data generated by a sinusoidal physical model.

(d)

[–30, 370] by [–15, 15]

Y2 = 8.73 sin 10.034x + 0.622 - 0.05 This is another fairly good fit, which indicates that the residuals are not random. There is a periodic variation that is most probably due to physical causes.

(e) The first regression indicates that the data are nearly sinusoidal. The second regression indicates that the variation of the data around the predicted values is also nearly sinusoidal. Periodic variation around periodic models is a predictable consequence of bodies orbiting bodies, but ancient astronomers had a difficult time reconciling their data with their simpler models of the universe.

SECTION 5.5 Exploration 1 1. If BC … AB, the segment will not reach from point B to the dotted line. On the other hand, if BC 7 AB, then a circle of radius BC will intersect the dotted line in a unique point. (Note that the line only extends to the left of point A.) 3. The second point 1C22 is the reflection of the first point 1C12 on the other side of the altitude. 5. If BC Ú AB, then BC can only extend to the right of the altitude, thus determining a unique triangle.

Quick Review 5.5 1. bc/d

3. ad/b

5. 13.314

7. 17.458°

9. 224.427°

Exercises 5.5 1. C = 75°; a L 4.5; c L 5.1 3. B = 45°; b L 15.8; c L 12.8 5. C = 110°; a L 12.9; c L 18.8 7. C = 77°; a L 4.1; c L 7.3 9. B L 20.1°; C L 127.9°; c L 25.3 11. C L 37.2°; A L 72.8°; a L 14.2 13. zero 15. two 17. two 19. B1 L 72.7°; C1 L 43.3°; c1 L 12.2; B2 L 107.3°; C2 L 8.7°; c2 L 2.7

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 889

SELECTED ANSWERS

21. 23. 25. 31. 35.

889

A1 L 78.2°; B1 L 33.8°; b1 L 10.8; A2 L 101.8°; B2 L 10.2°; b2 L 3.4 (a) 6.691 6 b 6 10 (b) b L 6.691 or b Ú 10 (c) b 6 6.691 (a) No: this is an SAS case. (b) No: only two pieces of information given 27. No triangle is formed 29. No triangle is formed 33. A1 L 24.6°; B1 L 80.4°; a1 L 20.7; A2 L 5.4°; B2 L 99.6°; a2 L 4.7 A = 99°; a L 28.3; b L 19.1 Cannot be solved with Law of Sines (an SAS case). 37. (a) 54.6 ft (b) 51.9 ft 39. L 24.9 ft 41. 1.9 ft 43. L 108.9 ft sin A sin B sin A a 45. 36.6 mi to A; 28.9 mi to B 47. True. By the Law of Sines, , which is equivalent to = = . a b sin B b 49. C 51. A 53. (b) Possible answers: a = 1, b = 13, c = 2 (or any set of three numbers proportional to these). (c) Any set of three identical numbers. 55. (a) h = AB sin A (b) BC 6 AB sin A (c) BC Ú AB or BC = AB sin A (d) AB sin A 6 BC 6 AB 57. AC L 8.7 mi; BC L 12.2 mi; h L 5.2 mi

SECTION 5.6 Exploration 1 1. 8475.742818 paces2 3. 0.0014714831 square miles 5. The estimate of “a little over an acre” seems questionable, but the roughness of their measurement system does not provide firm evidence that it is incorrect. If Jim and Barbara wish to make an issue of it with the owner, they would be well advised to get some more reliable data.

Quick Review 5.6 1. A L 53.130°

3. A L 132.844° x 2 + y 2 - 81 x2 + y2 - 81 b 5. (a) cos A = (b) A = cos-1 a 2xy 2xy 7. One answer: 1x - 121x - 22 9. One answer 1x - i21x + i2 = x 2 + 1

Exercises 5.6

1. A L 30.7°; C L 18.3°, b L 19.2 3. A L 76.8°; B L 43.2°, C L 60° 5. B L 89.3°; C L 35.7°, a L 9.8 7. A L 28.5°; B L 56.5°, c L 25.1 9. No triangles possible 11. A L 24.6°; B L 99.2°, C L 56.2° 13. B1 L 72.9°; C1 L 65.1°, c1 L 9.487; B2 L 107.1°; C2 L 30.9°, c2 L 5.376 15. No triangle 17. L 222.33 ft 2 19. L 107.98 cm2 21. L 8.18 23. No triangle is formed 25. L 216.15 27. L 314.05 29. L 1.445 radians 31. L 374.1 in.2 33. L 498.8 in.2 35. L 130.42 ft 37. (a) L 42.5 ft (b) The home-to-second segment is the hypotenuse of a right triangle, so the distance from the pitcher’s rubber to second base is 60 12 - 40 L 44.9 ft. (c) L 93.3° 39. (a) tan-1 11/32 L 18.4° (b) L 4.5 ft (c) L 7.6 ft 41. L 12.5 yd 43. L 37.9° 45. True. By the Law of Cosines, b 2 + c2 - 2bc cos A = a 2, which is a positive number. Since b 2 + c2 - 2bc cos A 7 0, it follows that b 2 + c2 7 2bc cos A. 47. B 49. C 51. Area = 1nr 2/22 sin 1360°/n2 30.2 - 15.1 37.2 - 12.4 53. (a) Ship A: 55. 6.9 in.2 = 15.1 knots; Ship B: = 12.4 knots (b) 35.18º (c) 34.8 nautical mi 1 hr 2 hr

CHAPTER 5 REVIEW EXERCISES 1. sin 200° 3. 1 5. cos 3x = cos 12x + x2 = cos 2x cos x - sin 2x sin x = 1cos2 x - sin2 x2 cos x - 12 sin x cos x2 sin x

= cos3 x - 3 sin2 x cos x = cos3 x - 311 - cos2 x2cos x = cos3 x - 3 cos x + 3 cos3 x = 4 cos3 x - 3 cos x

7. tan2 x - sin2 x = sin2 x a 9. csc x - cos x cot x = 11.

1 - cos2 x cos2 x

b = sin2 x #

sin2 x cos2 x

= sin2 x tan2 x

1 cos x 1 - cos2 x sin2 x - cos x # = = = sin x sin x sin x sin x sin x

11 + tan u211 - cot u2 + 11 + cot u211 - tan u2 1 + tan u 1 + cot u + = 1 - tan u 1 - cot u 11 - tan u211 - cot u2

=

11 + tan u - cot u - 12 + 11 + cot u - tan u - 12 11 - tan u211 - cot u2

=

0 = 0 11 - tan u211 - cot u2

6965_SE_Ans_833-934.qxd

13. cos2

Page 890

2 1 1 + sec t t 1 1 + cos t sec t = c" ba b = 11 + cos t2 d = 11 + cos t2 = a 2 B2 2 2 sec t 2 sec t

cos f sin f cos f cos f sin f sin f cos2 f sin2 f + = a ba b + a ba b = + 1 - tan f 1 - cot f 1 - tan f cos f 1 - cot f sin f cos f - sin f sin f - cos f

= 17.

3:40 PM

SELECTED ANSWERS

890

15.

1/25/10

cos2 f - sin2 f = cos f + sin f cos f - sin f

11 - cos y22 11 - cos y22 11 - cos y22 1 - cos y ƒ 1 - cos y ƒ = = = = 2 2 B 1 + cos y D 11 + cos y211 - cos y2 D 1 - cos y D sin y ƒ sin y ƒ

=

1 - cos y

ƒ sin y ƒ

; since 1 - cos y Ú 0, we can drop that absolute value.

tan u + tan 13p/42 tan u + 1-12 3p tan u - 1 b = = = 4 1 - tan u tan 13p/42 1 - tan u 1 -12 1 + tan u 1 - cos b cos b 1 1 = = csc b - cot b 21. tan b = 2 sin b sin b sin b 19. tan au +

1 sin2 x 1 - sin2 x cos2 x = = = cos x cos x cos x cos x cos x 25. Many answers are possible, for example, 1cos x - sin x211 + 4 sin x cos x2. 23. Yes: sec x - sin x tan x =

27. Many answers are possible, for example, 1 - 4 sin2 x cos2 x - 2 sin x cos x. p 5p + np or + np, n = 0, " 1, "2, . . . 29. 12 12 p 3p 31. - + np 33. tan 1 35. x L 1.12 37. x L 1.15 39. p/3, 5p/3 41. 43. No solutions 4 2 5p 7p 11p p b´a , 2p b 45. c 0, b ´ a , 47. 1p/3, 5p/32 49. y L 5 sin 13x + 0.92732 51. C = 68°, b L 3.9, c L 6.6 6 6 6 6 53. No triangle is formed. 55. C = 72°; a L 2.9, b L 5.1 57. A L 44.4°, B L 78.5°, C L 57.1° 59. L 7.5 61. (a) L 5.6 6 b 6 12 (b) b L 5.6 or b Ú 12 (c) b 6 5.6 63. L 0.6 mi 65. 1.25 rad 1 67. (a) sin u + sin 2u (b) u = 60°; L 1.30 square units 2 u 69. (a) h = 4000 sec - 4000 miles (b) L 35.51° 2 71. Area of circle - area of hexagon = 256p - 6 # 6413 L 139.140 cm2. 73. 405p/24 L 53.01 cm3 u + v u - v 75. (a) By the product-to-sum formula in Exercise 74c, 2 sin cos 2 2 u + v 1u v2 1 u + v + u - v + sin b = sin u + sin v. = 2 # asin 2 2 2 u - v - 1u + v2 u + v u - v + u + v 1 u - v (b) By the product-to-sum formula in Exercise 74c, 2 sin cos = 2 # asin + sin b 2 2 2 2 2 u - v u + v = sin u + sin 1 - v2 = sin u - sin v. (c) By the product-to-sum formula in Exercise 74b, 2 cos cos 2 2 u + v 1u v2 1 u + v + u - v = 2 # acos + cos b = cos v + cos u = cos u + cos v. (d) By the product-to-sum formula in 2 2 2

Exercise 74a, -2 sin

u + v - 1u - v2 u - v u + v + u - v 1 u + v sin = - 2 # a cos - cos b = - 1cos v - cos u2 = cos u - cos v. 2 2 2 2 2

77. (a) Any inscribed angle that intercepts an arc of 180° is a right angle. (b) Two inscribed angles that intercept the same arc are congruent. (d) Because ∠A¿ and ∠A are congruent,

opp a = . hyp d sin A (e) Of course. They both equal by the Law of Sines. a

(c) In right ¢A¿BC, sin A¿ =

sin A sin A¿ a/d 1 = = = . a a a d

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 891

SELECTED ANSWERS

Chapter 5 Project 1.

[–2, 34] by [–0.1, 1.1]

5. One possible model is y = 0.5 sin (0.2(x + 8.4)) + 0.5.

SECTION 6.1 Exploration 1 1. 85, 29

3. 86, - 79

Quick Review 6.1 1.

9 13 ; 4.5 2

3. - 5.36; - 4.50

5. 33.85°

9. 180° + tan-1 15/22 L 248.20°

7. 60.95°

Exercises 6.1 1. Both vectors represent 83, - 29 by the HMT Rule. 5. 85, 29; 129 15. 8- 3, 89

7. 8- 5, 19; 126

17. 84, - 99

2 1 , i 15 15 29. L 816.31, 7.619 25. (a) h

3. Both vectors represent 8- 2, - 29 by the HMT Rule.

9. 8-2, - 249; 2 1145

19. 8-4, - 189

21. -

1

25

11. 8-11, -79; 1170

i +

2

25

j

23. -

1

25

i -

13. 81, 79 2

25

j

2 1 4 5 4 5 i + j ,i (b) i j 27. (a) h 15 15 141 141 141 141 31. L 8 -14.52, 44.709 33. 5; L 53.13° 35. 5; L 306.87° 37. 7; 135°

(b)

39. 812, - 129

41. L 8- 223.99, 480.349 43. (a) L 8 - 111.16, 305.409 (b) L 362.85 mph; bearing L 337.84° 45. (a) L 83.42, 9.409 (b) The horizontal component is the (constant) horizontal speed of the basketball as it travels toward the basket. The vertical component is the vertical velocity of the basketball, affected by both the initial speed and the downward pull of gravity. 47. L 82.20, 1.439 49. ƒ F ƒ L 100.33 lb and u L - 1.22° 51. L 342.86°; L 9.6 mph 53. L 13.66 mph; L 7.07 mph 55. True. u and - u have the same length but opposite directions. Thus, the length of -u is also 1 57. D 59. A

SECTION 6.2 Exploration 1 1. 8 -2 - x, -y9, 82 - x, - y9

3. Answers will vary.

Quick Review 6.2 1. 113

3. 1

Exercises 6.2 1. 72

3. -47

5. 83, 139

7. 8 -1, - 139

9. h

6 4 , i 113 113

5. 30 7. - 14 9. 13 11. 4 13. L 115.6° 15. L 64.65° 17. 165° 19. 135° 21 21 17 82 29 82 21. L 94.86° 25. - 83, 19; - 83, 19 + 27. 29. 47.73°, 74.74°, 57.53° 8 -1, 39 89, 29; 89, 29 + 8- 2, 99 10 10 10 85 85 85 31. L -20.78 33. Parallel 35. Neither 37. Orthogonal 39. (a) 14, 02 and 10, -32 (b) 14.6, -0.82 or 13.4, 0.82 53 8 i 41. (a) 17, 02 and 10, - 32 (b) L 17.39, - 0.922 or 16.61, 0.922 43. 8 -1, 49 or h , 45. L 138.56 pounds 13 13 47. (a) L 415.82 pounds (b) L 1956.30 pounds 49. 14,300 foot-pounds 51. L 21.47 foot-pounds 53. L 85.38 foot-pounds 55. 100 139 L 624.5 foot-pounds 61. False. If one of u or v is the zero vector, then u # v = 0 but u and v are not perpendicular. 5 1 31129 63. D 65. A 67. (a) 2 # 0 + 5 # 2 = 10 and 2 # 5 + 5 # 0 = 10 (b) 85, -29; 862, 1559 (c) ƒ w2 ƒ = 29 29 29 ƒ 2x0 + 5y0 - 10 ƒ ƒ ax0 + by0 - c ƒ (d) d = (e) d = 129 2a2 + b2

891

6965_SE_Ans_833-934.qxd

892

1/25/10

3:40 PM

Page 892

SELECTED ANSWERS

SECTION 6.3 Exploration 1 1.

3. t = 12

5. Tmin … - 2 and Tmax Ú 5.5

[–10, 5] by [–5, 5]

Exploration 2 3.

[0, 450] by [0, 80]

[0, 450] by [0, 80]

[0, 450] by [0, 80]

[0, 450] by [0, 80]

Quick Review 6.3 1. (a) 8 - 3, -29

(b) 84, 69

5.

(c) 87, 89

8 8 1x + 32 or y - 6 = 1x - 42 7 7 9. 20p rad/sec

3. y + 2 =

7. x 2 + y 2 = 4

[–3, 7] by [–7, 7]

Exercises 6.3 1. (b) 3 - 5, 54 by 3 - 5, 54 5. (a) t -2 -1 x 0 1 y 1/2 -2

3. (a) 3 - 5, 54 by 3 -5, 54 5. (b) 0 1 2 2 und.

3 4

y

4 5/2

5

a4, 5 b 2 a0, – 1 b 2

7.

(3, 4)

8

x

(1, –2)

9.

[–10, 10] by [–10, 10]

[–10, 10] by [–10, 10]

11. y = x - 1: line through 10, - 12 and 11, 02 2

13. y = - 2x + 3, 3 … x … 7: line segment with endpoints 13, -32 and 17, -112

17. y = x 3 - 2x + 3: cubic polynomial 19. x = 4 - y 2 2 parabola that opens to left with vertex at 14, 02 21. t = x + 3, so y = , on domain: 3-8, -32 ´ 1-3, 24 23. x2 + y2 = 25, x + 3 circle of radius 5 centered at 10, 02 25. x2 + y2 = 4, three-fourths of a circle of radius 2 centered at 10, 02 (not in Quadrant II) 27. x = 6t - 2; y = - 3t + 5 29. x = 3t + 3, y = 4 - 7t, 0 … t … 1 31. x = 5 + 3 cos t, y = 2 + 3 sin t, 0 … t … 2p 15. x = 1y - 12 : parabola that opens to right with vertex at 10, 12

33. 0.5 6 t 6 2

35. -3 … t 6 - 2

37. (b) Ben is ahead by 2 ft.

39. (a) y = - 16t 2 + 1000

(c) 744 ft

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 893

SELECTED ANSWERS

41. (a) 0 6 t 6 p/2

(b) 0 6 t 6 p

893

43. (a) About 2.80 sec (b) L 7.18 ft 45. (a) Yes (b) 1.59 ft p p 47. No 49. v L - 10.00 ft/sec or 551.20 ft/sec 51. x = 35 cos a t b and y = 50 + 35 sin a t b 6 6 53. (a) When t = p 1or 3p, or 5p, etc.2, y = 2. This corresponds to the highest points on the graph. (b) 2p units 55. (no answer) 57. (no answer) 59. True. Both correspond to the rectangular equation y = 3x + 4. 61. A 63. D

65. (a)

(c) p/2 6 t 6 3p/2

(b) a

(d) 1x - h22 + 1y - k22 = a2; circle of radius a centered

(c)

at 1h, k2 (e) x = 3 cos t - 1; y = 3 sin t + 4

[–6, 6] by [–4, 4]

[–6, 6] by [–4, 4]

67. (a) Jane is traveling in a circle of radius 20 feet and origin 10, 202, which yields x1 = 20 cos 1nt2 and y1 = 20 + 20 sin 1nt2. Since the Ferris wheel is making one revolution 12p2 every 12 seconds, 2p = 12n, so n = y1 = 20 + 20 sin a

p tb , in radian mode. 6

2p p p = . Thus, x1 = 20 cos a tb and 12 6 6

(b) Since the ball was released at 75 ft in the positive x-direction and gravity acts in the negative

y-direction at 16 ft/s2, we have x2 = at + 75 and y2 = - 16t2 + bt, where a is the initial speed of the ball in the x-direction and b is the initial speed of the ball in the y-direction. The initial velocity vector of the ball is 60 8cos 120°, sin 120°9 = 8-30, 30 139, so a = - 30 and b = 3013. As a result, x2 = - 30t + 75 and y2 = - 16t3 + 130132t are the parametric equations for the ball. (c)

Jane and the ball will be close to each other but not at the exact same point at t = 2.2 seconds.

2 2 p p tb + 30t - 75 b + a20 + 20 sin a t b + 16t 2 - 30 13t b 6 6 D (e) The minimum distance occurs at t L 2.2 seconds, when d1t2 L 1.64 feet. Jane will have a good chance of catching the ball.

(d) d1t2 =

a 20 cos a

[–50, 100] by [–50, 50]

69. About 4.11 ft

71. (a) (no answer) (b) (no answer)

73. t =

1 2 1 1 3 , ;t = , , 3 3 4 2 4

SECTION 6.4 Exploration 1 1. (no answer)

3. 1 - 2, p/32, 12, p/22, 13, 02, 11, p2, 14, 3p/22

Quick Review 6.4 1. (a) II (b) III

3. 7p/4, - 9p/4

7. 1x - 322 + y 2 = 4

5. 500°, - 200°

Exercises 6.4 3 313 1. a - , b 2 2

(b)

3. 1 - 1, - 132

7.

y 5 a3,

5! a 3, b 2 6

! b 2

3 3 , 2 3 2 , a 2

a–

(0, ! ) (0, 2! )

5. (a)

3

x

a3,

5

p/2

5p/6

p

4p/3

2p

312/2

3

3/2

0

- 313/2

0

(2, 30°) 30°

13.

9.

4π 3

4! b 3 ! b 4

up/4 r

2π 5

1 O

O 4π b 3

9. L 11.14

a–1,

2π b 5

11.

2 O

120° O 2 (–2, 120°)

6965_SE_Ans_833-934.qxd

894

1/25/10

3:40 PM

Page 894

SELECTED ANSWERS

3 3 15. a , 13b 4 4

17. 1- 2.70, 1.302

19. 12, 02

p p + 2npb and a -2, + 12n + 12pb, n an integer 6 6 p 5p p 3p 27. (a) a 12, b or a - 12, b (b) a 12, b or a - 12, b 4 4 4 4

21. 10, - 22

25. 11.5, - 20° + 360n°2 and 1-1.5, 160° + 360n°2, n an integer

9p 13p 29. (a) 1129, 1.952 or 1- 129, 5.092 b or a - 12, b. 4 4 (c) The answers from part (a), plus 1129, 8.232 or 1- 129, 11.382 31. (b)

(c) The answers from part (a), and also a 12, (b) 1 - 129, - 1.192 or 1129, 1.952

37. x2 + a y +

35. x = 3 — a vertical line 39. x2 + a y with radius 15

23. a 2,

3 3 9 3 b = — a circle centered at a 0, - b with radius 2 4 2 2

1 2 1 1 1 b = — a circle centered at a 0, b with radius 2 4 2 2 45. r =

43. r = 2/cos u = 2 sec u

33. (c)

2

[–5, 5] by [–5, 5]

41. 1x + 222 + 1y - 122 = 5 — a circle centered at 1 - 2, 12

5 2 cos u - 3 sin u

[–5, 5] by [–5, 5]

47. r 2 - 6r cos u = 0, so r = 6 cos u

49. r 2 + 6r cos u + 6r sin u = 0, so r = - 6 cos u - 6 sin u

[–3, 9] by [–4, 4]

[–12, 6] by [–9, 3]

a p a 3p a 5p a 7p , b, a , b, a , b , and a , b 12 4 12 4 12 4 12 4 55. False. 1r, u2 = 1r, u + 2np2 for any integer n. These are all distinct polar coordinates. 57. C 59. A 61. (a) If u1 - u2 is an odd integer multiple of p, then the distance is ƒ r1 + r2 ƒ . If u1 - u2 is an even integer multiple of p, then the distance is ƒ r1 - r2 ƒ . (b) 63. L 6.24 65. L 7.43 67. x = ƒ1u2 cos 1u2, y = ƒ1u2 sin 1u2 69. x = 51cos u21sin u2, y = 5 sin2 u 71. x = 4 cot u, y = 4 51. 2 13 L 3.46 mi

53. a

SECTION 6.5 Quick Review 6.5

p 3p p 3p 5p 7p 1. Minimum: -3 at x = e , f ; Maximum: 3 at x = 50, p, 2p6 3. Minimum: 0 at x = e , , , f ; Maximum: 2 2 2 4 4 4 4 2 2 at x = 50, p, 2p6 5. No; no; yes 7. sin u 9. cos u - sin u

Exercises 6.5 1. (a)

u

0

p/4

p/2

3p/4

p

5p/4

3p/2

7p/4

r

3

0

-3

0

3

0

-3

0

(b) a0,

5

! 3! 5! 7! , , , 4 4 4 4

(3, !)

6

a–3,

y b 5

! b 2

a–3,

3! b 2

(3, 0) x 5

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 895

SELECTED ANSWERS

3. k = p

895

5. k = 2p

[–5, 5] by [–3, 3]

[–5, 5] by [–4, 3]

7. r3 is graph (b) 9. Graph (b) is r = 2 - 2 cos u 11. Graph (a) is r = 2 - 2 sin u 13. Symmetric about the y-axis 15. Symmetric about the x-axis 17. All three symmetries 19. Symmetric about the y-axis 21. Maximum r is 5—when u = 2np for any integer n. 23. Maximum r is 3 1along with -32—when u = 2np/3 for any integer n. 25. Domain: All reals Range: r = 3 Continuous Symmetric about the x-axis, y-axis, and origin Bounded Maximum r-value: 3 No asymptotes

[–6, 6] by [–4, 4]

33. Domain: All reals Range: 30, 84 Continuous Symmetric about the x-axis Bounded Maximum r-value: 8 No asymptotes

[–6, 12] by [–6, 6]

41. Domain: All reals Range: 30, q 2 Continuous No symmetry Unbounded Maximum r-value: none No asymptotes Graph for u Ú 0:

[–45, 45] by [–30, 30]

27. Domain: u = p/3 Range: 1- q , q 2 Continuous Symmetric about the origin Unbounded Maximum r-value: none No asymptotes

[–4.7, 4.7] by [–3.1, 3.1]

35. Domain: All reals Range: 33, 74 Continuous Symmetric about the x-axis Bounded Maximum r-value: 7 No asymptotes

[–7, 11] by [–6, 6]

3p p d ´ cp, d 2 2 Range: 30, 14 Continuous on domain Symmetric about the origin Bounded Maximum r-value: 1 No asymptotes

43. Domain: c 0,

[–1.5, 1.5] by [–1, 1]

29. Domain: All reals Range: 3- 2, 24 Continuous Symmetric about the y-axis Bounded Maximum r-value: 2 No asymptotes

[–3, 3] by [–2, 2]

37. Domain: All reals Range: 3- 3, 74 Continuous Symmetric about the x-axis Bounded Maximum r-value: 7 No asymptotes

[– 4, 8] by [– 4, 4]

31. Domain: All reals Range: 31, 94 Continuous Symmetric about the y-axis Bounded Maximum r-value: 9 No asymptotes

[–9, 9] by [–2.5, 9.5]

39. Domain: All reals Range: 30, 24 Continuous Symmetric about the x-axis Bounded Maximum r-value: 2 No asymptotes

[–3, 1.5] by [–1.5, 1.5]

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 896

SELECTED ANSWERS

896

45. 56, 2, 6, 26 47. 53, 5, 3, 5, 3, 5, 3, 5, 3, 56 49. r1 and r2 51. r2 and r3 53. (a) A 4-petal rose curve with 2 short petals of length 1 and 2 long petals of length 3 (b) Symmetric about the origin (c) Maximum r-value: 3 55. (a) A 6-petal rose curve with 3 short petals of length 2, and 3 long petals of length 4 (b) Symmetric about the x-axis (c) Maximum r-value: 4 57. (no answer) 59. (no answer) 61. False. The spiral r = u is unbounded. 63. D 65. B 67. (e) Domain: All reals Range: 3- ƒ a ƒ , ƒ a ƒ 4 Continuous Symmetric about the x-axis Bounded Maximum r-value: ƒ a ƒ No asymptotes 69. (a) For r1: 0 … u … 4p (or any interval that (b) r1: 10 (overlapping) petals; is 4p units long). For r2: same answer. r2: 14 (overlapping) petals 71. Starting with the graph of r1, if we rotate counterclockwise (centered at the origin) by p/4 radians 145°2, we get the graph of r2; rotating r1 counterclockwise by p/3 radians 160°2 gives the graph of r3.

[–5, 5] by [–5, 5]

[–5, 5] by [–5, 5]

[–5, 5] by [–5, 5]

73. (no answer)

SECTION 6.6 Quick Review 6.6 1. 2 + 3i, 2 - 3i

3. -4 - 4i

5. u =

5p 6

7.

4p 3

9. 1

Exercises 6.6 1.

y

–2 + 2i

1 + 2i i

x 3–i

p p p 2p p 2p + i sin b + i sin b 5. 2 12 a cos + i sin b 7. 4 a cos 9. L 1131cos 0.59 + i sin 0.592 2 2 4 4 3 3 p 3 13 p 3 12 16 13. 15. 5/2 - 15/22 13i 17. 19. 141cos 155° + i sin 155°2 3 acos + i sin b i - i 6 6 2 2 2 2 23p 23p 2 5 1 23. 1cos 30° - i sin 30°2 25. 21cos p + i sin p2 27. (a) 5 + i; - i (b) Same as part (a) 15 acos + i sin b 12 12 3 2 2 12 12 (a) 18 - 4i; L 0.35 + 0.41i (b) Same as part (a) 31. 33. 4 12 + 4 12 i 35. - 4 - 4i 37. -8 + i 2 2 -1 + 13 i - 1 - 13 i 4p 10p 16p 4p 10p 16p 41. 2 3 3acos 3 3 a cos + i sin b, 2 + i sin b, 2 3 3 acos + i sin b ; ;2 32 9 9 9 9 9 9 2 34 2 34

3. 3 acos 11. 21. 29. 39.

43. L 2 3 51cos 1.79 + i sin 1.792, L 2 3 51cos 5.97 + i sin 5.972 3 51cos 3.88 + i sin 3.882, L 2 p 3p 3p 7p 7p 9p 9p p 45. cos + i sin , cos + i sin , - 1, cos + i sin , cos + i sin 5 5 5 5 5 5 5 5 p 13p 5p 37p 49p p 13p 5p 37p 49p 47. 2 5 2acos 5 2 acos + i sin b , 2 + i sin b, 2 5 2 acos + i sin b, 2 5 2 acos + i sin b, 2 5 2 acos + i sin b 30 30 30 30 6 6 30 30 30 30

6965_SE_Ans_833-934.qxd

6/10/10

5:04 PM

Page 897

SELECTED ANSWERS

5 2 acos 49. 2

p p 9p 9p 13p 13p 17p 17p + i sin b , 2 5 2i, 2 5 2 a cos + i sin b, 2 5 2 acos + i sin b, 2 5 2 a cos + i sin b 10 10 10 10 10 10 10 10

53. 12acos

p p 17p 17p + i sin b , -1 + i, 12a cos + i sin b 12 12 12 12

8 2 acos 51. 2

55.

1 + i 2 64

,2 6 2 a cos

2 6 2 acos

57. 1, -

p p 9p 9p 17p 17p 25p 25p + i sin b , 2 8 2 a cos + i sin b, 2 8 2 a cos + i sin b, 2 8 2 acos + i sin b 16 16 16 16 16 16 16 16

7p 7p 11p 11p 5p 5p 19p 19p + i sin b, 2 6 2 acos + i sin b, 2 6 2 a cos + i sin b, 2 6 2 a cos + i sin b, 12 12 12 12 4 4 12 12

23p 23p + i sin b 12 12

1 13 ! i 2 2

59. ! 1,

13 1 13 1 ! i, - ! i 2 2 2 2

y

y

0.5

0.5 0.5

x

61. -8; - 2 and 1 ! 13i

0.5

x

65. False. For example, the complex number 1 + i has infinitely many trigonometric forms.

p p 9p 9p + i sin b. Here are two: 12a cos + i sin b , 12acos 67. B 69. A 4 4 4 4 71. (a) (no answer) (b) r 2 (c) cos 12u2 + i sin 12u2 (d) (no answer) 73. Set the calculator for rounding to 2 decimal places. In part (b), use Degree mode. (a) (b) (c)

75. x1t2 = 1 132t cos 10.62t2 y1t2 = 1 132t sin 10.62t2

79. 1, -

13 13 1 1 + i, - i 2 2 2 2

81. -1,

13 1 13 1 + i, i 2 2 2 2

[–7, 2] by [0, 6]

83. - 1, L 0.81 + 0.59i, 0.81 - 0.59i, -0.31 + 0.95i, - 0.31 - 0.95i

CHAPTER 6 REVIEW EXERCISES 1. 8 -2, -39

3. 137

5. 6

7. 83, 69; 3 15

9. 8-8, - 39; 173

11. (a) h -

3 5 13. (a) tan-1 a b L 0.64, tan-1 a b L 1.19 (b) L 0.55 15. L 1-2.27, -1.062 4 2 2p 2p 19. a 1, + 12n + 12pb and a - 1, + 2npb, n an integer 3 3

2 1 , i 15 15

17. 112, - 122

(b) h

3 6 ,i 15 15

897

6965_SE_Ans_833-934.qxd

898

1/25/10

3:40 PM

Page 898

SELECTED ANSWERS

3 3 21. (a) a - 113, p + tan-1 a - b b L 1- 113, 2.162 or a 113, 2p + tan-1 a - b b L 1113, 5.302 2 2 3 3 (b) a 113, tan-1 a - b b L 1 113, - 0.982 or a - 113, p + tan-1 a - b b L 1- 113, 2.162 2 2

3 3 (c) The answers from part (a), and also a - 113, 3p + tan-1 a - b b L 1 - 113, 8.442 or a 113, 4p + tan-1 a - b b L 1 113, 11.582 2 2 23. (a) 15, 02 or 1-5, p2 or 15, 2p2 (b) 1- 5, -p2 or 15, 02 or 1-5, p2 (c) The answers from part (a), and also 1-5, 3p2 or 15, 4p2

3 29 29 3 25. y = - x + : line through a 0, b with slope m = 27. x = 21y + 122 + 3: parabola that opens to right with vertex 5 5 5 5 at 13, -12 29. y = 1x + 1: square root function starting at 1 - 1, 02 31. x = 2t + 3, y = 3t + 4 33. a = - 3, b = 4, ƒ z 1 ƒ = 5

7p 7p 7p + i sin b . Other representations would use angles + 2np, n an integer. 4 4 4 41. L 1343cos 15.252 + i sin 15.2524. Other representations would use angles L 5.25 + 2np, n an integer. 35. 3 13 + 3i

37. - 1.25 - 1.2513 i

39. 312a cos

3 43. 121cos 90° + i sin 90°2; 1cos 330° + i sin 330°2 4 5p 243 12 243 12 5p + i sin b (b) i 45. (a) 243 acos 4 4 2 2 8 18 acos 49. 2

47. (a) 1251cos p + i sin p2

(b) -125

p p 9p 9p 17p 17p 25p 25p + i sin b , 2 8 18 a cos + i sin b, 2 8 18 a cos + i sin b, 2 8 18 a cos + i sin b 16 16 16 16 16 16 16 16 y

1 x

1

51. 1, cos

2p 2p 4p 4p 6p 6p 8p 8p + i sin , cos + i sin , cos + i sin , cos + i sin 5 5 5 5 5 5 5 5 y

0.5 x

0.5

53. (b)

55. (a)

63. a x + 65. r = -

57. Not shown

59. (c)

61. x2 + y2 = 4—a circle with center 10, 02 and radius 2

3 2 13 113 3 b + 1y + 122 = —a circle of radius with center a - , - 1b 2 4 2 2 4 = - 4 csc u sin u

[–10, 10] by [–10, 10]

67. r = 6 cos u - 2 sin u

[–3, 9] by [–5, 3]

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 899

SELECTED ANSWERS

69.

899

71.

[–7.5, 7.5] by [–8, 2]

[–3, 3] by [–2.5, 1.5]

Domain: All reals Range: 3-2, 24 Continuous Symmetric about the y-axis Bounded Maximum r-value: 2 No asymptotes

Domain: All reals Range: 3 -3, 74 Continuous Symmetric about the y-axis Bounded Maximum r-value: 7 No asymptotes 73. (a) (no answer)

(b) (no answer) (c) (no answer)

(d)

[–9, 2] by [–6, 6]

75. (a) L 8 - 463.64, 124.239

(b) L 508.29 mph; bearing L 283.84°

2

79. (a) h = - 16t + 245t + 200 (c)

77. (a) L 826.91 pounds (b) 2883.79 pounds

(b) Graph and trace: x = 17 and y = - 16t 2 + 245t + 200 (d) 924 ft (e) L 1138 ft; t L 7.66 (f ) About 16.09 sec with 0 … t … 16.1 (upper limit may vary) on 30, 184 by 30, 12004. This graph will appear as a vertical line from about (17, 0) to about (17, 1138). Tracing shows how the arrow begins at a height of 200 ft, rises to over 1000 ft, then falls back to the ground.

[0, 18] by [0, 1200]

81. x = 40 sin a

2p 2p t b, y = 50 - 40 cos a tb 15 15

83. (a)

(b) All 4’s should be changed to 5’s. 85. t L 1.06 sec, x L 68.65 ft 87. (a) L 77.59 ft (b) L 4.404 sec 89. L 17.65 ft [–7.5, 7.5] by [–5, 5]

Chapter 6 Project Answers are based on the sample data shown in the table. 1.

3.

[–0.1, 2.1] by [0, 1.1]

5.

[–0.1, 2.1] by [–1.1, 1.1]

[0, 1.1] by [–1.1, 1.1]

6965_SE_Ans_833-934.qxd

900

1/25/10

3:40 PM

Page 900

SELECTED ANSWERS

SECTION 7.1 Exploration 1 1.

[0, 10] by [–5, 5]

Quick Review 7.1 1. y =

5 2 - x 3 3

2 3. x = - , x = 1 3

5. 0, 2, -2

7. y = 1-4x + 62/5

Exercises 7.1 1. (a) No

(b) Yes (c) No

13. 1 -3/2, 27/22 and 11/3, 2/32 19. 18, - 22

31. One solution

21. 14, 22

3. 19, -22

5. 150/7, -10/72

33. Infinitely many solutions

39. 1 -1.2, 1.62 and 12, 02

7. 1- 1/2, 22

17. a

35. L 10.69, - 0.372

41. L 12.05, 2.192 and 1- 2.05, 2.192

45. (a) y L - 0.2262x 2 + 3.9214x + 22.7333

(b) y L

[–2, 10] by [–5, 50]

47. (a) y L 317.319x + 12894.513

9. No solution

42.2253

37. L 1-2.32, -3.162, 10.47, - 1.772 and 11.85, -1.082

43. Demand curve Supply curve 13.75, 143.752

1 + 0.8520e -0.3467x

(c) Quadratic: Never. Logistic: About 2015. (d) The quadratic regression predicts that the expenditures will eventually be zero. (e) The logistic regression predicts that the expenditures will eventually level off at about 42.225 billion dollars.

[–2, 10] by [–5, 50]

(b) y L 46.853x + 5567.528

[0, 20] by [0, 30000]

(c) About 1963

[0, 20] by [0, 10000]

65. (a) y = 13/22 24 - x 2, y = - 13/22 24 - x 2

49. L 5.28 m * L 94.72 m 51. Current speed L 1.06 mph; rowing speed L 3.56 mph 53. Medium: $0.79; large: $0.95 55. a = 2/3 and b = 14/3 57. (a) 300 miles 59. False. A system of two linear equations in two variables has either 0, 1, or infinitely many solutions. 61. C 63. D

(b)

[– 4.7, 4.7 by –3.1, 3.1]

67. 1 ; 12/3, 10/32

11. 1;3, 92

- 1 - 3 189 3 - 189 -1 + 3 189 3 + 189 , b and a , b 10 10 10 10 25. Infinitely many solutions 27. 10, 12 and 13, -22 29. No solution

15. 10, 02 and 13, 182

23. No solution

9. -4x - 6y = - 10

69. 12.5 units

L 1- 1.29, 2.292 or 11.91, -0.912

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 901

SELECTED ANSWERS

901

SECTION 7.2 Exploration 1 1. A = c

1 -1 d; B = c 4 2

2 5

2 d 5

Exploration 2

3. c

-1 d 2

8 11

1. det A = - a12a21a33 + a13a21a32 + a11a22a33 - a13a22a31 - a11a23a32 + a12a23a31 3. Since each term in the expansion contains an element from each row and each column, at least one factor in each term is a zero. Therefore, the expansion will be the sum of n zero terms, or zero.

Quick Review 7.2 1. 13, 22; 1x, - y2

3. 1 - 2, 32; 1y, x2

Exercises 7.2 1. 2 * 3; not square 11. (a) c (d) C

3 -3

- 18 6 13

19. (a) c

0 d 1

2 - 11

-10 (b) C 8 4

5 -4 -2

3. 3 * 2; not square (b) c

2 -5 S 5

5. 13 cos u, 3 sin u2

1 1

6 d 9

(c) c

6 -3

5. 3 * 1; not square

9 d 15

-3 15. (a) C 1 S 4

-1 (b) C 1 S -4

4 (b) C -5 -2

-5 -6 S 6

2 d 12

-15 12 S 6

31. a = - 2, b = 0

8 4 -8

(d) c

1 4

6 21. (a) C 3 8

35. c

-1 1

7. 3

(d) C -7 7 -1

-1 2S -12

-2 3S -1

-1 27. (a) C 2 1 1.5 d -1

9. cos a cos b - sin a sin b

9. 4 1 1 -7 13. (a) C - 2 0 S (b) C 2 -1 0 5

15 d 22

-6 (c) C 3 S 0

25. Not possible; 318, 144

33. AB = BA = I2

7. sin a cos b + cos a sin b

17. (a) c

(b) C 3 0 2

2 5 - 18

4 1S 1

-4 -11

1 0 -3

1 (b) C 1 4

37. No inverse

1 -2 S 2 - 18 d - 17

3 0S 10 2 0 3

1 2S -1

-9 (c) C 0 6 (b) c

61. 3x

y4c

c 0

0 d 1

(b) L 10.37 1.372

55. A # A-1 = I2

57. 3x

y4c

-1 0

39. No inverse

0 d 1

63. False. It can be negative. For example, the determinant of A = c

71. (a) A # A-1 = A-1 # A = I2

1 2

(c) It is the inverse of A.

73. (b) The constant term equals -det A.

29. a = 5, b = 2

41. - 14

Exploration 1 1. x + y + z must equal 60 L. 3. The number of liters of 35% solution must equal twice the number of liters of 55% solution. 3.75 5. C 37.5 S 18.75

Quick Review 7.3 3. 38 L

5. 1 - 1, 62

y4c

0 d is -1. -1

0 -1

43. B

1 R -1/3

(b) 1A - C2B T -1 d 0

65. B

67. D

(c) The coefficient of x 2 is the opposite of the sum of the elements of the main diagonal in A.

SECTION 7.3

1. 12.8 L

59. 3x

-12 d -26

23. (a) 3- 84

45. (a) The distance from city X to city Y is the same as the distance from city Y to city X. (b) Each entry represents the distance from city X to city X. 47. (a) 3382 227.504 49. (a) AB T or BAT

51. (a) L 11.37 0.372

5 0

3 -3 S 3

7. y = - z + w + 1

9. c

- 0.5 0.5

- 0.75 d 0.25

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 902

SELECTED ANSWERS

902

Exercises 7.3 1. a

25 7 , , - 2b 2 2

2 9. C 1 0

-6 2 -8

4 -3 S 4

3. 11, 2, 12

15 9 7 7. a , , 4, - b 2 2 2

5. No solution - 10 2 1

0 11. C 1 -3

10 -3 S -2

13. R12

15. 1 -32R2 + R3

For Exercises 17–20, possible answers are given. 1 17. C 0 0 2 25. C -1 3

-1 - 1.2 S 1

3 1 0

-3 1 0

1 -4 -1

1 19. C 0 0 1 -3 S 2

2 1 0

3 0 1

2 27. C 1 0

-4 -0.6 S -9.2

1 21. C 0 0

-5 0 2

-1 1 -1

1 -2 -3

0 1 0

2 -1 0

1 2S 0

23. c

1 0

0 1

-1 2

3 d -1

-3 4S 5

In Exercises 29–32, the variable names are arbitrary. 31. 2x + z = 3 29. 3x + 2y = - 1 -x + y = 2 - 4x + 5y = 2 2y - 3z = - 1 1 33. 12, -1, 42; C 0 0 37. No solution 45. c

2 1

-2 1 0

1 -1 1

39. 12 - z, 1 + z, z2

5 x -3 dc d = c d -2 y 1

55. 10, -10, 12

8 -5 S 4

71. (a) y L 1.0404x + 69.4047

2x + 4y = 3

59. a 2 -

65. No solution

43. 1z + w + 2, 2z - w - 1, z, w2 49. 1- 2, 32

3 1 z, - z - 4, z b 2 2

51. 1-2, -5, - 72

61. 1 -2w - 1, w + 1, - w, w2

67. ƒ1x2 = 2x 2 - 3x - 2

(b) y L 3.1290x + 144.8465

[–10, 50] by [–50, 300]

93. (a) C1x2 = x 2 - 8x + 13

41. No solution

47. 3x - y = - 1

57. 13, 3, - 2, 02

63. 1 -w - 2, -z + 0.5, z, w2

35. 1-2, 3, 12

(c) 2020

53. 1-1, 2, - 2, 32

69. ƒ1x2 = 1 -c - 32x 2 + x + c, for any c

73. 825 children, 410 adults, 165 senior citizens 75. $14,500 CDs, $5500 bonds, $60,000 growth funds 77. $0 CDs, $38,983.05 bonds, $11,016.95 growth fund 79. 22 nickels, 35 dimes, and 17 quarters 81. 116/3, 220/32 85. False. The determinant of the matrix must be not equal to zero 87. D 89. D

[–10, 50] by [–50, 300]

(b)

(c) 4 ; 13

(d) det A = C102 = 13

(e) a11 + a22 = 14 - 132 + 14 + 132 = 8

[–1, 8.4] by [–3.1, 3.1]

SECTION 7.4 Exploration 1 1. (a) 3 = A2

(b) 2 = A1

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 903

SELECTED ANSWERS

903

Quick Review 7.4 1.

3x - 5

3.

2

x - 4x + 3

4x 2 + 6x + 1 x 3 + 2x 2 + x

3 x - 2

5. 3x 2 - 2 +

Exercises 7.4

7. 1x + 121x - 321x 2 + 42

9. A = 3, B = - 1, C = 1

A1 A2 A3 A4 A5 B1x + C1 -3 4 3 + + 2 + 3 + + + + 5. 7. 2 2 x x - 1 x + 4 x - 2 x x 1x - 12 x2 + 9 x + 1 2 x + 2 1 2 1 2 -1 3x - 1 1 -1 + 2 + + + + 2 + 9. 11. 13. 15. 2 3 2 2 x - 2 x + 3 x - 5 x - 3 x 1x - 22 1x - 22 1x + 32 x + 2 1x + 22 2 3 1 -1/2 1 -2 -2 5 2 -1 + 2 + + + + + 17. 19. 21. 23. 2 25. x 2x x + 2 x - 3 x + 4 x + 3 2x - 1 x - 1 x + 1 1x + 122 1x 2 2x -1 1 -1 -x 2 + 2 + 2 + + 2 27. 29. 2 31. x x - 3 x - 1 x x + 2 1x + 222 x + x + 1 1.

A1 A2 A3 + + x x - 2 x + 2

x + 5

3.

3 -2 + x 1 x + 1 x - 1 2 Graph of 12x + x + 32/1x 2 - 12:

33. 2 +

2

;

[– 4.7, 4.7] by [–10, 10]

3 x - 2 -2 ; + x x2 + x x + 1

Graph of y = 1x 3 - 22/1x 2 + x2

[–4.7, 4.7] by [–10, 15]

37. (c)

Graph of -2/1x + 12:

[– 4.7, 4.7] by [–10, 10]

[–4.7, 4.7] by [–10, 10]

35. x - 1 +

Graph of 3/1x - 12:

39. (d)

Graph of y = x - 1:

Graph of y = 3/1x + 12:

[–4.7, 4.7] by [–10, 15]

47. True. The behavior of ƒ near x = 3 is the same as the behavior of y = 49. E

51. B

53. (a) A = 3

[– 4.7, 4.7] by [–10, 15]

-3 3 + 45. 1b - a21x - a2 1b - a21x - b2

1 -1 + 43. ax a1x - a2

41. (a)

[– 4.7, 4.7] by [–10, 15]

Graph of y = - 2/x:

(b) B = - 2, C = 2

1 1 = - q. and limx :3 x - 3 x - 3

55. b/1x - 122

SECTION 7.5 Quick Review 7.5 1. 13, 02; 10, - 22

3. 120, 02; 10, 502

y

5. 130, 602

y

5

50

5

x

50

x

7. 110, 1402

9. 13, 32

2x - 1 1x 2 + 122 -2 + 1 x + 1 2 - 122

6965_SE_Ans_833-934.qxd

1/26/10

11:36 AM

Page 904

SELECTED ANSWERS

904

Exercises 7.5 1. Graph (c); boundary included y 7.

3. Graph (b); boundary included y 9.

5

5

9

5

x

5

5. Graph (e); boundary included y 11.

x

5

Boundary x = 4 included 13.

y

15.

5

17.

y 10

6

5

x

10

Boundary x 2 + y 2 = 9 excluded

Boundary y = 2x excluded 21.

y

Boundary y = sin x excluded 23.

y

16

y 90

5

5 10

x

x 90

25.

27. x 2 + y 2 … 4 y Ú -x2 + 1

y 9

9

49. y1 = 2 1 -

B

x

33. 35. 37. 39. 41.

x2 x2 ; y2 = - 2 1 9 9 B

x

x

6

19.

Boundary y = x 2 + 1 excluded

Boundary 2x + 5y = 7 included

y

x

1 29. y … - x + 5 2 3 y … - x + 9 2 x Ú 0 y Ú 0

x

31. The minimum is 0 at 10, 02; the maximum is 880/3 at 1160/3, 80/32.

The minimum is 162 at 16, 302; there is no maximum. The minimum is 24 at 10, 122; there is no maximum. L13.48 tons of ore R and L 20.87 tons of ore S; $1926.20 x operations at Refinery 1 and y operations at Refinery 2 such that 2x + 4y = 320 with 40 … x … 120. False. It is a half-plane. 43. A 45. D 51.

y 5

5

x

6965_SE_Ans_833-934.qxd

1/25/10

3:40 PM

Page 905

SELECTED ANSWERS

905

CHAPTER 7 REVIEW EXERCISES 1. (a) c

1 8

2 d 3

1 7. C 2 -2

2 -3 1

1 17. C 0 0

0 1 0

(b) c

-3 0

-3 -3 4 S; C 2 -1 1

0 0 1

4 d -3

(c) c

2 1 -2

8 - 11 S 5

-6 d 0

2 -8

4 -3 S -1

19. 11, 22

57.

11 d -6

9. AB = BA = I4

29. 19/4, - 3/4, -7/42

37. Demand curve L 17.57, 42.712 Supply curve

49.

-7 4

21. No solution

27. 1 - 2z + w + 1, z - w + 2, z, w2 1 -2, 1, 3, - 12 43. 1x, y2 L 12.27, 1.532

(d) c

3. c

-3 0

-7 -12

-2 0 11. D 10 -3

-5 -1 24 -7

y

51.

31. No solution

39. 1x, y2 L 10.14, - 2.292

59.

5

5

3x - 4 2 + 2 x + 1 x + 1

6 1 -27 8

-1 0 T 4 -1

23. 1- z - w + 2, w + 1, z, w2

53. (c)

13. 20

1 15. C 0 0

0 1 0

2 -1 S 0

25. No solution

33. 1- w + 2, z + 3, z, w2

35.

47.

1 2 + x + 1 x - 4

55. (b)

61.

y 90

5. 33 74; not possible

41. 1x, y2 = 1-2, 12 or 1x, y2 = 12, 12

45. 1a, b, c, d2 = 117/840, - 33/280, -571/420, 386/352

-1 1 2 + + x + 2 x + 1 1x + 122

11 d; not possible 24

63.

y

y 5

7

x

5 8 90

x

x

Corners at 10, 902, 190, 02, 1360/13, 360/132 Boundaries included

Corners at L 10.92, 2.312 and L 15.41, 3.802 Boundaries included

Corners at L 1- 1.25, 1.562 and L 11.25, 1.562 Boundaries included

65. The minimum is 106 at 110, 62; there is no maximum. 67. The minimum is 205 at 110, 252; the maximum is 292 at 14, 402. 69. (a) L 12.12, 0.712 (b) L 1 -0.71, 2.122 71. (a) y L 11.6327x + 975.4457 (b) y L 21.4916x + 880.8866 (c) About 1989 and 7 months

77. $160,000 at 4%, $170,000 at 6.5%, $320,000 at 9% 79. Pipe A: 15 hours; Pipe B: L 5.45 hours; Pipe C: 12 hours 81. n must be equal to p. [–5, 50] by [850, 1650]

[–5, 50] by [850, 1650]

73. (a) N = 3200 400 600 2504

Chapter 7 Project 1.

(b) P = 3$80 $120 $200 $3004

(c) NP T = $259,000

75. Answers will vary.

412.574 315.829 ; Females: y L ; 145, 642; 1 + 10.956e -0.01539x 1 + 9.031e -0.01831x This represents the time when the female population became greater than the male population. (159, 212); This represents the time when the male population will again become greater than the female population. 7. Approx. 49.1% male and 50.9% female 3. Yes; no; no

[–1, 20] by [110, 160]

Males: y L 1.7585x + 119.5765 Females: y L 1.6173x + 126.4138

5. Males: y L

x

6965_SE_Ans_833-934.qxd

906

1/25/10

3:40 PM

Page 906

SELECTED ANSWERS

SECTION 8.1 Exploration 1 1. The axis of the parabola with focus (0, 1) and directrix y = - 1 is the y-axis because it is perpendicular to y = - 1 and passes through 10, 12. The vertex lies on this axis midway between the directrix and the focus, so the vertex is the point 10, 02. 3. 51 -2 16, 62, 1 -2 15, 52, 1- 4, 42, 1- 2 13, 32, 1- 2 12, 22, 1- 2, 12, 10, 02, 12, 12, 1212, 22, 12 13, 32, 14, 42, 12 15, 52, 1216, 626

Exploration 2 1.

3.

y

5.

y

10

7. Downward

y 10

10 y=4

y=4

y=4 10

V(2, 1)

x

10

F(2, –2)

V(2, 1)

x

10

F(2, –2)

A(–4, –2)

F(2, –2)

x B(8, –2)

x=2

x=2

Quick Review 8.1 1. 113

5. y + 6 = - 1x - 122

3. y = ; 2 1x

9. ƒ1x2 = - 21x + 122 + 3

7. Vertex: (1, 5); ƒ1x2 can be obtained from g1x2 by stretching x 2 by 3, shifting up 5 units, and shifting right 1 unit.

[–3, 4] by [–2, 20]

Exercises 8.1 3 3 1. Vertex: 10, 02; focus: a 0, b ; directrix: y = - ; focal width: 6 2 2

3. Vertex: 1-3, 22; focus: 1-2, 22; directrix: x = - 4; focal width: 4

1 1 4 5. Vertex: 10, 02; focus: a 0, - b ; directrix: y = ; focal width: 3 3 3 15. x 2 = 20y

17. y 2 = 8x

2

19. x 2 = - 6y

9. (a)

21. 1y + 422 = 81x + 42

2

25. 1y - 32 = - 81x - 42

7. (c)

27. 1x - 22 = 161y + 12

2

11. y 2 = - 12x

13. x 2 = - 16y

23. 1x - 322 = 61y - 5/22

29. 1y + 42 = - 101x + 12

31.

y

5

5

33.

35.

y

37.

y

x

39.

10

5

5

x

6

x [– 4, 4] by [–2, 18]

[–8, 2] by [–2, 2]

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 907

SELECTED ANSWERS

41.

43.

45.

[–2, 6] by [– 40, 5]

[–10, 15] by [–3, 7]

907

47.

[–22, 26] by [–19, 13]

[–13, 11] by [–10, 6]

49. Completing the square, the equation becomes 1x + 122 = y - 2, a parabola with vertex 1-1, 22, focus 1- 1, 9/42, and directrix y = 7/4. 51. Completing the square, the equation becomes 1y - 222 = 81x - 22, a parabola with vertex 12, 22, focus 14, 22, and directrix x = 0. 53. 1y - 222 = - 6x 55. 1x - 222 = - 41y + 12 57. The derivation only requires that p is a fixed real number. 59. The filament should be placed 1.125 cm from the vertex along the axis of the mirror. 61. The electronic receiver is located 2.5 units from the vertex along the axis of the parabolic microphone. 63. Starting at the leftmost tower, the lengths of the cables are roughly 65. False. Every point on a parabola is the same distance from its 79.44, 54.44, 35, 21.11, 12.78, 10, 12.78, 21.11, 35, 54.44, and 79.44. focus and its directrix. 67. D 69. B 71. (a)–(c) (d) Parabola y

F l

73. (a)

Axis

P A x

Generator

(b)

(c)

Axis

Generator

Cylinder

Circle

Single line

Plane

(d)

Plane

Line

Two parallel lines

SECTION 8.2 Exploration 1 1. x = - 2 + 3 cos t and y = 5 + 7 sin t; cos t =

1x + 222 1y - 522 y - 5 x + 2 and sin t = ; cos2 t + sin2 t = 1 yields the equation + = 1. 3 7 9 49

3. Example 1: x = 3 cos t and y = 2 sin t Example 2: x = 2 cos t and y = 113 sin t Example 3: x = 3 + 5 cos t and y = - 1 + 4 sin t 5. Example 1: x = 3 cos t, y = 2 sin t; cos t =

y y2 x2 x , sin t = ; cos2 t + sin2 t = 1 yields + = 1, or 4x 2 + 9y 2 = 36. 3 2 9 4

y y2 x2 x , sin t = + = 1. ; sin2 t + cos2 t = 1 yields 2 13 4 113 1x - 322 1y + 122 y + 1 x - 3 Example 3: x = 3 + 5 cos t, y = - 1 + 4 sin t; cos t = , sin t = ; cos2 t + sin2 t = 1 yields + = 1. 5 4 25 16 Example 2: x = 2 cos t, y = 113 sin t; cos t =

6965_SE_Ans_833-934.qxd

6/10/10

5:05 PM

Page 908

SELECTED ANSWERS

908

Exploration 2 3. a = 8 cm, b L 7.75 cm, c = 2 cm, e = 0.25, b/a L 0.97; a = 7 cm, b L 6.32 cm, c = 3 cm, e L 0.43, b/a L 0.90; a = 6 cm, b L 4.47 cm, c = 4 cm, e L 0.67, b/a L 0.75 5.

[–0.3, 1.5] by [0, 1.2]

b/a = 21 - e2

Quick Review 8.2 3 3. y = ! 24 - x 2 2

1. 161

5. x = 8

7. x = 2, x = - 2

9. x =

3 ! 115 2

Exercises 8.2 1. Vertices: 14, 02, 1- 4, 02; Foci: 13, 02, 1 - 3, 02 7. (d)

Foci: 11, 02, 1 - 1, 02

9. (a)

11.

3. Vertices: 10, 62, 10, - 62; Foci: 10, 32, 10, - 32 13.

y

5

17.

y

y

10

10

15.

5. Vertices: 12, 02, 1- 2, 02;

x

5

x

19.

8

4

21.

y2 x2 + = 1 9 4 1y - 222

x [–9.4, 9.4] by [–6.2, 6.2]

23. x 2/25 + y 2/21 = 1

25.

[–17, 4.7] by [–3.1, 3.1]

y2 x2 + = 1 25 16

27.

y2 x2 + = 1 36 16

29.

y2 x2 + = 1 25 16

1x - 122

= 1 33. 1x - 322/9 + 1y + 422/5 = 1 35. 1y + 222/25 + 1x - 322/9 = 1 36 16 37. Center: 1 -1, 22; vertices: 1-6, 22, 14, 22; foci: 1 -4, 22, 12, 22 39. Center: 17, -32; vertices: 17, 62, 17, - 122; foci: 17, - 3 ! 1172 31.

+

41.

43.

45. Vertices: 11, -42, 11, 22; foci: 11, - 1 ! 152; eccentricity:

[–8, 8] by [–6, 6]

x = 2 cos t, y = 5 sin t

[–8, 2] by [0, 10]

x = 2 13 cos t - 3, y = 15 sin t + 6

47. Vertices: 1 -7, 12, 11, 12; foci: 1 -3 ! 17, 12; eccentricity: 51.

y2

a

2

+

x2

b2

= 1

15 3

17 13 foci: 14, -8 ! 132; eccentricity: 4 2

53. a = 237,086.5, b L 236,571, c = 15,623.5, e L 0.066

49.

1x - 222

55. L 1347 Gm, L 1507 Gm

16

+

1y - 322 9

= 1

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 909

SELECTED ANSWERS

57. a - c 6 1.511.3922 = 2.088 Gm

59. 1 ; 151.75, 02 L 1 ;7.19, 02

909

61. 1-2, 02, 12, 02

63. (a) Approximate solutions: 1 "1.04, - 0.862, 1" 1.37, 0.732

1 + 1161 294 + 21161 - 1 + 1161 294 - 21161 ,b, a " , b 8 16 8 16 65. False. The distance is a11 - e2. 67. C 69. B (b) a "

71. (a) When a = b = r, A = pab = prr = pr 2 and P L p12r2 # 13 - 114r214r2/12r22 = p12r2 # 13 - 22 = 2pr. 73. (a) (b) y 2/4 + 1x - 322 = 1

(b) Answers will vary.

[–4.7, 4.7] by [–3.1, 3.1]

SECTION 8.3 Exploration 1 1. x = - 1 + 3/cos t = - 1 + 3 sec t and y = 1 + 2 tan t; sec t = 1x + 122

-

1y - 122

y - 1 x + 1 and tan t = ; sec2 t - tan2 t = 1 yields the equation 3 2

= 1. 9 4 3. Example 1: x = 3/cos t, y = 2 tan t; Example 2: x = 2 tan t, y = 15/cos t; Example 3: x = 3 + 5/cos t, y = - 1 + 4 tan t; Example 4: x = - 2 + 3/cos t, y = 5 + 7 tan t 5. Example 1: x = 3/cos t = 3 sec t, y = 2 tan t; sec t = x/3, tan t = y/2; sec2 t - tan2 t = 1 yields x 2/9 - y 2/4 = 1, or 4x 2 - 9y 2 = 36. Example 2: x = 2 tan t, y = 15/cos t = 15 sec t; tan t = x/2, sec t = y/15; sec2 t - tan2 t = 1 yields y 2/5 - x 2/4 = 1. Example 3: x = 3 + 5/cos t = 3 + 5 sec t, y = - 1 + 4 tan t; sec t = 1x - 32/5, tan t = 1y + 12/4; sec2 t - tan2 t = 1 yields 1x - 322/25 - 1y + 122/16 = 1. Example 4: x = - 2 + 3/cos t = - 2 + 3 sec t, y = 5 + 7 tan t; sec t = 1x + 22/3, tan t = 1y - 52/7;

sec2 t - tan2 t = 1 yields 1x + 222/9 - 1y - 522/49 = 1.

Quick Review 8.3 1. 1146

4 3. y = " 2x 2 + 9 3

5. No solution

7. x = 2, x = - 2

9. a = 3, c = 5

Exercises 8.3 1. Vertices: 1" 4, 02; foci: 1 " 123, 02 11.

13.

y

15

20

3. Vertices: 10, "62; foci: 10, "72

5. Vertices: 1"2, 02; foci: 1 " 17, 02

15.

y

x

20

9. (a)

17.

y

15

7. (c)

4

x

3

x [–9.4, 9.4] by [–6.2, 6.2]

19.

[–9.4, 9.4] by [–3.2, 9.2]

[–9.4, 9.4] by [–6.2, 6.2]

35.

23. x 2/4 - y 2/5 = 1

21.

1x + 122 4

-

1y - 222 5

= 1

37.

1y - 622 25

-

1x + 322 75

= 1

27.

y2 x2 = 1 25 75

31.

1y - 122 4

-

25. y 2/16 - x 2/209 = 1 29.

1x - 222 9

y2 x2 = 1 144 25 = 1

33.

1x - 222 9

-

1y - 322 16

= 1

6965_SE_Ans_833-934.qxd

910

6/10/10

5:06 PM

Page 910

SELECTED ANSWERS

39. Center: 1 -1, 22; vertices: 111, 22, 1 -13, 22; foci: 112, 22, 1- 14, 22 43. 45.

41. Center: 12, -32; vertices: 12, 52, 12, -112; foci: 12, -3 ! 11452 47.

[–12.4, 6.4] by [–0.2, 12.2]

[–14.1, 14.1] by [–9.3, 9.3]

[–9.4, 9.4] by [–5.2, 7.2]

Vertices: 13, -22, 13, 42;

y2 5y 2 x2 x2 113 53. 2 - 2 = 1 = 1 foci: 13, 1 ! 1132; e = 4 16 3 a b 55. a = 1440, b = 600, c = 1560, e = 13/12; The Sun is centered at 11560, 02. 57. A bearing and distance of about 40.29° and 1371.11 miles, respectively

49.

51.

[–9.4, 9.4] by [–6.2, 6.2]

Vertices: 10, 12, 14, 12; foci: 12 ! 113, 12; e = 59. 1 -2, 02, 14, 3132

113 2

61. (a)

Four solutions: 1!2.13, !1.812

29 21 , ! 10 b A 641 A 641

(b) a ! 10

[–9.4, 9.4] by [–6.2, 6.2] [–9.4, 9.4] by [–4.2, 8.2]

63. True, because c - a = ae - a. 69. (a–d) y

65. B

67. B

(e) x 2/9 - y 2/16 = 1

5

73. Answers will vary. 75. Answers will vary.

x

SECTION 8.4 Quick Review 8.4 1. cos 2a = 5/13

3. cos 2a = 1/2

Exercises 8.4 1. y = - 5 ! 2 -x 2 + 6x + 7

[–6.4, 12.4] by [–11.2, 1.2]

5. a = p/4

7. cos a = 2/ 15

3. y = 4 ! 212x + 2

[–19.8, 17.8] by [–8.4, 16.4]

9. sin a = 1/112

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 911

SELECTED ANSWERS

7. y = 8/1x - 12

5. y = 4/x

[–10, 12] by [–12, 12]

[–9.4, 9.4] by [–6.2, 6.2]

11. y =

9. y =

1 1x - 4 " 2-23x 2 + 28x + 882 6

[– 4.7, 4.7] by [–3.1, 3.1]

y2 x2 = 1 9 16 19. 1x¿, y¿2 = 15, -3 - 152

1 1x - 1 " 231- x 2 + 6x + 922 4

13. x 2 = - 4y

15.

17. 1x¿, y¿2 = 14, -12

[–2, 8] by [–3, 3]

21. Hyperbola:

1y - 122 9

-

y!

1x + 122 4

= 1;

1y¿22 9

-

1x¿22 4

= 1

23. Parabola: 1x + 122 = y - 2; 1x¿22 = y¿ y!

8

25

5

x!

5

25. Ellipse:

1y + 222 9

y!

+

1x - 122 4

= 1;

1y¿22 9

+

1x¿22 4

= 1

27. Parabola: 1y - 222 = 81x - 22; 1y¿22 = 8x¿ y!

8

4

12

29. Hyperbola:

x!

x!

8

x!

1x + 122 1y¿22 1x¿22 y2 = 1; = 1 4 2 4 2 y!

8

8

x!

31. The horizontal distance from O to P is x = h + x¿ = x¿ + h, and the vertical distance is y = k + y¿ = y¿ + k. 33. 13 12/2, 7 12/2) 35. L 1-5.94, 2.382

911

6965_SE_Ans_833-934.qxd

912

1/25/10

3:41 PM

Page 912

SELECTED ANSWERS

37. Hyperbola:

1x¿22 16

-

1y¿22 16

= 1

39. Ellipse:

1y¿22 20

+

1x¿22 4

41. Ellipse: y =

= 1

[–9.4, 9.4] by [–6.2, 6.2]

[–9.4, 9.4] by [–6.2, 6.2]

[–9.4, 9.4] by [–6.2, 6.2]

10x " 2290 - 11x 2 9

a L 0.954 L 54.65° 43. - 24 6 0; ellipse

45. 0; parabola

47. -48 6 0; ellipse

53. In the “old” coordinate system, the center is 10, 02, the vertices are a 13, 32 and 1 - 3, - 32.

63. (a) y = ; x

55. Answers vary.

49. 12 7 0; hyperbola

312 3 12 312 3 12 , b and a ,b , and the foci are 2 2 2 2

57. True, because there is no xy-term.

(b) y = 2x + 3/2, y = 1 -1/22x + 21/2

69. Answers vary. Intersecting lines:

[–4.7, 4.7] by [–3.1, 3.1]

51. -12 6 0; ellipse

59. B

61. A

67. Answers will vary. Parallel lines:

[– 4.7, 4.7] by [–3.1, 3.1]

A plane containing the axis of a cone intersects the cone.

A degenerate cone is created by a generator that is parallel to the axis, producing a cylinder. A plane parallel to a generator of the cylinder intersects the cylinder and its interior.

One line:

No graph:

[–4.7, 4.7] by [–3.1, 3.1]

[– 4.7, 4.7] by [–3.1, 3.1]

A plane containing a generator of a cone intersects the cone.

A plane parallel to a generator of a cylinder fails to intersect the cylinder. Also, a degenerate cone is created by a generator that is perpendicular to the axis, producing a plane. A second plane perpendicular to the axis of this degenerate cone fails to intersect it.

Circle:

Point:

[–4.7, 4.7] by [–3.1, 3.1]

A plane perpendicular to the axis of a cone intersects the cone but not its vertex.

[– 4.7, 4.7] by [–3.1, 3.1]

A plane perpendicular to the axis of a cone intersects the vertex of the cone.

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 913

SELECTED ANSWERS

SECTION 8.5 Exploration 1 e = 0.7, e = 0.8: an ellipse; e = 1: a parabola; e = 1.5, e = 3: a hyperbola The graphs have a common focus, 10, 02, and a common directrix, the line x = 3. As e increases, the graphs move away from the focus and toward the directrix.

[–12, 24] by [–12, 12]

Quick Review 8.5 5p 7p ,u = 6 6 7. Foci: 1; 15, 02; vertices: 1 ;3, 02 1. r = - 3

3. u =

Exercises 8.5 1. r =

5. The focus is 10, 42, and the directrix is y = - 4.

9. Foci: 1; 5, 02; vertices: 1; 4, 02

2 ; Parabola 1 - cos u

3. r =

[–10, 20] by [–10, 10]

12 ; Ellipse 5 + 3 sin u

5. r =

7 ; Hyperbola 3 - 7 sin u

[–5, 5] by [– 4, 2]

[–7.5, 7.5] by [–7, 3]

7. e = 1, parabola; directrix: x = 2

9. e = 1, parabola; directrix: y = -

5 = - 2.5 2

2 5 , ellipse; directrix: y = 4 13. e = = 0.4, ellipse; directrix: x = 3 6 5 15. (b); 3 -15, 54 by 3 - 10, 104 17. (f); 3-5, 54 by 3-3, 34 19. (c); 3-10, 104 by 3- 5, 104 11. e =

21. r =

12 5 + 3 cos u

23. r =

3 2 + sin u

25. r =

15 2 + 3 cos u

27. r =

33.

31.

91y - 4/322

29. r =

[–13, 14] by [–13, 5]

[–6, 14] by [–7, 6]

e = 0.4, a = 5, b = 121, c = 2

12 2 + 3 sin u 35.

e =

6 5 + 3 cos u

[–3, 12] by [–5, 5]

1 , a = 8, b = 413, c = 4 2

e =

5 , a = 3, b = 4, c = 5 3

3x 2 = 1 39. y 2 = 41x + 12 64 16 41. Perihelion distance L 0.54 AU; aphelion distance L 35.64 AU 43. (a) v L 1551 m/sec = 1.551 km/sec 45. True. For a circle, e = 0, so the equation is r = 0, which graphs as a point. 47. D 49. B 37.

+

Perihelion Distance (AU) 51. (c) Planet Mercury 0.307 Venus 0.718 Earth 0.983 Mars 1.382 Jupiter 4.953 Saturn 9.020

Aphelion Distance (AU) 0.467 0.728 1.017 1.665 5.452 10.090

(b) About 2 hr 14 min

(d) The difference is greatest for Saturn.

55. 5r - 3r cos u = 16 Q 5r = 3x + 16. So, 25r 2 = 251x 2 + y 22 = 13x + 1622. 25x 2 + 25y 2 = 9x 2 + 96x + 256 Q 16x 2 - 96x + 25y 2 = 256. Completing the square yields

1x - 322 25

+

y2 = 1, the desired result. 16

913

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 914

SELECTED ANSWERS

914

SECTION 8.6 Quick Review 8.6 1. 21x - 222 + 1y + 322 5. h

-4 5 , i 141 141

3. P lies on the circle of radius 5 centered at 12, -32.

7. Circle of radius 5 centered at 1- 1, 52

Exercises 8.6 1.

9. Center: 1 -1, 32; radius: 2

3.

z

5. 7. 9. 11.

z

8

(3, 4, 2)

8

13. 1x - 522 + 1y + 122 + 1z + 222 = 64 15. 1x - 122 + 1y + 322 + 1z - 222 = a

y

y

x

153 21a - 122 + 1b + 322 + 1c - 222 11, - 1, 11/22 1x - 1, y + 4, z + 32

(1, –2, –4)

8 x

17.

19.

z

21.

z

(0, 0, 3)

5 (0, 0, 3)

(0, 9, 0) y 10

5

y

(0, –2, 0)

(3, 0, 0)

10 x

23. 8- 2, 4, - 89 25. - 84 27. - 20

z

6

5

6 x

(9, 0, 0)

5

y

(6, 0, 0)

10 x

3 12 4 31. 8 -3, 4, - 59 33. v = - 195.01i - 7.07j + 68.40k ,- , i 13 13 13 35. r = 82, - 1, 59 + t83, 2, -79; x = 2 + 3t, y = - 1 + 2t, z = 5 - 7t 37. r = 86, -9, 09 + t81, 0, - 49; x = 6 + t, y = - 9, z = - 4t 1 11 39. 130 41. r = 8 -1, 2, 49 + t81, 4, - 79 43. x = - 1 + 3t, y = 2 - 6t, z = 4 - 3t 45. x = t, y = 6 - 7t, z = - 3 + t 2 2 47. Scalene 29. h

49. (a)

(b) The z-axis; a line through the origin in the direction k

z 5

5 5 x

51. (a)

z 5

y

5 5 x

y

(b) The intersection of the xzplane 1y = 02 and the plane x = - 3; a line parallel to the z-axis through 1 - 3, 0, 02

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 915

SELECTED ANSWERS

53. 57. 59. 67.

915

r = 8x 1 + 1x 2 - x 12t, y1 + 1y2 - y12t, z 1 + 1z 2 - z 12t9 55. Answers vary. True. The equation can be viewed as an equation in three variables, where the coefficient of z is zero. The surface is an elliptical cylinder. B 61. C 65. 8 - 1, -5, - 39 i * j = 81, 0, 09 * 80, 1, 09 = 80 - 0, 0 - 0, 1 - 09 = 80, 0, 19 = k

CHAPTER 8 REVIEW EXERCISES 1.

y

3.

y

5.

10

7 x

8 x

9

7.

Vertex: 1-2, 12; focus: 1 -2, 02; directrix: y = 2; focal width: 4 9.

y

11.

y

6

x

Hyperbola; center: 10, 02; vertices: 1" 5, 02; foci: 1 " 161, 02 15. (h)

17. (f)

19. (c)

Ellipse; center: 12, - 12; vertices: 16, -12, 1-2, - 12; foci: 15, - 12, 1 - 1, -12 25.

y 7

10

6

Hyperbola;

1x - 122 3

-

10

x

1y - 222

29. See proof on pages 582–583.

y

y 11

x

Parabola; 1x - 322 = y + 12

x

x

Hyperbola; center: 1 -3, 52; vertices: 1- 3 " 312, 52; foci: 1 - 3 " 146, 52 23.

y 40

27.

y

4

10

21.

Ellipse; center: 10, 02; vertices: 10, "2 122; foci: 10, " 132

20

10

10

x

7

Vertex: 10, 02; focus: 13, 02; directrix: x = - 3; focal width: 12

13. (b)

y

2

3

= 1 31.

x

Parabola; 1y - 222 = 6ax +

17 b 6

10

–10

15

x [0, 25] by [0, 17]

–15

Hyperbola;

1y + 422 30

-

1x - 322 45

= 1

Ellipse; 1 y = 38x + 5 " 2 - 8x 2 + 200x - 4554 12

6965_SE_Ans_833-934.qxd

916

1/25/10

3:41 PM

Page 916

SELECTED ANSWERS

33.

35.

[–8, 12] by [–5, 15]

Hyperbola; y = 37. y 2 = 8x 47.

1x - 222 9

-

[–24, 20] by [–20, 15]

3x 2 - 5x - 10 2x - 6

Hyperbola; y =

39. 1x + 322 = 121y - 32 1y - 122 16

= 1

55.

41.

y2 x2 + = 1 169 25

49. x 2/25 + y 2/4 = 1 57.

[–8, 3] by [–10, 10]

1 37x + 20 " 225x 2 + 272x + 2804 4

43. x 2/9 + 1y - 222/5 = 1

51. 1x + 222 + 1y - 422 = 1

Ellipse; 61.

41x - 1/22 9 63. 65. 67. 69.

y2 x2 = 1 25 11

53. x 2/9 - y 2/25 = 1

59.

[–3, 3] by [–2, 2]

Parabola; y 2 = - 81x - 22

45.

y2 + = 1 2

[–8, 8] by [–11, 0]

Hyperbola;

811y + 49/922 196

-

9x 2 = 1 245

169 80, - 3, - 29 - 13 83/5, - 4/5, 09

71. 1x + 122 + y 2 + 1z - 322 = 16

[–20, 4] by [–8, 8]

Parabola; y 2 = - 41x - 12 73. r = 8-1, 0, 39 + t8 -3, 1, - 29 75. 10, 4.52 79. At apogee, v L 2633 m/sec; at perigee, v L 9800 m/sec

Chapter 8 Project Answers are based on the sample data provided. 1. 3. With respect to the graph of the ellipse, the point 1h, k2 represents the center of the ellipse. The value a is the semimajor axis, and b is the semiminor axis.

[0.4, 0.75] by [–0.7, 0.7]

5. The parametric equations for the sample data set are x 1T L 0.131 sin 14.80T + 2.102 + 0.569 and y1T L 0.639 sin 14.80T - 2.652.

[–0.1, 1.4] by [–1, 1]

[0.4, 0.75] by [–0.7, 0.7]

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 917

SELECTED ANSWERS

SECTION 9.1 Exploration 1 1. 6

3. No

Quick Review 9.1 1. 52

3. 6

5. 10

7. 11

9. 64

Exercises 9.1 1. 6 3. 120 19. Combinations

5. 12 7. 362,880 (ALGORITHM) 21. Combinations 23. 19,656,000

9. 34,650 25. 36

11. 1716 27. 2300

13. 24 15. 30 17. 120 29. 17,296 31. 37,353,738,800

n! 45. D 47. B . a! b! 51. (a) 12 (b) There are 12 factors of 5 in 50!, one in each of 5, 10, 15, 20, 30, 35, 40, and 45 and two in each of 25 and 50. Each factor of 5, when paired with one of the 47 factors of 2, yields a factor of 10 and consequently a 0 at the end of 50! 55. 3 57. L 20,123 years 33. 41

35. 7776

37. 511

39. 12

41. 1024

43. True. Both equal

SECTION 9.2 Exploration 1 1. 1, 3, 3, 1; These are (in order) the coefficients in the expansion of 1a + b23. 3. 51 5 10 10 5 16; These are (in order) the coefficients in the expansion of 1a + b25.

Quick Review 9.2 1. x 2 + 2xy + y 2

3. 25x 2 - 10xy + y 2

7. u 3 + 3u 2v + 3uv 2 + v 3

5. 9s 2 + 12st + 4t 2

9. 8x 3 - 36x 2y + 54xy 2 - 27y 3

Exercises 9.2 1. a 4 + 4a 3b + 6a 2b 2 + 4ab 3 + b 4 5. x 3 + 3x 2y + 3xy 2 + y 3 11. 1

13. 364

3. x 7 + 7x 6y + 21x 5y 2 + 35x 4y 3 + 35x 3y 4 + 21x 2y 5 + 7xy 6 + y 7

7. p 8 + 8p 7q + 28p 6q 2 + 56p 5q 3 + 70p 4q 4 + 56p 3q 5 + 28p 2q 6 + 8pq 7 + q 8 5

15. 126,720

4

3

17. ƒ1x2 = x - 10x + 40x - 80x + 80x - 32

19. h1x2 = 128x 7 - 448x 6 + 672x 5 - 560x 4 + 280x 3 - 84x 2 + 14x - 1 23. 35. 37. 41.

9. 36

2

21. 16x 4 + 32x 3y + 24x 2y 2 + 8xy 3 + y 4

x - 6x y + 15x y - 20x y + 15xy - 6x y + y + 15x -8 + 90x -6 + 270x -4 + 405x -2 + 243 25. x True. The signs of the coefficients are determined by the powers of the 1 -y2. C 39. A (a) 1, 3, 6, 10, 15, 21, 28, 36, 45, 55 (b) They appear diagonally down the triangle, starting with either of the 1’s in row 2. (c) (d) 3

5/2 1/2

2

3/2 3/2

2

1/2 5/2

3

-10

n n n n n n n n 43. 2n = 11 + 12n = a b 1n10 + a b 1n - 111 + a b 1n - 212 + Á + a b101n = a b + a b + a b + Á + a b 0 1 2 n 0 1 2 n

SECTION 9.3 Exploration 1 1.

Antibodies present

3.

p = 0.997 Antibodies present

p = 0.006

p = 0.006 p = 0.994

Antibodies absent

+ p = 0.00598

p = 0.994

p = 0.003

– p = 0.00002

p = 0.015

+ p = 0.01491

p = 0.985

– p = 0.97909

Antibodies absent

5. L 0.286

917

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 918

SELECTED ANSWERS

918

Quick Review 9.3 1. 2

3. 8

5. 2,598,960

7. 120

9.

1 12

Exercises 9.3 1. 1/9 3. 5/12 5. 1/4 7. 5/12 9. (a) No; the numbers do not add up to 1. (b) Yes; assuming the gerbil cannot be in more than one compartment at a time, the proportions cannot sum to more than 1. 11. 0.4 13. 0.2 15. 0.7 17. 0.09 19. 0.08 21. 0.64 23. 1/134,596 25. 5/3542 27. (a) (b) 0.3 (c) 0.2 (d) 0.2 (e) yes 0.3

0.2

0.3

A

B 0.2

29. 41. 51. 57.

0.64 31. 3/5 33. 19/30 35. (a) 0.67 (b) 0.33 39. (a) 86/127 (b) 91/127 1/36 43. 1/1024 45. 1/1024 47. 45/1024 49. 1023/1024 False. A sample space consists of outcomes, which are not necessarily equally likely. 53. D (a) Type of Bagel (b) L 0.051 Probability Plain

0.37

Onion

0.12

Rye

0.11

Cinnamon Raisin

0.25

Sourdough

0.15

(c) 62/127 55. A

59. (a) L 2 (b) Yes (c) L 1.913% 61. (a) $1.50 (b) 1/3

SECTION 9.4 Quick Review 9.4 1. 19

3. 80

5. 10/11

7. 2560

9. 15

Exercises 9.4 3 4 5 6 7 101 1. 2, , , , , ; 3. 0, 6, 24, 60, 120, 210; 999,900 5. 8, 4, 0, -4; -20 7. 2, 6, 18, 54; 4374 2 3 4 5 6 100 11. Diverges 13. Converges to 0 15. Converges to -1 17. Converges to 0 19. Diverges 21. (a) 4 (b) 42 (c) a1 = 6 and an = an - 1 + 4 for n Ú 2 (d) an = 6 + 41n - 12 23. (a) 3

(b) 22

25. (a) 3

(b) 4374

(c) a1 = - 5 and an = an - 1 + 3 for n Ú 2 (c) a1 = 2 and an = 3an - 1 for n Ú 2

9. 2, - 1, 1, 0; 3

(d) an = - 5 + 31n - 12

(d) an = 2 # 3n - 1

(c) a1 = 1 and an = - 2an - 1 for n Ú 2 (d) an = 1-22n - 1 3 29. a1 = - 20; an = an - 1 + 4 for n Ú 2 31. a1 = " , r = "2, and an = 31" 22n - 2 2 33. 35. 27. (a) -2

(b) - 128

[0, 5] by [–2, 5]

[0, 10] by [–10, 100]

37. 700, 702.3, 704.6, 706.9, Á , 815, 817.3 39. 775 41. 9 43. True. The common ratio r must be positive, so the sign of the first term determines the sign of every number in the sequence. 45. A 47. E 49. (b) 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 51.(b) an : 2p as n : q 55. a1 = 31 14, a2 = 31 24, a3 = 32 34, a4 = 33 54, a5 = 35 84, a6 = 38 134, a7 = 313 214. The entries in the terms of this sequence are successive pairs of terms from the Fibonacci sequence.

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 919

SELECTED ANSWERS

919

SECTION 9.5 Exploration 1 1. 45

3. 1

5.

1 3

Exploration 2 1. 1 + 2 + 3 + Á + 99 + 100 3. 101 5. The sum in 4 involves two copies of the same progression, so it doubles the sum of the progression. The answer is 5050.

Quick Review 9.5 1. 22

3. 27

Exercises 9.5 1. a 16k - 132 11

k=1

17. 155

19.

5. 512

7. -40

3. a k 2 n+1 k=1

5. a 61 -22k q

k=0

8 11 - 2-122 L 2.666 3

(b) 1, -1, 2, -2, 3, - 3; divergent

9. 55

7. 18

9. 3240

21. - 196,495,641

25. Yes; 12

(b) 20,00011.12 (c) $370,623.34 37. (a) 120; 1 + 0.07/12 41. False. The series might well diverge.

13. 24,573

15. 50.411 - 6-92 L 50.4

23. (a) 0.3, 0.33, 0.333, 0.3333, 0.33333, 0.333333; convergent

27. No

n

11. 975

29. Yes; 1

31. 707/99

(b) $20,770.18

33. -

17,251 999

35. (a) 1.1

39. L 24.05 m

diverge, but a 1n + 1 -n22 is constant and converges to 0. 8

n=1

43. A 45 C 47. (a) Heartland: 19,237,759 people; Southeast: 42,614,977 people (b) Heartland: 517,825 mi2; Southeast: 348,999 mi2 (c) Heartland: L 37.15 people/mi2; Southeast: L 122.11 people/mi2 49. n Fn Sn Fn + 2 - 1 1 2 3 4 5 6 7 8 9

1 1 2 3 5 8 13 21 34

1 2 4 7 12 20 33 54 88

1 2 4 7 12 20 33 54 88

Conjecture: Sn = Fn + 2 - 1

SECTION 9.6 Exploration 1 Start with the rightmost peg if n is odd and the middle peg if n is even.

Exploration 2 1. Yes

3. Still all prime

Quick Review 9.6 1. n 2 + 5n

3. k 3 + 3k 2 + 2k

Exercises 9.6

5. 1k + 123

7. 5; t + 4; t + 5

2k 2k + 2 1 9. ; ; 2 3k + 1 3k + 4

1. Pn: 2 + 4 + 6 + Á + 2n = n 2 + n. P1 is true: 2112 = 12 + 1. Now assume Pk is true: 2 + 4 + 6 + Á + 2k = k 2 + k. Add 21k + 12 to both sides: 2 + 4 + 6 + Á + 2k + 21k + 12 = k 2 + k + 21k + 12 = k 2 + 3k + 2 = k 2 + 2k + 1 + k + 1 = 1k + 122 + 1k + 12, so Pk + 1 is true. Therefore, Pn is true for all n Ú 1. 3. Pn: 6 + 10 + 14 + Á + 14n + 22 = n12n + 42. P1 is true: 4112 + 2 = 112112 + 42. Now assume Pk is true: 6 + 10 + 14 + Á + 14k + 22 = k12k + 42. Add 41k + 12 + 2 = 4k + 6 to both sides: 6 + 10 + 14 + Á + 14k + 22 + 341k + 12 + 24 = k12k + 42 + 4k + 6 = 2k 2 + 8k + 6 = 1k + 1212k + 62 = 1k + 12321k + 12 + 44, so Pk + 1 is true. Therefore, Pn is true for all n Ú 1.

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 920

SELECTED ANSWERS

920

5. Pn: an = 5n - 2. P1 is true: a1 = 5 # 1 - 2 = 3. Now assume Pk is true: ak = 5k - 2. To get ak + 1, add 5 to ak; that is, ak + 1 = 15k - 22 + 5 = 51k + 12 - 2. This shows that Pk + 1 is true. Therefore, Pn is true for all n Ú 1.

7. Pn: an = 2 # 3n - 1. P1 is true: a1 = 2 # 31 - 1 = 2 # 30 = 2. Now assume Pk is true: ak = 2 # 3k - 1. To get ak + 1, multiply ak by 3; that is, ak + 1 = 3 # 2 # 3k - 1 = 2 # 3k = 2 # 31k + 12 - 1. This shows that Pk + 1 is true. Therefore, Pn is true for all n Ú 1. 111 + 12 k1k + 12 1k + 121k + 22 9. P1: 1 = . Pk: 1 + 2 + Á + k = . Pk + 1: 1 + 2 + Á + k + 1k + 12 = . 2 2 2 1 1 1 1 1 k 11. P1: # = . P: + # + Á + = . 1 2 1 + 1 k 1#2 2 3 k1k + 12 k + 1 1 1 1 1 k + 1 . Pk + 1: # + # + Á + + = 1 2 2 3 k1k + 12 1k + 121k + 22 k + 2 13. Pn: 1 + 5 + 9 + Á + 14n - 32 = n12n - 12. P1 is true: 4112 - 3 = 1 # 12 # 1 - 12. Now assume Pk is true: 1 + 5 + 9 + Á + 14k - 32 = k12k - 12. Add 41k + 12 - 3 = 4k + 1 to both sides: 1 + 5 + 9 + Á + 14k - 32 + 341k + 12 - 34 = k12k - 12 + 4k + 1 = 2k 2 + 3k + 1 = 1k + 1212k + 12 = 1k + 12321k + 12 - 14, so Pk + 1 is true. Therefore, Pn is true for all n Ú 1. 1 1 1 n 1 1 15. Pn: # + # + Á + . = . P is true: # = 1 2 2 3 n1n + 12 n + 1 1 1 2 1 + 1 1 1 k 1 Now assume Pk is true: # + # + Á + . = 1 2 2 3 k1k + 12 k + 1 1 1 1 1 1 k 1 Add to both sides: # + # + Á + + = + 1k + 121k + 22 1 2 2 3 k1k + 12 1k + 121k + 22 k + 1 1k + 121k + 22 k1k + 22 + 1 1k + 121k + 12 k + 1 k + 1 = = = = , so Pk + 1 is true. Therefore, Pn is true for all n Ú 1. 1k + 121k + 22 1k + 121k + 22 k + 2 1k + 12 + 1 17. Pn: 2n Ú 2n. P1 is true: 21 Ú 2 # 1 (in fact, they are equal). Now assume Pk is true: 2k Ú 2k.

Then 2k + 1 = 2 # 2k Ú 2 # 2k = 2 # 1k + k2 Ú 21k + 12, so Pk + 1 is true. Therefore, Pn is true for all n Ú 1.

19. Pn: 3 is a factor of n 3 + 2n. P1 is true: 3 is a factor of 13 + 2 # 1 = 3. Now assume Pk is true: 3 is a factor of k 3 + 2k. Then 1k + 123 + 21k + 12 = 1k 3 + 3k 2 + 3k + 12 + 12k + 22 = 1k 3 + 2k2 + 31k 2 + k + 12. Since 3 is a factor of both terms, it is a factor of the sum, so Pk + 1 is true. Therefore, Pn is true for all n Ú 1. a111 - r n2 21. Pn: The sum of the first n terms of a geometric sequence with first term a1 and common ratio r Z 1 is . 1 - r 1 k a111 - r 2 a111 - r 2 P1 is true: a1 = . Now assume Pk is true so that a1 + a1r + Á + a1r k - 1 = . 1 - r 11 - r2 a111 - r k2 a111 - r k2 + a1r k11 - r2 Add a1r k to both sides: a1 + a1r + Á + a1r k - 1 + a1r k = + a1r k = 11 - r2 1 - r a1 - a1r k + a1r k - a1r k + 1 a1 - a1r k + 1 = = , so Pk + 1 is true. Therefore, Pn is true for all positive integers n. 1 - r 1 - r 23. Pn: a k = n

n1n + 12

k=1

2

. P1 is true: a k = 1 = 1

k=1

Add 1k + 12 to both sides, and we have a i = =

1k + 1211k + 12 + 12

25. 125,250

2

27.

k k1k + 12 1#2 . Now assume Pk is true: a i = . 2 2 i=1

k+1

k1k + 12

i=1

2

+ 1k + 12 =

k1k + 12 2

+

21k + 12 2

=

1k + 121k + 22 2

, so Pk + 1 is true. Therefore, Pn is true for all n Ú 1.

1n - 321n + 42

29. L 3.44 * 10 10

31.

n1n 2 - 3n + 82

33.

n1n - 121n 2 + 3n + 42

2 3 4 35. The inductive step does not work for 2 people. Sending them alternately out of the room leaves 1 person (and one blood type) each time, but we cannot conclude that their blood types will match each other. 37. False. Mathematical induction is used to show that a statement Pn is true for all positive integers. 39. E 41. B 43. Pn: 2 is a factor of 1n + 121n + 22. P1 is true because 2 is a factor of 122132. Now assume Pk is true so that 2 is a factor of 1k + 121k + 22. Then 31k + 12 + 1431k + 12 + 24 = 1k + 221k + 32 = k 2 + 5k + 6 = k 2 + 3k + 2 + 2k + 4 = 1k + 121k + 22 + 21k + 22. Since 2 is a factor of both terms of this sum, it is a factor of the sum, and so Pk + 1 is true. Therefore, Pn is true for all positive integers n. 45. Given any two consecutive integers, one of them must be even. Therefore, their product is even. Since n + 1 and n + 2 are consecutive integers, their product is even. Therefore, 2 is a factor of 1n + 121n + 22.

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 921

SELECTED ANSWERS

921

47. Pn: Fn + 2 - 1 = a Fk. P1 is true since F1 + 2 - 1 = F3 - 1 = 2 - 1 = 1, which equals a Fk = 1. 1

n

k=1

k=1

Now assume that Pk is true: Fk + 2 - 1 = a Fi. Then F1k + 12 + 2 - 1 = Fk + 3 - 1 = Fk + 1 + Fk + 2 - 1 k

i=1

= 1Fk + 2 - 12 + Fk + 1 = a a Fi b + Fk + 1 = a Fi, so Pk + 1 is true. Therefore, Pn is true for all n Ú 1. k

k+1

i=1

i=1

49. Pn: a - 1 is a factor of a n - 1. P1 is true because a - 1 is a factor of a - 1. Now assume Pk is true so that a - 1 is a factor of a k - 1. Then a k + 1 - 1 = a # a k - 1 = a1a k - 12 + 1a - 12. Since a - 1 is a factor of both terms in the sum, it is a factor of the sum, and so Pk + 1 is true. Therefore, Pn is true for all positive integers n. 51. Pn: 3n - 4 Ú n for n Ú 2. P2 is true since 3 # 2 - 4 Ú 2. Now assume that Pk is true: 3k - 4 Ú k. Then 31k + 1) - 4 = 3k + 3 - 4 = 13k - 42 + 3 Ú k + 3 Ú k + 1, so Pk + 1 is true. Therefore, Pn is true for all n Ú 2. 53. Use P3 as the anchor and obtain the inductive step by representing any n-gon as the union of a triangle and an 1n - 12-gon.

SECTION 9.7 Exploration 1

1. The average is about 12.8.

3. Alaska, Colorado, Georgia, Texas, and Utah

Quick Review 9.7 1. L 15.48%

3. L 14.44%

5. L 1723

7. $235 thousand

9. 1 million

Exercises 9.7 1.

0 5 8 9 1 3 4 6 2 3 6 8 3 3 9 4 5 6 1 61 is an outlier.

7.

3. 6 6 7 7

Males 3 0 7 8 1 2

Life expectancy (years)

Frequency

60.0–64.9

2

65.0–69.9

4

70.0–74.9

6

13.

5. 8 2

8 3

3

3

Males

3

9.

19.

[1965, 2008] by [–1000, 11000]

7

7

9

9

11.

[50, 80] by [–1, 9]

[0, 60] by [–1, 5]

17.

15.

[–1, 25] by [–5, 60]

3

Females

3 0 6 8 8 8 7 6 5 8 3 2 2 1 7 1 2 7 5 6 8 0 0

[1965, 2008] by [–1000, 11000]

[–1.5, 17] by [–2, 80]

After approaching parity in 1985, the top PGA player’s earnings have grown much faster than the top LPGA player’s earnings, even if the unusually good years for Tiger Woods (1999 and 2000) are not considered part of the trend.

21.

The top male’s earnings appear to be growing exponentially, with unusually high earnings in 1999 and 2000. Since the graph only shows the earnings of the top player (as opposed to a mean or median for all players), it can behave strangely if the top player has a very good year—as Tiger Woods did in 1999 and 2000.

The two home run hitters enjoyed similar success.

[–1, 25] by [–5, 60]

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 922

SELECTED ANSWERS

922

23. (a) 28 2 29 3 7 30 31 6 7 32 7 8 33 5 5 5 8 34 2 8 8 35 3 3 4 36 3 7 37 38 5

(b)

Interval

Frequency

25.0–29.9 30.0–34.9 35.0–39.9

25.

(c)

(d) Time is not a variable in the data.

3 11 6 [20, 45] by [–1, 13]

27. False. The empty branches are important for visualizing the distribution of the data.

29. C

31. A

35.

[0, 13] by [–15, 40]

[1890, 2010] by [–4, 40]

= CA; + = NY, . = TX

SECTION 9.8 Exploration 1 1. Figure (b)

Quick Review 9.8 1. x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 7. a x iƒi 8

i=1

9.

Exercises 9.8

1 50 1x i - x 22 50 ia =1

3.

1 1x + x 2 + x 3 + x 4 + x 5 + x 6 + x 72 7 1

5.

1 31x 1 - x22 + 1x 2 - x22 + Á + 1x 5 - x224 5

1. (a) statistic (b) parameter (c) statistic 3. 26.8 5. L 60.12 7. 3.9 million 9. L 15.2 satellites 11. 2 13. 30 runs/yr; L 29.8 runs/yr; Mays 15. What-Next Fashion 17. median: 87.85; mode: None 19. L 3.61 21. (a) L 6.42°C (b) L 6.49°C (c) The weighted average is the better indicator. 23. Willie Mays: Five-number summary: {4, 20, 31.5, 40, 52}; Range: 48; IQR: 20; No outliers; Mickey Mantle: Five-number summary: { 13, 21, 28.5, 37, 54}; Range: 41; IQR: 16; No outliers 27. (a) (b) 25. 528.2, 31.7, 33.5, 35.3, 38.56; 10.3; 3.6; No outliers

[–3, 80] by [–1, 2]

29. 39. 49. 51.

[–3, 80] by [–1, 2]

3/11 31. (a) Mays (b) Mays 33. s L 9.08; s2 = 82.5 35. s L 120.69; s2 L 14,566.59 37. s L 1.53; s2 L 2.34 No 41. (a) 68% (b) 2.5% (c) A parameter 43. False. The median is a resistant measure. 45. A 47. B There are many possible answers; examples are given. (a) 52, 2, 2, 3, 6, 8, 206 (b) 51, 2, 3, 4, 6, 48, 486 (c) 5- 20, 1, 1, 1, 2, 3, 4, 5, 66 No 55. 75.9 years 57. 5%

SECTION 9.9 Exploration 1 1. Correlation begins with a scatter plot, which requires numerical data from two quantitative variables (like height and weight). “Gender” is categorical. 3. The doctor’s “experiment” proves nothing about the effect of vanilla gum on headache pain unless we can compare these subjects with a similar group that does not use vanilla gum. Many headaches are gone in two hours anyway.

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 923

SELECTED ANSWERS

923

Quick Review 9.9 1.

1 6

3.

4 1 = 52 13

Exercises 9.9

5.

1 10

7. a

1 5 b = 0.00001 10

9. 1 - a

9 5 b = 0.40951 10

1. Incorrect. Intelligence might be associated with some quantitative variable, but beauty is categorical. 3. Incorrect. The high correlation coefficient does nothing to support Sean’s crazy theory, because the great blue whale (with a long name and a huge weight) is an unusual point that lies far away from the other three. 7. This is random (technically pseudo-random, but it should suffice). 5. Incorrect. Marcus is OK with his first observation, but not with 9. This is not a random sample of all Reno citizens. All fifty are likely his second. While his linear model is a bad fit, he should not to be from early in the alphabet. conclude that “there is no significant mathematical relationship.” 11. This is not random, nor does it apparently try to be. In fact, check out this sinusoidal fit: 13. Voluntary response bias. The students most likely to respond were those who felt strongly about suggestions for improvement, so the rate of negative responses was probably higher than the parameter. He could have gotten a less biased response with an in-class census of all his students (ideally in multiple-choice form so that their handwriting would not betray their identities). 15. Undercoverage bias. The survey systematically excluded the students who were not actually eating in the dining hall, so the sample statistic was bound to be higher than the population parameter. A better method would have been to choose a random sample from the student body first, then seek them out for the survey (perhaps in their homerooms). 17. Response bias. The question was designed to elicit a negative response, and it never even mentioned stop signs. The 97% was much higher than it would have been with a simple question like, “Should citizens be allowed to ignore stop signs?” 19. Observational study. No treatment imposed. 21. Observational study. No treatment imposed. 23. Experiment. 25. Using random numbers, select 12 of the 24 plots to get the new fertilizer. Use the original fertilizer on the other 12 plots. Compare the yields at harvest time. 27. This requires three treatments. Split the 24 plots randomly into three groups of 8: new fertilizer 1, new fertilizer 2, and original fertilizer. 29. Fatigue may be a factor after they have driven 20 golf balls. They could gather the data on different days, or they could randomly choose half the golfers to drive the new ball first. 31. The music assignment should be randomized, not left to the choice of the mother. Otherwise, the mother’s music preference (with possible lifestyle implications) becomes a potentially significant confounding variable. 33. One possible solution: Use the command “randInt 11, 500, 502” to choose 50 random numbers from 1 to 500. If there are any repeat numbers in the list, use “randInt 11, 5002” to pick additional numbers until you have a sample of 50. 35. One possible solution: Enter the numbers 1 to 32 in list L1 using the command “seq1X, X, 1, 322 : L1” and enter 32 random numbers in list L2 using the command “rand1322 : L2.” Then sort the random numbers into ascending order, bringing L1 along for the ride, using the command “SortA1L2, L12.” The numbers in list L1 are now in random order. 37. One possible solution: Use the command “randInt 11, 5, 202” to generate 20 random numbers from 1 to 5. Let 1 and 2 designate donors with O-positive blood. Do this nine times and keep track of how many strings have fewer than four numbers that are 1 or 2. 39. One possible solution: Use the command “randInt 11, 62” to generate random numbers between 1 and 6. Push ENTER twice to get a roll of two dice. (Note that you do not want to generate random totals between 2 and 12. You learned in Section 9.3 that those totals are not equally likely.) 41. False. Observational studies can show strong associations, but experiments would be required to establish causation. 43. B (Note that this is the only quantitative variable among the choices.) 45. C 47. Answers will vary. Note that you should not expect all the counts to be exactly the same (that would suggest nonrandomness in itself), but “randomness” would predict an approximately equal distribution, especially for a large class. 49. (a) Correlation coefficient will increase; slope will remain about the same. (b) Correlation coefficient will increase; slope will increase. (c) Correlation coefficient will decrease; slope will decrease. 53. (a) The size of the hospital is not affecting the death rates of the patients. The lurk51. One possible picture: ing variable is the patient’s condition. Bigger hospitals tend to get the more critical cases, and critical cases have a higher death rate. (b) The number of seats is not affecting the speed of the jet. The lurking variable is the size of the aircraft. Larger jets generally have more seats and go faster. (c) The size of the shoe does not affect reading ability. The lurking variable is the age of the student. In general, older students have larger feet and read at a higher level. (d) The extra firemen are not causing more damage. The lurking variable is the size of the fire. Larger fires cause more damage and require more firefighters. (e) The salary is not generally affected by the player’s weight. The lurking variable is the player’s position on the team. Linemen weigh more and tend to earn less money than the (usually lighter) players in the so-called “skill” positions (e.g., quarterbacks, running backs, receivers, and defensive backs).

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 924

SELECTED ANSWERS

924

CHAPTER 9 REVIEW EXERCISES 1. 792 3. 18,564 5. 3,991,680 7. 43,670,016 9. 14,508,000 11. 8,217,822,536 13. 26 15. 325 17. (a) 5040; Meg Ryan (b) 778,377,600; Britney Spears 19. 32x 5 + 80x 4y + 80x 3y 2 + 40x 2y 3 + 10xy 4 + y 5 21. 243x 10 + 405x 8y 3 + 270x 6y 6 + 90x 4y 9 + 15x 2y 12 + y 15 23. 512a 27 - 2304a 24b 2 + 4608a 21b 4 - 5376a 18b 6 + 4032a 15b 8 - 2016a 12b 10 + 672a 9b 12 - 144a 6b 14 + 18a 3b 16 - b 18 25. -1320 27. 51, 2, 3, 4, 5, 66 29. 513, 16, 31, 36, 61, 636 31. 5HHH, HHT, HTH, HTT, THH, THT, TTH, TTT6 33. 5HHH, TTT6 35. 1/64 37. 1/4 39. 0.25 41. 0.24 43. 0.64 45. (a) 0.5 (b) 0.15 (c) 0.35 (d) L 0.43 47. 0, 1, 2, 3, 4, 5; 39 49. - 1, 2, 5, 8, 11, 14; 32 51. - 5, -3.5, -2, - 0.5, 1, 2.5; 11.5 53. -3, 1, -2, -1, -3, - 4; - 76 55. Arithmetic with d = - 2.5; an = 14.5 - 2.5n 57. Geometric with r = 1.2; an = 10 # 11.22n - 1 59. Arithmetic with d = 4.5; an = 4.5n - 15.5 61. an = 31 -42n - 1; r = - 4 63. -4 65. -985.5 67. 21/8 69. 59,048 71. 3280.4 73. 75. $27,441.91 77. Converges; 6 79. Diverges 81. Converges; 3

83. a 15k - 132 21

k=1

[0, 15] by [0, 2]

85. a 12k + 122 or a 12k - 122

91. Pn:1 + 3 + 6 + Á +

q

q

k=0

k=1

n1n + 12 2

=

n1n + 121n + 22 6

Now assume Pk is true: 1 + 3 + 6 + Á +

n13n + 52 2 111 + 12

. P1 is true:

k1k + 12 2

=

89. 4650 =

2 k1k + 121k + 22

111 + 1211 + 22 6

.

. 6 1k + 121k + 22

k1k + 121k + 22 1k + 121k + 22 + = + 2 2 6 2 1k + 1211k + 12 + 1211k + 12 + 22 k 1 k + 3 , so Pk + 1 is true. = 1k + 121k + 22a + b = 1k + 121k + 22a b = 6 2 6 6 Therefore, Pn is true for all n Ú 1. 93. Pn: 2n - 1 … n!. P1 is true: 21 - 1 … 1! (they are equal). Now assume Pk is true: 2k - 1 … k!. Add

1k + 121k + 22

87.

2

to both sides: 1 + 3 + 6 + Á +

k1k + 12

Then 21k + 12 - 1 = 2 # 2k - 1 … 2 # k! … 1k + 12k! = 1k + 12!, so Pk + 1 is true. Therefore, Pn is true for all n Ú 1.

95. (a)

9 1 2 10 6 7 11 4 5 5 7 7 12 0 2 4 6 7 7 13 5 6 14 1 6 7 7 8 15 4 8 16 1 4 17 0 6 18 19 20 21 9 22 23 4

97. (a) 12 0 0 4 4 13 1 1 2 6 7 9 14 0 3 4 8 15 6 16 3 17 7 9 18 0 19 0 1 7 20 2 21 22 23 0

(b) Price

90,000– 99,999 100,000–109,999 110,000–119,999 120,000–129,999 130,000–139,999 140,000–149,999 150,000–159,999 160,000–169,999 170,000–179,999 210,000–219,999 230,000–239,999

(a) Length (in seconds) 120–129 130–139 140–149 150–159 160–169 170–179 180–189 190–199 200–209 210–219 220–229 230–239

Frequency

(c)

2 2 5 6 2 5 2 2 2 1 1

[8, 24] by [–1, 7]

Frequency 4 6 4 1 1 2 1 3 1 0 0 1

(c)

[120, 240] by [0, 7]

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 925

SELECTED ANSWERS

925

99. Five-number summary: 59.1, 11.7, 13.1, 15.4, 23.46; Range: 14.3; IQR: 3.7; s L 3.19, s2 L 10.14; Outliers: 21.9 and 23.4 101. Five-number summary: 5120, 131.5, 143.5, 179.5, 2306; Range: 110; IQR: 48; s = 29.9, s2 = 891.4; No outliers 103. (a) (b) 105. (a) (b)

107. Earlier 4 0 0 9 2 1 8 4 3 0 3 7

12 13 14 15 16 17 18 19 20 21 22 23

Later 4 1 6 7

[100, 250] by [–1, 1]

[8, 4] by [–1, 1]

[8, 24] by [–1, 1]

The songs released in the earlier years tended to be shorter.

109.

Again, the data demonstrates that songs appearing later tended to be longer.

6 9 0 0 1 7 2

[–1, 25] by [100, 250]

[100, 250] by [–1, 1]

111. 1 9 36 84 126 126 84 36 9 1 113. (a) L 0.922 (b) L 0.075 115. Incorrect. Intelligence might be associated with some quantitative variable, but strength is categorical. 117. This will work. 119. Voluntary response bias

0

Chapter 9 Project Answers are based on the sample data shown in the table. 3. 1. 5 5 9 6 1 1 2 3 3 3 4 4 4 4 6 5 6 6 6 7 8 8 9 9 9 7 0 0 1 1 1 2 2 3 7 5 [59, 78] by [–1, 7] 66 or 67 inches

5. The data set is well distributed and probably does not have outliers. 7.

[56, 78] by [–1, 7]

SECTION 10.1 Exploration 1 1. 3

3. They are the same.

Quick Review 10.1 1. -4/7

3. y = 13/22 x + 6

5. y - 4 =

Exercises 10.1

3 1x - 12 4

1. 12 mi per hour 3. 3 5. 4a 7. 1 9. No tangent 11. 4

(b) 48 ft/sec

9. -

1 21h + 22

13. 12

[–7, 9] by [–1, 9]

15. (a) 48

7. h + 4

[–10, 11] by [–12, 2]

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 926

SELECTED ANSWERS

926

17. (a) -4 (b) y - 2 = - 41x + 12 y (c)

19. (a) 1 (b) y = x - 5 y (c)

19

5

4

x

x

4

21. - 1; 1; none 23. - 4 25. - 12 27. Does not exist 29. - 3 31. 6x + 2 33. (a) 9 ft/sec; 15 ft/sec (b) ƒ1x2 = 8.94x2 + 0.05x + 0.01, x = time in seconds (c) L 35.9 ft

[–0.1, 1] by [–0.1, 8]

35. (a)

(b) Since the graph of the function does not have a definable slope at x = 2, the derivative of f does not exist at x = 2. (c) Derivatives do not exist at points where functions have discontinuities.

y 9

x

5

37. (a)

(b) Since the graph of the function does not have a definable slope at x = 2, the derivative of f does not exist at x = 2. (c) Derivatives do not exist at points where functions have discontinuities.

y 3

5

39. Possible answer: y 10

–1

–10

5

x

x

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 927

SELECTED ANSWERS

43. The slope of the line is a; ƒ¿1x2 = a. 45. False; the instantaneous velocity is a limit of average velocities. It is nonzero when the ball is moving. 47. D 49. C

41. Possible answer: y 5

–1

927

5

x

–5

51.

(a) No, there is no derivative because the graph has a corner at x = 0. (b) No

53.

[–4.7, 4.7] by [–3.1, 3.1]

(a) No, there is no derivative because the graph has a vertical tangent (no slope) at x = 0. (b) Yes, x = 0. [–4.7, 4.7] by [–3.1, 3.1]

55. (a) 48 ft/sec (b) 96 ft/sec 57. y 1

10

x

SECTION 10.2 Exploration 1 1. 0.1 gal; 1 gal

3. 0.000000001 gal; 1 gal

Quick Review 10.2 1.

1 1 9 25 9 49 81 25 , , , 2, , , , 8, , 8 2 8 8 2 8 8 2

3.

65 2

5.

505 2

7. 228 miles

9. 4,320,000 ft3

Exercises 10.2 1. 195 mi 1 13. c0, d , c 2 3 15. c1, d , c 2 17. (a) y

3. 540,000 ft3 5. 2176 km 7. 13; Answers will vary. 9. 13; Answers will vary. 1 3 3 , 1 d , c1, d , c , 2 d 2 2 2 3 5 5 7 7 , 2 d , c2, d , c , 3 d , c3, d, c , 4 d 2 2 2 2 2 (b) y (c) y

18

18

18

1

2

3

4

5

x

1

2

3

RRAM: 30

4

5

x

1

2

LRAM: 14

3

4

5

x

11. 32.5

(d) Average: 22

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 928

SELECTED ANSWERS

928

19. (a)

(b)

y 5

(c)

y 5

1

2

3

4

x

5

1

2

3

4

5

x

1

RRAM: 10 21. 20 23. 37.5 25. 16.5 27. 2p 39. 8k + 12 41. 24 + 4k 45. 64 ft

29. 2

3

4

5

x

LRAM: 10 31. 2

33. 1

35. 4

37. 4

[0, 2] by [–50, 0]

[0, 3] by [0, 50]

(b) t = 1.5 sec

(c) 36 ft

(b) 33.86 ft

51. True; the exact area is given by the limit as n : q . L0

2

49. (a)

47. (a)

57.

(d) Average:10

y 5

1

1x - 12 dx = -

1 2

59. True

61. False

53. A

55. C

63. False

SECTION 10.3 Exploration 2 1. 50; 0

Quick Review 10.3 1. (a) -

3 1 (b) (c) Undefined 16 64

9.

3. (a) x = - 2 and x = 2

(b) y = 2

5. (b)

y

7. (a) [-2, q 2

(b) None

5

4

x

Exercises 10.3 1 13. (a) Division by zero (b) 3 6 15. (a) Division by zero (b) - 4 17. (a) The square root of negative numbers is not defined in the real plane. (b) The limit does not exist. 19. -1 21. 0 23. 2 25. ln p 27. (a) 3 (b) 1 (c) None 29. (a) 4 (b) 4 (c) 4 31. (a) True (b) True (c) False (d) False (e) False (f ) False (g) False (h) True (i) False (j) True 33. (a) L 2.72 (b) L 2.72 (c) L 2.72 35. (a) 6 (b) -4 (c) 16 (d) -2 1. -4

3. 7

5. 17

7. 0

9. a2 - 2

11. (a) Division by zero

(b) -

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 929

SELECTED ANSWERS

37. (a)

39. (a)

y

y

9

8

4

4

x

(b) 0; 0 (c) 0 41. 2

x

(b) 0; 3 (c) Does not exist; lim- ƒ1x2 Z lim+ ƒ1x2 x:0

43. 0

57. q ; x = - 2 75. B

45. 1

47. 0; 0

59. q ; x = 5

49. q ; 1

61. 3

x:0

51. (a) q

(b) - q

65. q

67. 0

63. 1

53. (a) Undefined (b) 0 55. - q ; x = 3 1 69. Undefined 71. 73. False; lim ƒ1x) = 5 x :3 2

77. C

79. (a)

81. (a)

y

83. (a)

y 2

1

π

x 1

x

[–2, 25] by [0, 60]

(b) 1- 1, 0) ´ (0, 1) (c) x = 1 (d) x = - 1

(b) 1 -p, 0) ´ (0, p) (c) x = p (d) x = - p

(b) ƒ1x2 L

57.71

, where x = the number 1 + 6.39e -0.19x of months; lim ƒ1x2 L 57.71 x: q

(c) It’s about 58,000. 85.

87.

y 5

x= 4

y 5

x= 2

y= 2 x 15 8

x

x= 1

89. (d)

929

n

A

4 8 16 100 500 1,000 5,000 10,000 100,000

4 3.3137 3.1826 3.1426 3.1416 3.1416 3.1416 3.1416 3.1416

Yes, A : p as n : q .

(e)

n

A

4 8 16 100 500 1,000 5,000 10,000 100,000

36 29.823 28.643 28.284 28.275 28.274 28.274 28.274 28.274

As n : q , A : 9p.

(f) One possible answer: lim A = lim nh2 tan a

n: q

n: q

180° b n

= h2 lim n tan a n: q

= h2p = ph2

180° b n

As the number of sides of the polygon increases, the distance between h and the edge of the circle becomes progressively smaller. As n : q , h : radius of the circle.

6965_SE_Ans_833-934.qxd

930

1/25/10

3:41 PM

Page 930

SELECTED ANSWERS

91. (a)

21x + 22 2x + 4 = = 2 x + 2 x + 2 (c) y = 2 (b) y =

y 5

5

93. (a)

x

1x - 121x2 + x + 12 x3 - 1 = x - 1 x - 1 = x2 + x + 1 (c) y = x2 + x + 1 (b) y =

y 8

5

x

SECTION 10.4 Exploration 1 1. 1.364075504

3. 10 sin x dx; sum1seq1sin10 + K*p/502*p/50, K, 1, 5022 = 1.999341983; fnInt1sin1X2, X, 0, p2 = 2 p

Quick Review 10.4 1. 5

3. 2/3

5. 3

7. L 0.5403

9. L 1.000

Exercises 10.4 1. - 4 3. - 12 5. 0 21. 106.61 mi 23. (a) -50 ft/sec (b)

7. L 1.0000

9. L - 3.0000

11. 64/3

13. 2

15. L 0

(c) s1t2 = - 16.08t 2 + 0.36t + 499.77 (d) L -47.88 ft/sec (e) L 179.28 ft/sec

[–1, 6] by [0, 550]

25. (a)

Midpoint

¢s/¢t

0.25 0.75 1.25 1.75 2.25 2.75 3.25 3.75 4.25 4.75 5.25

-10 -20 -40 -60 -70 -90 -100 -120 -140 -150 -170

(b)

(c) Approximately -47.95 ft/sec; this is close to the results in Exercise 23.

[0, 6] by [–180, 20]

17. 1

19. L 3.1416

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 931

SELECTED ANSWERS

27. 100 ft 31. (b)

N

LRAM

RRAM

10 20 50 100

15.04 16.16 16.86 17.09

19.84 18.56 17.82 17.57

33. (b)

Average 17.44 17.36 17.34 17.33

N

LRAM

RRAM

Average

10 20 50 100

98.24 101.76 103.90 104.61

112.64 108.96 106.78 106.05

105.44 105.36 105.34 105.33

37. (b)

N

LRAM

RRAM

10 20 50 100

1.08 1.04 1.02 1.01

0.92 0.96 0.98 0.99

RRAM

7.84 8.56 9.02 9.17

11.04 10.16 9.66 9.49

Average 9.44 9.36 9.34 9.33

N

LRAM

RRAM

10 20 50 100

7.70 7.81 7.87 7.89

8.12 8.02 7.95 7.93

Average 7.91 7.91 7.91 7.91

(c) fnInt gives 7.91, the same result as N100.

(c) fnInt gives 105.33; at N100, the average is 105.3344. 39. (b)

LRAM

(c) fnInt gives 9.33; at N100, the average is 9.3344.

(c) fnInt gives 17.33; at N100, the average is 17.3344. 35. (b)

N 10 20 50 100

931

41. (b)

Average 1.00 1.00 1.00 1.00

N

LRAM

RRAM

10 20 50 100

0.56 0.58 0.59 0.59

0.62 0.61 0.60 0.60

Average 0.59 0.59 0.59 0.59

(c) fnInt = 0.59, the same result as N100.

(c) fnInt gives 1, the same result as N100.

43. True; the notation NDER refers to a symmetric difference quotient using h = 0.001. 45. B 47. C 49. (a) 4x + 3 (b) 3x2 (c) 11.002, 11 (d) The symmetric method provides a closer approximation to ƒ¿122 = 11. (e) 12.006001; 12.000001; symmetric 51. The values of ƒ10 + h2 and ƒ10 - h2 are the same 53. (a) 4 (b) L 19.67 57. (b)

(c) y L x 3

A1x2

x 0.25 0.5 1 1.5 2 2.5 3

0.0156 0.125 1 3.375 8 15.625 27

(d) The exact value of A1x2 for any x greater than zero appears to be x 3. (e) A¿1x2 = 3x 2

[–2, 5] by [–5, 30]

[0, 5] by [–5, 30]

CHAPTER 10 REVIEW EXERCISES 1. (a) 2 1 17. 4 25. -1

(b) Does not exist

3. (a) 2

(b) 2

5. -1

7. -7

9. 0

11. 0

13. - q

19. ƒ has vertical asymptotes at x = - 1 and x = - 5; f has a horizontal asymptote at y = 0. x3 - 1 x - 1 27. y = L 3

if x Z 1

29. - 9

31. (a) 8.01

15. q 21. - 8

23. -

1 9

(b) 8

if x = 1

33. 1; y = x - 1 35. 10x + 7 37. LRAM: 42.2976; RRAM: 40.3776; 41.3376 39. (a) (b) 1995 to 1996: 8.4 cents per year; 2002 to 2003: 23.3 cents per year (c) 2004 to 2005 (d) 1997 to 1998 (e) y = 8.371x + 78.694 (f ) y = 0.0794x 3 - 0.9206x 2 + 2.9743x + 112.5684

[–5, 20] by [0, 300] [–5, 20] by [0, 300]

(g) 1997:1.8; 1998:3.5; 1999:5.7; 2000:8.4 (h) 504.6 cents per gallon. Likely too high.

[–5, 20] by [0, 300]

6965_SE_Ans_833-934.qxd

1/25/10

3:41 PM

Page 932

SELECTED ANSWERS

932

Chapter 10 Project

3. y = 271661.8371 # 11.0557797t2 5. Regression model predictions: 2,382,109; 4,099,161; 7,053,883. The web site predictions are probably more reasonable, since the scatter plot in Question 1 of this project suggests that growth in recent years has been fairly linear.

1.

[–5, 35] by [0, 2000000]

APPENDIX A Appendix A.1 1. 9 or -9 23. 1.3 41.

3. 4

5.

25. 2.1

2 3 x 2y 2 y

4 4 or 3 3

27. 12 12

59. 3a 2b 21b Ú 02

73. 1x - 2 ƒ y ƒ 21x

75. 6

9. - 6

45. 2x 5/3y 1/3

4 3

13. 4

63.

ƒxƒ xƒyƒ

79. 7

15. 2.5

33. x 2 ƒ y ƒ 2 4 3y 2

17. 729 35. 2x 2 2 53

49. 2 3 x -5 = 1/2 3 x5

47. 2 4 a 3b

61. 4x 4y 2 77. =

11. -

31. ƒ x ƒ y 2 12x

29. -5 2 32

43. 1a + 2b22/3

57. a -17/30

7. 12

65. 3y 2/ ƒ x 3 ƒ 81. 6

51. 2 4 2x

67. ƒ x ƒ 2 4 6x 2y 2/2

69.

19. 0.25

21. - 2

37. 22 34 53. 2 8 xy

2x2 3x y

39.

2 5 x3 x

55. 2 15 a 71. 0

83. L 3.48 sec

n

n

85. If n is even, then there are two real nth roots of a 1when a 7 02: 1a and - 1a.

Appendix A.2

1. 3x 2 + 2x - 1; degree 2 3. -x 7 + 1; degree 7 5. No 7. Yes 9. 4x 2 + 2x + 4 11. 3x 3 - x 2 - 9x + 3 3 2 2 3 2 2 13. 2x - 2x + 6x 15. -12u + 3u 17. -15x - 5x + 10x 19. x + 3x - 10 21. 3x 2 + x - 10 23. 9x 2 - y 2 2 2 3 2 2 3 6 2 3 2 4 3 25. 9x + 24xy + 16y 27. 8u - 12u v + 6uv - v 29. 4x - 9y 31. x + 2x - 5x + 12 33. x + 2x - x 2 - 2x - 3 2 3 2 35. x - 2 37. u - v, u Ú 0 and v Ú 0 39. x - 8 41. 51x - 32 43. yz1z - 3z + 22 45. 1z + 721z - 72 47. 18 + 5y218 - 5y2 49. 1y + 422 51. 12z - 122 53. 1y - 221y 2 + 2y + 42 55. 13y - 2219y 2 + 6y + 42 57. 11 - x211 + x + x 22 59. 1x + 221x + 72 61. 1z - 821z + 32 63. 12u - 5217u + 12 65. 13x + 5214x - 32 67. 12x + 5y213x - 2y2 69. 1x - 421x 2 + 52 71. 1x 2 - 321x 4 + 12 73. 1c + 3d212a - b2 75. x1x 2 + 12 2 77. 2y13y + 42 79. y14 + y214 - y2 81. y11 + y215 - 2y2 83. 215x + 4215x - 22 85. 212x + 5213x - 22 87. 12a - b21c + 2d2 89. 1x - 321x + 221x - 22 91. 12ac + bc2 - 12ad + bd2 = c 12a + b2 - d 12a + b2 = 12a + b21c - d2; Neither of the groupings 12ac - bd2 and 1 - 2ad + bc2 have a common factor to remove.

Appendix A.3

5 30 5 1 3. 5. 7. 9. All real numbers 11. x Ú 4 or 34, q 2 13. x Z 0 and x Z - 3 15. x Z 2 and x Z 1 3 77 6 10 17. x Z 0 19. 8x 2 21. x 2 23. x 2 + 7x + 12 25. x 3 + 2x 2 27. 1x - 221x + 72 cancels out during simplification; the restriction indicates that the values 2 and - 7 were not valid in the original expression. 29. No factors were removed from the expression. 31. 1x - 32 ends up in the numerator of the simplified expression; the restriction reminds us that it began in the denominator so that 3 is not allowed. 1.

33.

6x 2 ,x Z 0 5

45.

x + 1 ,x Z 1 3

55. 63.

31x - 32 28

35.

x2 ,x Z 0 x - 2

47. -

37. -

1 , x Z 1 and x Z - 3 x - 3

, x Z 0 and y Z 0

57.

39.

x , x Z 0 and y Z 0 41x - 32

x 2 + xy + y 2 , x Z y, x Z 0, and y Z 0 x + y

69. a + b, a Z 0, b Z 0, and a Z b

y + 5 4z 2 + 2z + 1 1 x2 - 3 41. 43. ,y Z 6 ,z Z , x Z -2 y + 3 z + 3 2 x2 21x - 12 1 1 2 49. 51. , y Z 5, y Z - 5, and y Z 53. x y x 2

z ,z Z 3 z + 3

71.

65.

59.

1 x + 3 , x Z 4 and x Z x - 3 2

1 , x Z -y xy

73.

x + y xy

2x - 2 x + 5 67. -

61.

1 , x Z 0 and x Z - 3 3 - x

2x + h x 21x + h22

,h Z 0

6965_SE_Ans_833-934.qxd

1/26/10

11:37 AM

Page 933

SELECTED ANSWERS

933

SECTION C Appendix C.1 1. (a) False statement (b) Not a statement (c) False statement (d) Not a statement (e) Not a statement (f ) Not a statement (g) True statement (h) Not a statement (i) Not a statement (j) Not a statement 3. (a) There is no natural number x such that x + 8 = 11. (b) There exists a natural number x such that x + 0 Z x. (c) There is no natural x such that x 2 = 4. (d) There exists a natural number x such that x + 1 = x + 2. 5. (a) The book does not have 500 pages. (b) Six is not less than eight. (c) 3 # 5 Z 15 (d) No people have blond hair. (e) Some dogs do not have four legs. (f ) All cats have nine lives. (g) Some squares are not rectangles. (h) All rectangles are squares. (i) There exists a natural number x such that x + 3 Z 3 + x. (j) For all natural numbers x, 3 # 1x + 22 Z 12. (k) Not every counting number is divisible by itself and 1. (l) All natural numbers are divisible by 2. (m) For some natural number x, 7. (a) F (b) T (c) T (d) F (e) F (f ) T (g) F (h) F (i) F (j) F 9. (a) R ´ S (b) Q ¨ Q (c) R ´ Q 5x + 4x Z 9x. (d) P ¨ 1R ´ S2 11. (a) The statements ' 1 p ¡ q2 and ' p ¿ ' q are equivalent, and the statements ' 1 p ¿ q2 and ' p ¡ ' q are equivalent. (b) The corresponding DeMorgan Laws for sets are P ´ Q = P ¨ Q and P ¨ Q = P ´ Q. The analogy comes from letting p mean “x is a member of P” and letting q mean “x is a member of Q.” Then, for the first law, ' 1 p ¡ q2 means “x is a member of P ´ Q,” which is equivalent to “x is a member of P ¨ Q,” which translates into ' p ¿ ' q. 13. (a) Today is not Wednesday or the month is not June. (b) I did not eat breakfast yesterday, or I did not watch television yesterday. (c) It is not true that both it is raining and it is July.

Appendix C.2

1. (a) p : q (b) ' p : q (c) p : ' q (d) p : q (e) ' q : ' p (f ) q 4 p 3. (a) Converse: If you’re good in sports, then you eat Meaties; Inverse: If you don’t eat Meaties, then you’re not good in sports; Contrapositive: If you’re not good in sports, then you don’t eat Meaties. (b) Converse: If you don’t like mathematics, then you don’t like this book; Inverse: If you like this book, then you like mathematics; Contrapositive: If you like mathematics, then you like this book. (c) Converse: If you have cavities, then you don’t use Ultra Brush toothpaste; Inverse: If you use Ultra Brush toothpaste, then you don’t have cavities; Contrapositive: If you don’t have cavities, then you use Ultra Brush toothpaste. (d) Converse: If your grades are high, then you’re good at logic; Inverse: If you’re not good at logic, then your grades aren’t high; Contrapositive: If your grades aren’t high, then you’re not good at logic. 5. (a) T (b) T (c) F (d) F (e) T (f) F 7. No 9. If a number is not a multiple of 4, then it is not a multiple of 8. 11. (a) p is false. (b) p is false. (c) q can be true, and in fact q true and p false makes p : q true and is the only way for q : p to be false. 13. (a) Helen is poor. (b) Some freshmen are intelligent. (c) If I study for the final, then I will look for a teaching job. (d) There exist triangles that are isosceles. 15. (a) If a figure is a square, then it is a rectangle. (b) If a number is an integer, then it is a rational number. (c) If a figure has exactly three sides, then it may be a triangle. (d) If it rains, then it is cloudy.

6965_SE_App_Indx_pp935-938.qxd

2/2/10

12:27 PM

Page 935

Applications Index Note: Numbers in parentheses refer to exercises on the pages indicated

Biology and Life Science Analyzing height data, 733 Angioplasty, 114 Average deer population, 403, 447 Bacterial growth, 263(57), 266, 272(39) Biological research, 208(71) Blood pressure, 358(76) Blood type, 691(35) Capillary action, 368(65) Carbon dating, 263(58), 272(40) Circulation of blood, 194(65) Comparing age and weight, 170(49), 172(67) Finding faked data, 453(76) Focusing a lithotripter, 597, 600(59, 60) Galileo’s gravity experiment, 66–67 Half-life, 676(38) HIV testing, 667(38) Humidity, 157, 224 Indirect measurements, 389, 448(35) Latitude and longitude, 327–328(63–70) Life expectancy, 170(50), 701(3–8), 708 ff., 716(55, 56) Medicare expenditures, 527(45), 575(70) Modeling a rumor, 270 National Institutes of Health spending, 249(89) Penicillin use, 302(53, 54) Personal income, 527(46) Salmon migration, 455, 463 Satellite photography, 117(33) Sleep cycles, 397(44) Spread of flu, 272(45), 316(76, 93) Temperature conversion, 126(34) Temperatures of selected cities, 144 Testing positive for HIV, 664 Weather balloons, 117(31) Weight and pulse rate of selected mammals, 183(55), 291(65) Weights of loon chicks, 711, 712, 714(17) Weight loss, 397(36)

Business Analyzing an advertised claim, 648 Analyzing profit, 194(64) Analyzing the stock market, 78(61) Beverage business, 171(59) Breaking even, 233(33), 249(90) Buying a new car, 650(38) Cash-flow planning, 58(39) Cell phone antennas, 149(29, 30) City government, 649(12) Company wages, 243(56) Comparing prices, 65, 531 Costly doll making, 170(52) Defective baseball bats, 732(113)

Defective calculators, 667(34) Defective light bulbs, 732(114) Depreciation of real estate, 33 Electing officers, 649(11) Equilibrium price, 525 Estimating personal expenditures with linear models, 524 Food prices, 528(53) Forming committees, 649(27) Gross domestic product data, 772(24) Interior design, 149(36) International finance, 138(65) Inventory, 541(48) Investment planning, 37(46) Investment returns, 149(38), 150(40) Investment value, 18(35, 36) Job interviews, 649(32) Job offers, 149(28) Linear programming problems, 568 ff., 575(65–68) Management planning, 171(58, 60) Manufacturing swimwear, 348(60) Manufacturing, 150(47, 49), 555(74) Maximizing profit, 572(40) Maximum revenue, 529(69, 70) Minimizing operating cost, 569 Mining ore, 571(37) Modeling depreciation, 160 Monthly sales, 397(35) Nut mixture, 528(54) Painting houses, 714(14) Percent discounts, 66 Pizza toppings, 647, 650(39) Planning a diet, 572(38) Planning for profit, 243(58) Predicting cafeteria food, 667(30) Predicting demand, 163 Predicting economic growth, 376(72) Predicting revenue, 166–167 Producing gasoline, 572(39) Production, 541(46, 47), 714(15) Profit, 541(49) Purchasing fertilizer, 568 Quality control, 716(57, 58) Questionable product claims, 643 Real estate appreciation, 37(45) Rental van, 528(57) Renting cars, 667(33) Salad bar, 650(37) Salary package, 528(58) Sale prices, 149(27) Starbucks Coffee, 156 Stocks and matrices, 576(73) Straight-line depreciation, 170(51) Supply and demand, 206(57), 207(58), 555(81, 82) Telephone area codes, 641, 648 Total revenue, 170(55), 171(59)

Train tickets, 554(73) U.S. motor vehicle production, 18(29) World motor vehicle production, 18(30) Yearly average gasoline prices, 776(39)

Chemistry and Physics Accelerating automobile, 753(46) Acid mixtures, 230–231, 233(31, 32), 249(93) Airplane speed, 528(52) Angular speed, 323 Archimedes’ Principle, 207–208(69, 70) Atmospheric pressure, 267, 272(41, 42) Boyle’s Law, 58(38), 183(51), 227(71) Charles’s Law, 183(52) Chemical acidity (pH), 296, 301(47, 48), 316(82) Combining forces, 465(48–50), 516(76) Diamond refraction, 183(53) Drug absorption, 315(73) Earthquake intensity, 290(52), 295, 301(45, 46), 316(81) Escape velocity, 610(62) Finding a force, 471, 473(45–47, 48), 516(77) Finding the effect of gravity, 463 Flight engineering, 465(43, 44), 515(74, 75) Light absorption, 281(60), 316(89) Light intensity, 227(72), 290(53, 54) Magnetic fields, 427(75) Mixing solutions, 149(31, 32), 552, 555(78) Moving a heavy object, 465(46, 47) Newton’s Law of Cooling, 151(51, 52), 296 ff., 301(49–52), 317(95, 96) Oscillating spring, 374, 376(71), 396(29) Physics experiment, 76(23) Potential energy, 303(69) Radioactive decay, 266, 271(33, 34), 316(79, 80) Reflective property of a hyperbola, 607 Reflective property of an ellipse, 597 Reflective property of a parabola, 586 Refracted light, 348(55, 56) Resistors, 234(39), 244(62), 249(92) Rowing speed, 528(51) Sound intensity, 251, 280, 281(59), 290(51) Sound waves, 319, 393 Speed of light, 1, 35 Temperature, 359(79, 80) Tuning fork, 396(30) Windmill power, 183(54) Work, 470–471, 473(49–56), 516(78), 754(50)

Construction and Engineering Architectural design, 395(23), 449(42) Architectural engineering, 208(72), 248(86) Box with maximum volume, 140, 149(33) Box with no top, 59(46), 194–195(66–68) Building construction, 541(50) Building specifications, 37(49) Cassegrain telescope, 611(70)

935

6965_SE_App_Indx_pp935-938.qxd

936

2/2/10

12:27 PM

Page 936

APPLICATIONS INDEX

Civil engineering, 394(15), 395(21) Construction engineering, 449(39) Designing a baseball field, 448(36) Designing a box, 191, 241, 243(59) Designing a bridge arch, 589(64) Designing a flashlight mirror, 588(59) Designing a satellite dish, 588(60) Designing a softball field, 448(37) Designing a suspension bridge, 588(63) Dimensions of a Norman window, 47(61) Draining a cylindrical tank, 154(62–64) Elliptical pool table, 639(77) Ferris wheel design, 440(42) Garden design, 336(66) Grade of a highway, 37(48), 398(49) Gun location, 610(58) Height of a ladder, 47(60) Industrial design, 249(94, 95) Landscape design, 170(57) Measuring a baseball diamond, 445–446 Packaging a satellite dish, 141, 149(35) Parabolic headlights, 588(62), 639(76) Parabolic microphone, 586, 588(61), 639(75) Patio construction, 676(40) Residential construction, 149(34) Storage container, 248(87) Surveying a canyon, 439(37), 451(62) Surveyor’s calculations, 449(38) Volume of a box, 194 (66–68), 248(85) Weather forecasting, 439(38)

Data Collectors (CBR™, CBL™) Analyzing a bouncing ball, 318 Ellipses as models of pendulum motion, 517, 601(73, 74), 640 Free fall with a ball, 167, 180, 753(45), 769 ff. Height of a bouncing ball, 250 Light intensity, 183(57), 227(72) Motion detector distance, 207(62), 216(51, 52), 402 Tuning fork, 396(30), 397(43), 398(51)

Financial Annual percentage rate (APR), 310 Annual percentage yield (APY), 307, 312(57, 58), 316(87, 88) Annuities, 308 ff., 311(49, 50), 313(69, 70), 316(83, 84), 317(97, 98) 685(37, 38), 731(75, 76) Business loans, 576(77) Car loan, 310, 312(51, 52) Cellular phone subscribers, 79(63) Comparing salaries, 702(23), 714(25) Comparing simple and compound interest, 312(60), 317(99) Compound interest, 304 ff., 311(21–30), 315(63–65) Consumer price index, 63, 146, 149(24) Currency conversion, 126(33) Employee benefits, 150(48) Future value, 315(66), 681 House mortgage, 312(53–56), 316(85, 86) Investments, 555(75–77), 576(76)

IRA account, 311(47, 48) Loan payoff (spreadsheet), 312–313(67, 68) Major League Baseball salaries, 76(25–28) Per capita income, 714(16) Present value, 315(67, 68) Savings account, 684–685(35, 36) Size of continent, 713(10)

Geometry Area of a sector, 328(71, 72) Area of regular polygon, 444 Beehive cells, 452(68) Cone-shaped pile of grain, 142 Conical tank of water, 149(37) Connecting algebra and geometry, 39(69–71), 77(50, 52), 173(82–84), 243(57) Constructing a cone, 243(60) Designing a juice can, 231, 241, 243(61) Designing a running track, 323, 327(52) Designing a swimming pool, 234(38) Designing rectangles, 234(40) Diagonals of a parallelogram, 16–17 Diagonals of a regular polygon, 146 Dividing a line segment in a given ratio, 466(61) Dividing a line segment into thirds, 20(64) Finding a maximum area, 170(54) Finding a minimum perimeter, 231 Finding dimensions of a painting, 170(56) Finding dimensions of a rectangular cornfield, 528(50) Finding dimensions of a rectangular garden, 521, 528(49) Industrial design, 234(37) Inscribing a cylinder inside a sphere, 155(67) Inscribing a rectangle under a parabola, 155(68) Maximizing area of an inscribed trapezoid, 452(67) Maximizing volume, 433(55) Measuring a dihedral angle, 446 Medians of a triangle, 466(62) Minimizing perimeter, 234(35) Mirrors, 337(75) Page design, 234(36) Pool table, 337(76) Tire sizing, 326(46) Tunnel problem, 433(56)

Mathematics Angle of depression, 388, 395(24) Angle of elevation, 348(59), 395(22) Approximation and error analysis, 349(77) Calculating a viewing angle, 384, 385(53) Characteristic polynomial, 543(72, 73) Computing definite integrals from data, 770–771 Connecting geometry and sequences, 677(51) Curve fitting, 554(67–70), 574(45, 46) Cycloid, 484(53) Designing an experiment, 722 Diagonals of a regular polygon, 650(52) Distance from a point to a line, 474(67) Eigenvalues of a matrix, 556(93, 94) Epicycloid, 516(83) Expected value, 668(61), 669(62)

Finding a random sample, 727(33) Finding area, 401(104) Finding derivatives from data, 745(34), 769–770, 771(23) Finding distance, 336(65), 393(3), 394(4, 13), 395(19, 20, 24), 401(97, 98, 100), 439(38), 440(41, 45, 46), 452(64) Finding distance from a velocity, 771(21, 22), 772(27, 28) Finding distance traveled as area, 749, 751, 753(49), 768 Finding height, 334, 336(61, 62, 64), 389, 393(1, 2), 394(5–9, 11, 12, 14), 395(22), 401(95, 96, 101, 102), 440(40, 44), 452(63) Finding height of pole, 438, 440(39, 43) Fitting a parabola to three points, 551 Graphing polar equations parametrically, 493(67–71) Harmonic series, 686(52) Hyperbolic functions, 420(80) Hypocycloid, 484(54) Limits and area of a circle, 765(89) Locating a fire, 437 Normal distribution, 303(67) Parametrizing circles, 476, 485(65) Parametrizing lines, 478, 485(66) Permuting letters, 649(7, 9, 10) Polar distance formula, 493(61) Reflecting graphs with matrices, 538 Rotating with matrices, 538, 542(51), 575(69) Scaling triangles with matrices, 539 Solving triangles, 435, 443–444, 489 Symmetric matrices, 541(45) Taylor polynomials, 349(79, 80) Television coverage, 367(63) Testing inequalities on a calculator, 27(69) Tower of Hanoi, 687–688 Transformations and matrices, 542(57–61)

Miscellaneous Acreage, 446 ACT scores, 715(42) Advanced Placement Calculus exam scores, 714(18, 19) Arena seating, 676(39) Barry Bonds home runs, 696–697, 714(22, 27) Basketball attendance, 576(74) Basketball lineups, 651(58) Baylor School grade point averages, 97(78) Beatles songs, 731(97), 732(101, 105, 107–110) Blood donors, 727(37) Casting a play, 646, 651(56) Chain letter, 651(53) Choosing chocolates, 660, 661 Comparing cakes, 76(20) Computer graphics, 519, 539 Computer imaging, 117(34) Converting to nautical miles, 324 Ebb and flow of tides, 356, 358(75) Flower arrangements, 533 Football kick, 449(41) Graduate school survey, 667(39) Graduation requirement, 667(41)

6965_SE_App_Indx_pp935-938.qxd

2/2/10

2:46 PM

Page 937

APPLICATIONS INDEX

Hank Aaron home runs, 698, 701(2, 16), 702(22), 706 Home remodeling, 576(78) Horse racing, 730(46) Indiana Jones and the final exam, 649(36), 667(40) Investigating an athletic program, 668(60) Length of days, 377(88) License plate restrictions, 644, 649(23, 24), 729(9) Lighthouse coverage, 367(45), 385(54) Loose change, 555(79) LPGA golf, 702(18–20) Mark McGwire home runs, 696–697, 701(15), 702(22), 714(22, 26), 715(39) Melting snowball, 117(32) Mickey Mantle home runs, 701(12, 14), 702(21), 714(13, 23, 31, 32) Modeling a musical note, 393 Modeling illumination of the Moon, 454 Modeling temperature, 396(33), 401(105) Number of cassettes, 700, 715(36) Number of CDs, 699, 714(28), 715(35) Number of Wineries, 235(44) Pageant finalists, 647 Pepperoni pizza, 108(67) Pell Grants, 248(88) PGA golf, 702(17, 19, 20) Piano lessons, 667(29) Picking lottery numbers, 647 Popular web sites, 731(96), 732(100, 104) Radar tracking system, 491 Radio advertising, 727(32) Real estate prices, 731(95), 732(99) Roger Maris home runs, 701(1, 2), 705 ff. SAT Math scores, 61(33) SAT scores, 531, 715(41) Scaling grades, 127(47) Selecting customers, 727(34) Shooting free throws, 665, 724–725 Simulate a spinner, 727(38) Simulate rolling two six-sided dice, 727(39) Simulate drawing cards, 727(40) Soccer field dimensions, 47(59) Solar collection panel, 336(63) Stepping stones, 76(21) Swimming pool, 576(79) Swimming pool drainage, 234(41) Telephone numbers, 648 Testing effects of music, 727(31) Testing golf balls, 726(29) Testing soft drinks, 727(30) Time-rate problem, 234(42) Tracking airplanes, 492(51) Tracking ships, 492(52) Travel planning, 58(36) Truck deliveries, 576(75) Vacation money, 555(80) Visualizing a musical note, 360(89) Warren Moon passing yardage, 731(98), 732(102, 106) White House Ellipse, 579, 598 Willie Mays home runs, 701(11, 13), 702(21), 714(13, 23, 31, 32)

Wind speeds, 702(24), 714(24, 29, 30) Women’s 100-meter freestyle, 154(66) Yahtzee, 649(35), 667(36)

Motion Airplane velocity as a vector, 461–462 Air conditioning belt, 327(55) Automobile design, 326(44) Average speed, 745(33) Ball drop, 745(34) Baseball throwing machine, 171(62) Bicycle racing, 326(45), 328(74) Bouncing ball, 685(39) Bouncing block, 358(77) Calculating the effect of wind velocity, 462, 465(43, 44) Capture the flag, 483(38) Current affecting ship’s path, 465(53) Damped harmonic motion, 348(57) Falling rock, 475, 772(27) Famine relief air drop, 483(39) Ferris wheel motion, 358(73), 396(31, 32, 34) Field goal kicking, 516(86) Fireworks planning, 171(63) Foucault Pendulum, 327(54) Free-fall motion, 76(22), 167, 171(61), 184(68), 746(55, 56) Hang time, 516(87) Harmonic motion, 390–392, 427(74) Height of an arrow, 516(79) Hitting a baseball, 480, 483(40, 43–45), 484(49), 516(88) Hitting golf balls, 484(50) Hot-air balloon, 367(46), 386(55) Landscape engineering, 171(64) Launching a rock, 248(84) LP turntable, 359(78) Mechanical design, 395(27, 28) Mechanical engineering, 327(53) Motion of a buoy, 358(72) Navigation, 37(47), 62(82), 326(43, 48–50), 328(73), 390, 395(17, 18, 25, 26), 449(40), 450(53), 465(41, 42, 51, 52) Path of a baseball, 127(49) Pendulum, 348(58), 401(103), 402 Pilot calculations, 441(57) Projectile motion, 57, 58(33, 34), 62(81) Riding on a Ferris Wheel, 481, 484(51), 516(80, 81, 82) Rock toss, 485(63), 744(15), 753(47) Rocket launch, 744(16), 753(48) Rotating tire, 143, 149(39) Salmon swimming, 463 Ship’s propeller, 327(56) Shooting a basketball, 465(45) Simulating a foot race, 483(37) Simulating horizontal motion, 478, 483(37, 38) Simulating projectile motion, 479 Stopping distance, 194(63) Taming The Beast, 401(106) Throwing a ball at a Ferris Wheel, 485(67), 486(68), 517(89)

937

Throwing a baseball, 516(84, 85) Tool design, 326(47) Travel time, 149(25, 26), 771(21, 22) Tsunami wave, 358(74) Two Ferris Wheel problem, 486(69, 70) Two softball toss, 484(46) Using the LORAN system, 608, 610(57) Velocity in 3-space, 633, 636(33, 34) Windmill, 735, 743 Yard darts, 484(47, 48), 517(90)

Planets and Satellites Analyzing a comet’s orbit, 607 Analyzing a planetary orbit, 625 Analyzing the Earth’s orbit, 596 Dancing planets, 600(52) Elliptical orbits, 595, 639(79) Halley’s Comet, 600(58), 626(41) Icarus, 639(80) Kepler’s Third Law, 288 Lunar Module, 626(43) Mars satellite, 627(44) Mercury, 600(54) Modeling planetary data, 179, 184(67) Orbit of the Moon, 412(85) Orbit periods, 287 Planetary orbits, 627(51) Planetary satellites, 713(9) Rogue comet, 610(55, 56) Saturn, 600(55) Sungrazers, 600(57) Television coverage, 398(50), 452(69) The Moon’s orbit, 600(53) Uranus, 626(42) Venus and Mars, 600(56) Weather satellite, 639(78)

Population Alaska, 303(65) Anaheim, CA, 554(72) Anchorage, AK, 554(72) Arizona, 273(50) Austin, TX, 262–263(51–54) Bear population, 234(34) Columbus, OH, 262–263(51–54) Comparing populations, 702(25, 26) Dallas, TX, 261, 272(48) Deer population, 249(91), 272(46), 316(94) Detroit, MI, 265 Estimating population growth rates from data, 777 Florida, 269, 528(47) Garland, TX, 554(71) Georgia, 315(71) Guppy population, 316(78) Hawaii, 303(66), 575(71) Idaho, 575(71) Illinois, 315(72) Indiana 528(47) Los Angeles, CA, 272(43) Mexico, 273(58) Milwaukee, WI, 282(62) New York State, 263(56), 272(49)

6965_SE_App_Indx_pp935-938.qxd

938

2/2/10

2:46 PM

Page 938

APPLICATIONS INDEX

Ohio, 263(55) Pennsylvania, 269 Phoenix, AZ, 272(44) Population and matrices, 576(72) Population decrease, 315(74, 75) Population density, 685(47) Population growth, 271(29–32) Rabbit population, 316(77), 677(49), 765(83) Rain forest growth, 676(37) Richardson, TX, 554(71) San Antonio, 281(61) San Jose, CA, 260 Temperatures in Beijing, China, 703(35, 36), 714(20, 21)

U.S. population, 244(63), 268, 272(47), 273(57, 58) World population, 37(53)

U.S. Demographics Air travel, 76(24) Agriculture exports, 18(32) Agriculture surplus, 18(33) Americans’ income, 34, 37(50) Americans’ spending, 37(51) Analyzing U.S. Census data, 577 Annual housing cost, 244(64) Average hourly wage, 151(50) Causes of death, 667(35), 693, 694 Construction worker’s compensation, 170(53)

Crude oil imports, 154(65) Education budget, 10(53–56), 60(11) Employment statistics, 75(11–18) Exports to Canada, 18(34), 38(54) Exports to Mexico, 12 Female percentage of prison population, 65, 67 Imports from Mexico, 18(31), 37(52) Median women’s income, 172(68) Minimum hourly wage, 64 Patent applications, 171(65) Per capita federal aid, 696 Per capita income, 244(63) Prison population, 65 Remaining years, 234(43) Residents 65 or older, 694

6965_SE_Index_pp939-949.qxd

2/2/10

12:28 PM

Page 939

Index A Absolute maximum, 90 Absolute minimum, 90 Absolute value complex number, 504 inequality involving, 240 properties of, 13 real number, 11, 13 Absolute value equation, 45 Absolute value function, 101, 105, 132 Absolute value inequality, 54 Acceleration due to gravity, 167 Acute angle, 329 Acute triangle, 435 Addition property of equality, 21 of inequalities, 23 Additive identity of algebraic expression, 6 of complex number, 50 of matrix, 532 Additive inverse of algebraic expression, 6 of complex number, 50 of matrix, 532 properties of, 6 of real number, 6 Algebraic expression, 5. See also Rational expression; Real number constants and variables, 5 domain of, 791 expanded and factored, 6 properties of, 6 Algebraic function, 252 Algebraic models, 65 Ambiguous case of triangle, 435–437 law of sines, 435–437 Amplitude, 352 Anchor (mathematical induction), 687 Angle acute, 329 central, 320 complements, 406 coterminal, 338 degree-minute-second measure, 320–321 degree-radian conversion, 322 of depression, 388 dihedral, 446 directed, 487 direction, 460 of elevation, 388 initial and terminal sides, 338 obtuse, 435 positive and negative, 338 quadrantal, 342, 346 radian measure, 321 reference, 341, 346 of rotation, 338 standard position of, 329, 338

between vectors, 468 Angle of rotation, 614 Angular speed, 323 Annual percentage rate (APR), 310 Annual percentage yield (APY), 307 Annuity, 308 ordinary, 308 present and future value, 308–309 Aphelion, 595 APOLLONIUS OF PERGA (c 250–175 B.C.E.), 580 Approximate solution inequalities, 56 with table feature, 43–44 Arc length formula, 322–323 Arccos function, 380 ARCHIMEDES OF SYRACUSE (287–212 B.C.E.), 444, 749, 750, 755, 760 Arcsin function, 378 Arctan function, 381 Area, 748 Area of triangle, 444 Area problem, 749–750 ARGAND, JEAN ROBERT (1768–1822), 503 Argument of complex number, 504 Arithmetic sequence, 672–674 Arithmetic series, finite, 670 Arrow, 456 Associative property algebraic expression, 6 complex numbers, 50 Asymptote end behavior, 221 horizontal, 92, 93 of hyperbola, 603 slant, 221 vertical, 92, 93, 221 Atmospheric pressure, 267 Augmented matrix, 546 Average, weighted, 707 Average rate of change, 160, 740 Average velocity, 736 Axis of a cone, 580 of a conic, 581, 591, 602, 620 of ellipse, 565, 567 of hyperbola, 603, 605 of parabola, 164, 584 polar, 487 real, 503 x-, y-, and z-, 12, 629

B Back-to-back stemplot, 696 Bar chart, 693 Base, 7 change-of-base formula, 285 of exponential function, 252

of logarithmic function, 274 of natural logarithm, 277 Basic functions absolute value function, 101, 105, 132 cosine function, 101, 351 cubing function, 100, 176 exponential function, 100, 256 greatest integer function, 101 identity function, 99, 172 logistic function, 101 natural logarithm function, 100, 176 reciprocal function, 100 sine function, 100, 350 square root function, 100, 179 squaring function, 99, 172 Bearing, navigational, 321 Bel (B), 280 BELL, ALEXANDER GRAHAM (1847–1922), 280 BERNOULLI, JAKOB (1654–1705), 456 Best fit, line of. See Line of best fit Bias, 720 analyzing samples for, 720–721 response, 721 undercoverage, 721 voluntary response, 721 Biconditional, 808. See also Logic Binomial coefficients, 652–653 Binomial distribution, 664–665 Binomial probability, 665 Binomial theorem, 654 Binomials, 784–785 BLACKWELL, DAVID (b. 1919), 658 Blinded experiment, 721 Blocking, 722 Boundary of region, 565 Bounded function above, 88–89 below, 88–89 on an interval, 89 Bounded intervals, 4 Box-and-whisker plot. See Boxplot Boxplot, 709 Branches of hyperbola, 602

C Calculator-Based Laboratory System (CBL), 227 Calculator Based Ranger (CBR), 167 Cardioid curve, 498 Cartesian coordinate system, 12, 629 circles, 15 conversion with polar, 488 distance formula, 14, 630 midpoint formula, 14, 630 plotting data, 12 CASSEGRAIN, G., 607 Cassegrain telescope, 611 Categorical variable, 693 Census, 720

939

6965_SE_Index_pp939-949.qxd

940

1/25/10

10:54 AM

Page 940

INDEX

Center of circle, 15 of data, 704–705 of ellipse, 591 of hyperbola, 603 of sphere, 631 Central angle, 320 Chain rule, 812. See also Logic Change-of-base formula, 285 Characteristic polynomial, 543 Chord of a conic, 583, 603 Circle equation of, 15 parametric equations for, 475 and radian measure, 344 segment of, 446 unit, 344, 346 Circle graph, 693 Circular functions, 345 Closed interval, 4 Coefficient matrix, 546 Coefficient of determination, 146 Coefficient of term, 185 Cofactor, 535 Cofunction identities, 406 Column subscript (matrix), 530 Combinations, 646–647 Combinatorics, 642 combinations, 642 multiplication counting principle, 643 permutations, 644–646 tree diagram, 643 Common denominator, least, 793 compound fraction, 793–794 rational expressions, 793 Common difference of sequence, 672 Common logarithm, 275–276 Common ratio of sequence, 673 Commutative property of algebraic expressions, 6 complex numbers, 50 Complement (angle), 406 Completely factored polynomial, 786 Completing the square, 41 Complex conjugate, 51, 211–212 Complex conjugate zero, 211–212 Complex fraction, 793 Complex number, 49 absolute value of, 504 adding and subtracting, 49 argument of, 504 conjugates, 51 and coordinate plane, 49 exponents of, 50 modulus of, 504 multiplying and dividing, 50–51, 505–506 nth root, 508 and quadratic equations, 51 and roots, 508–510 standard form of, 49 trigonometric form of, 504 and vectors, 504 zeros of function, 210–211, 211–212 Complex plane, 503 Component form of vector, 456, 633 Components of a vector, 456

Composition of functions, 111, 132 Compound fraction, 793 Compound interest, 304, 307 Compound statement, 803. See also Logic Compounded annually, 304 Compounded continuously, 306 Conclusion, 807. See also Logic Conditional probability, 663 Conditionals (Implications), 807. See also Logic Confounding variable, 721 Conic section (conic), 580 defined as a ratio, 620 discriminant test, 671 ellipse, 591 focus coincident with pole, 621 hyperbola, 602 identifying, 617 parabola, 581 and polar equations, 621 rotation of, 615–616 and second-degree equation, 581 standard form of, 583, 591, 602 and transformations, 584, 594, 605 Conjugate, complex, 51 Conjugate axis of hyperbola, 603 Conjugate hyperbolas, 611 Conjunction, 803. See also Logic Constant, 5 of proportion, 174 of variation, 174 Constant function on an interval, 86–87 Constant percentage rate, 265 Constant term of polynomial, 161 Constraints, 568 Continuous at a point, 85 Continuous function, 102, 758 Control, 721 Convenience sample, 721 Coordinate plane Cartesian, 12 complex, 49 polar, 487 quadrants of, 12 Coordinates of a point, 3, 12, 629 Corner (vertex) point, 568 Correlation coefficient, 146, 162 Cosecant function, 329, 364 acute angle, 329 any angle, 340 graph of, 364 of special angles, 330 Cosine function, 329, 351 acute angles, 329 any angle, 340 cofunction identities, 442 of a difference identity, 421–422 graph of, 351 harmonic motion, 390–391 inverse, 380 law of cosines, 442–447 period of, 351, 353 special angles, 330 of a sum identity, 422 Cotangent function, 329, 362 acute angles, 329

any angle, 340 graph of, 362 of special angles, 330 Coterminal angle, 338 Counting. See Combinatorics Counting subsets of an n set, 647 Course, navigational, 321 Cross-product term, 637 Cube of difference, 785, 787 of sum, 785, 787 Cube root, 509–510, 779 Cubic inequality, 56 Cubic polynomial function, 185 graphing, 187 regression, 145 Cubing function, 100, 176 Cycloid, 484

D d’ALEMBERT, JEAN LE ROND (1717–1783), 755 Damped oscillation, 373–375 Damping, 373 Damping factor, 373 Data. See also Statistics definite integral from, 770 derivative from, 768–770 displaying, 693 function construction from, 143 Data analysis, 144 cubic regression, 145 exponential regression, 145 linear regression, 145 logarithmic regression, 145 logistic regression, 145 power regression, 145 quadratic regression, 145 quartic regression, 145 sinusoidal regression, 145 DE MOIVRE, ABRAHAM (1667–1754), 507 De Moivre’s Theorem, 507 Decay curve, 254 Decay factor, 254 Decibel, 251, 280 Decimal form of rational number, 2 Decomposition function, 113 partial fraction, 557 Decreasing function on an interval, 86–87 Deduction, 690 Deductive reasoning, 73 Definite integral, 750–751, 770 Degenerate conic section, 580 Degree, 320 of polynomial, 784 of polynomial function, 158 Degree-minute-second (DMS) angle measurement, 320–321 Degree of angle, 320 Degree-radian conversion equations, 322 Demand curve, 525 Denominator, 791 least common, 793 with linear factors, 558–560

6965_SE_Index_pp939-949.qxd

2/2/10

12:29 PM

Page 941

INDEX

with quadratic factors, 560–561 rationalizing, 781 Dependent events, 663 Dependent variable, 80 Derivative at a point, 740, 766 Derivative of a function, 740–742 from data, 768–770 numerical, 766 DESCARTES, RENÉ (1596–1650), 49, 732 Descartes’ Rule of Signs, 208 Descriptive statistics, 704 Determinant of matrix, 535–536 Difference of functions, 110 identity, 421–425 of sinusoids, 371–372 of two cubes, 787 of two squares, 415, 786 Differentiable, 740 Dihedral angle, 446 Direct reasoning, 811. See also Logic Direct variation, 174 Directed angle, 487 Directed distance, 487 Directed line segment, 456 Direction angle of vector, 460 Direction of vector, 456, 458, 460 Direction vector of a line, 456, 634 Directrix, 581, 620 Discontinuity infinite, 85 jump, 85 removable, 84–85 Discriminant of quadratic equation in x, 47 Discriminant of second-degree equation in x and y, 617, 671 Discriminant test for a second-degree equation in two variables, 617 Disjunction, 803. See also Logic Distance from changing velocity, 748–749 from constant velocity, 747 Distance formula in Cartesian space, 630 coordinate plane, 14 number line, 13 space, 630 Distinguishable permutations, 645 Distribution of data, 697 normal, 711 skewed left, 709 skewed right, 709 symmetric, 709 Distributive property algebraic expressions, 6 complex numbers, 50 Divergent series, 671, 682 Division, 6 of polynomials, 197 by zero, 791 Division algorithm for polynomials, 197 Domain algebraic expression, 791 function, 80 implied, 82 inverse function, 122

relevant, 82 Dot or inner product, 467, 633 on calculator, 468 properties of, 467, 633 Double-angle identity, 428 Double-blind experiment, 723 Double inequality, 25 dy/dx, 742

E e, base, 256–258 Eccentricity of conic ellipse, 595–597 hyperbola, 606–607 and polar coordinates, 620 Eigenvalues, 556 Eighth root, 510 Element (entry) of a matrix, 530 Element of a set, 2 Elementary row operations, 546–548 Elimination method, 522–524 Ellipse, 591, 620 center and focus, 591, 594 directrix of, 620 eccentricity of, 595–597 equation, 591 focal axis, 591 graph of, 592, 593 parametric equation for, 595 Pythagorean relation, 591, 594 semimajor and semiminor axes of, 592, 594 standard form of, 591, 594 transformation of, 594 vertices, 591, 592, 594 Ellipsoid of revolution, 597 Empirical probability, 668 Empty set, 659 End behavior, 93 asymptote, 221 of exponential function, 256 of logarithmic function, 274, 278 of polynomial, 187–189 of rational function, 221 of sequence, 671 Endpoints of interval, 4 Enumerative induction, 690 Equal complex numbers, 49 Equal fractions, 792 Equal matrices, 530 Equal vectors, 457, 633 Equally likely outcomes, 658 Equation, 21. See also Linear first-degree equation; Polar equation; Solving equations; Solving systems of equations; Trigonometric equation absolute value, 45 addition property, 21 algebraic solution, 21, 40, 69 approximate solutions, 43, 56 of circles, 15 of ellipse, 591, 594 extraneous solution, 229 graphical solution, 43, 69–70 matrix, 550 of parabola, 584

941

parametric, 475 polar, 490 properties of, 21 quadratic, 41 rational, 228 and relation, 114–116 second-degree, 581 Equation for a line in Cartesian space, 457 parametric form, 475–476 vector form, 457 Equation for a plane in Cartesian space, 632 Equation for a sphere in Cartesian space, 631 Equilibrium point, 525 Equilibrium price, 525 Equivalent directed line segments, 457 Equivalent equation, 21 inverse sine function, 379 inverse tangent function, 381 logarithmic function, 292 polar, 494 Equivalent inequalities, 24 Equivalent rational expression, 792 Equivalent systems of equations, 544 ERATOSTHENES OF CYRENE (276–194 B.C.E.), 320 Escape speed, 610 EULER, LEONHARD (1707–1783), 80, 100, 257 Even function, 90, 406–407 Event, 658 dependent, 663 independent, 661 multiplication principle, 661 Existential quantifiers, 802. See also Logic Expanded algebraic expressions, 6 Expected value, 668 Experiment, 721 blinded, 721 designing, 722–723 double-blind, 723 Explanatory variable, 642 Explicitly defined sequence, 670 Exponential decay factor, 266 Exponential function, 252, 254. See also Logarithmic function base of, 252, 256–258 end behavior, 255, 256 graphing, 255 growth and decay curve, 254 inverse of, 292 logistic growth and decay, 258–260 one-to-one, 292 regression equation, 145 solving equations, 292 Exponential growth factor, 266 Exponential notation, 7 Exponents, 7 base, 7 of complex numbers, 506 positive and negative, 7 properties, 7 rational, 781 Extended principle of mathematical induction, 692 Extracting square roots, 41 Extraneous solution, 229

6965_SE_Index_pp939-949.qxd

942

2/2/10

12:29 PM

Page 942

INDEX

F Factor, 721, 787–788. See also specific factors Factor Theorem, 199 Factored algebraic expressions, 6 Factorial notation, 654 Factoring polynomials, 785–788 common factors, 786 completely factored, 786 by grouping, 788 higher degree, 189 linear factors, 210 and long division, 197 special forms, 785–788 trinomials, 784 zero factor property, 41 Feasible point, 568 FERMAT, PIERRE DE (1601–1665), 658, 740 Fibonacci. See LEONARDO OF PISA Fibonacci number, 675 Fibonacci sequence, 675 Finite sequence, 670 Finite series arithmetic, 683 geometric, 683 First-degree (linear) equation in three variables, 632 First octant, 629 First quartile, 707 Fitting a line to data, 144 Five-number summary, 707–708 Focal axis of a conic, 620 ellipse, 591, 594 hyperbola, 602, 605 Focal chord of a parabola, 590 Focal length of a parabola, 583 Focal width of a hyperbola, 611 of a parabola, 583 Focus, 620 of ellipse, 591 of hyperbola, 602, 605 of parabola, 581 FOIL method, 784 Foot-pound, 471 Foucault pendulum, 327 Fourth root, 509 Fractional expressions, 791. See also Rational expression Fractions complex/compound, 793 equal, 792 operations with, 791, 792–793 partial, 557 reduced form of, 791 Free-fall motion, 167 Frequency distribution, 697 of observations, 697 of oscillations, 353, 391 of sinusoid, 353 Frequency table, 697 Function, 80. See also Exponential function; Graph; Logarithmic function; Polynomial function; Quadratic function; Rational function; Trigonometric function

average rate of change of, 740 bounded above, 88–89 bounded below, 88–89 bounded on an interval, 89 combining, 110 composition of, 111 constant on an interval, 86–87 continuous, 102, 758 from data, 143 decomposing, 113 decreasing on an interval, 86–87 definite integral of, 750–751 derivative of, 740–742 difference of, 110 differentiable, 740 domain, 80 end behavior, 93 evaluating, 80 even, 90–91 from formulas, 140–141 graph of, 81 from graphs, 141–142 identity, 99 implicitly defined, 115 increasing on an interval, 86–87 instantaneous rate of change of, 740 integrable, 750 inverse, 122–123 inverse relation, 121–122 linear, 159 local maximum and minimum, 90 lower bound, 89 maximum and minimum, 90 monomial, 175–176 notation, 80 numerical derivative of, 766 numerical integral of, 767 odd, 91 one-to-one, 122 periodic, 345 piecewise-defined, 104 point of discontinuity, 85 power, 174 probability, 659 product of, 110 quadratic, 164–166 quartic, 185 quotient of, 110 range, 80, 81 reciprocal, 219 step, 101 sum of, 110 symmetric difference quotient of, 766 tangent line of, 739 of two variables, 633 upper bound, 89 from verbal description, 142–143 y-intercept, 29, 221 zeros of, 69–70, 201, 210–211 Fundamental Theorem of Algebra, 210 Future value of annuity, 308

G GALILEI, GALILEO (1564–1642), 66, 167, 732

GAUSS, CARL FRIEDRICH (1777–1855), 49, 544, 678 Gaussian curve. See Normal curve Gaussian elimination, 544–545 General form of linear equation, 30 General second-degree equation, 581 Generator of a cone, 580 Geometric sequence, 672–674 Geometric series, finite, 680 Geosynchronous orbits, 398 Graph, 81. See also Function; Linear first-degree equation absolute value compositions, 132 of conics, 621–622 of cosecant function, 364 of cosine function, 351 of cotangent function, 362 of cubic function, 185 discontinuity, 84–85 of ellipse, 592, 593 of equation, 31 of equation in x and y, 31 of exponential function, 255–256 of a function, 81 functions from, 141–142 hidden behavior, 72 of histogram, 697 of hyperbola, 602, 605 of inequality, 24–25, 565 of inverse, 124 of leading term of polynomial, 188 limaçon curve, 498 of linear equation, 30 of linear system of equations, 522 local maximum and minimum, 90 of logarithmic function, 274, 277–278 of logistic growth, 259–260, 260–261 of parabola, 583 of parametric equations, 475–476 point-plotting method, 31 of polar equation, 495–496 of polynomial, 185 of quadratic function, 164–166 of quartic function, 185 of rational function, 221–222 of real number line, 3 of relation, 114–115 rose curve, 497 of secant function, 363 of second-degree equation, 612 of sequence, 674–675 of sine function, 350 of stemplot, 694 symmetry, 90–92 of system of inequalities, 566 of tangent function, 362 of timeplot, 697 Grapher, 31 ANS feature, 276, 675 approximating zeros, 69–70 complex numbers, 50 evaluating a function, 80 failure of, 72 hidden behavior, 72 NDER ƒ1a2, 766

6965_SE_Index_pp939-949.qxd

1/25/10

10:54 AM

Page 943

INDEX

NINT1ƒ1x2, x, a, b2, 767 parametric mode, 120 scientific notation, 8 sequence mode, 674–675 simulating motion with, 478–481 slope, 28, 36 summing sequences, 678 tables, 43–44 Graphical model, 66 Graphing utility. See Grapher Gravity, acceleration due to, 167 Growth constrained, 258 exponential, 266 inhibited, 258 logistic, 252 restricted, 252 unrestricted, 252 Growth curve, 254 Growth factor, 254

H Half-angle identity, 429–430 Half-life, 266 Half-plane, 565 Harmonic motion, 390–391 Head minus tail (HMT) rule, 457 HEINE, HEINRICH EDUARD (1821–1881), 755 Heron’s formula, 444–445 Hidden behavior of graph, 72 Higher-degree polynomial function, 158 end behavior, 187–189 local extremum, 187 zeros of, 201, 210–211 HIPPARCHUS OF NICAEA (190–120 B.C.E.), 320 HIPPOCRATES OF CHOIS (c 470–410 B.C.E.), 320 Histogram, 697 Horizontal asymptote, 92, 93 Horizontal component of vector, 460 Horizontal Line Test, 122 Horizontal stretch and shrink, 133–135, 352 Horizontal translation, 129–131 cofunction identities, 406 of quadratic function, 164–165 of sinusoid, 352 of tangent function, 362 Hubble Space Telescope, 607 humidity, relative, 224 Hyperbola, 602, 620 asymptotes of, 603 branches of, 602 center and focus, 602, 605 focal axis, 602, 605 parametric equations for, 606, 609 Pythagorean relation, 603, 605 reflective property of, 607 semiconjugate axis, 603, 605 semitransverse axis, 603, 605 standard form of, 603, 605 transformation of, 605 vertices, 602, 605 Hyperbolic cosine function, 273

Hyperbolic sine function, 273, 293 Hyperbolic tangent function, 273 Hyperbolic trig functions, 273, 420 Hyperboloid of revolution, 607 Hypocycloid, 484 Hypotenuse, 329 Hypothesis, 807. See also Logic

I Identity, trigonometric. See Trigonometric identity Identity function, 99, 172 Identity matrix, 534 Identity property, 6 Imaginary axis, 503 Imaginary axis of complex plane, 503 Imaginary number, 49 Imaginary part of complex number, 49 Imaginary unit i, 49 Implications (Conditionals), 807. See also Logic Implicitly defined function, 115 Implied domain, 82 Increasing function on an interval, 86–87 Independent event, 661 Independent variable, 80 Index of radical, 779 Index of summation, 678 Indirect reasoning, 811. See also Logic Individual, 693 Induction. See Mathematical induction Inductive hypothesis, 689 Inductive reasoning, 690 Inductive step, 689 Inequality absolute value, 54, 240 addition property, 23 double, 25 graph of, 24–25 linear, 23–25 polynomial, 236 properties of, 23 quadratic, 55, 236 radical, 240 rational, 239 symbols, 3 system of, 566–567 Inferential statistics, 704 Infinite discontinuity, 85 Infinite geometric series, 683 Infinite limit, 760 Initial point of a vector, 457 Initial point of directed line segment, 457 Initial side of angle, 338 Initial value of a function, 161 Instantaneous rate of change, 740 Instantaneous velocity, 738 Integer, 2 Integrable function, 750 Integral, definite, 770 Intercepted arc, 322 Interest annual percentage rate (APR), 310 annual percentage yield (APY), 307 annuity, 308 compounded annually, 307

943

compounded continuously, 306 compounded k times per year, 304 value of investment, 307 Interest rate, finding, 306 Intermediate Value Theorem, 190 Interquartile range, 707 Intervals of real numbers, 4 bounded, 4 closed, 4 endpoints, 4 half-open, 4 open, 5 unbounded, 5 Invariant under rotation, 617 Inverse additive, 6 multiplicative, 6 Inverse Composition Rule, 124 Inverse exponential function, 292 Inverse function, 122–123 Inverse matrix, 534 Inverse property, 6 Inverse Reflection Principle, 123 Inverse relation, 121–122, 124 Inverse trigonometric function cosine, 380 graph of, 378, 380–381 sine, 378 tangent, 380–381 Inverse variation, 174 Invertible linear system, 550 Irrational numbers, 2 Irrational zeros of polynomial, 201 Irreducible quadratic function, 51, 214 Isosceles right triangle, 330

J Joint variation, 184 Joule, 471 Jump discontinuity, 85

K KEPLER, JOHANNES (1571–1630), 179, 580 Kepler’s First Law, 595 Kepler’s laws, 179, 287–288, 601, 607 Kepler’s Third Law, 179 Knot, 390 kth term of a sequence, 670

L Latus rectum of a parabola, 590 Law of cosines, 442, 469 Law of detachment (modus ponens), 811. See also Logic Law of sines, 434–438 Leading coefficient of polynomial, 185, 784 of polynomial function, 158 Leading term of polynomial function, 185 Leading Term Test for polynomial end behavior, 188 Leaf of stemplot, 694 Least common denominator (LCD), 793 compound fraction, 793 rational expressions, 793

6965_SE_Index_pp939-949.qxd

944

1/25/10

10:54 AM

Page 944

INDEX

Least-square lines, 173 Left-hand limit, 758 Left rectangular approximation method (LRAM), 750 LEIBNIZ, GOTTFRIED WILHELM (1646–1716), 80, 732, 737, 742, 749, 750, 755 Leibniz notation, 742 Lemiscate curve, 499 Length of arc, 457 Length (magnitude) of directed line segment, 456 Length (modulus) of vector, 456 LEONARDO OF PISA (c 1170–1250), 675 Like terms of polynomials, 784 Limaçon curve, 497–499 Limit, 85. See also Asymptote; Continuous function; End Behavior of continuous function, 758 infinite, 760 at infinity, 748–749, 760 informal definition of, 755 left-hand, 758 one-sided, 759 at a point, 758 properties of, 756–757 right-hand, 758 two-sided, 759 Limit at a point, 738 Limit at infinity, 748–749, 760 Limit to growth, 258 Limits of infinite sequences, 671 Line graph, 699 Line of best fit, 144, 173 linear correlation, 145 linear regression, 145 modeling data, 144 scatter plot, 12 Line of sight, 388 Line of symmetry, 90–91, 166 Line of travel, 321 Linear combination of unit vectors, 460 Linear correlation, 145, 161 Linear (first-degree) equation in three variables, 632 Linear Factorization Theorem, 210, 213–214 Linear factors and partial fraction decomposition, 558–560 of polynomial, 210 Linear first-degree equation, 21 in 3 or more variables, 544 equivalent, 21 forms of, 30 general form of, 30 graphing, 31 parallel and perpendicular lines, 31–32 point-slope form, 29, 30 slope-intercept form, 30 slope of a line, 28 in x, 21 in x and y, 30 y, and z, 629 y-intercept, 29 Linear function, 159. See also Linear first-degree equation Linear inequality double, 25

equivalent, 24 graph of, 566 number line graph, 25 in x, 23 in x and y, 568 Linear programming, 567 Linear regression line, 145 Linear regression model, 145 Linear speed, 323 Linear system of equations, 629 Gaussian elimination, 544–545 graphs of, 522 matrices, 546–547, 550–551 substitution, 520 triangular form, 544–545 Lithotripter, 597, 600 Local extremum, 90 Local (relative) maximum and minimum, 90, 187 Local maximum value of a function, 90 Local minimum value of a function, 90 Locus (loci), 582 Logarithmic function. See also Exponential function with base 10, 275 with base b, 274 common logarithm, 275 end behavior, 274, 278 graph of, 274, 277–278 inverse rule, 274, 275 modeling data with, 145 natural logarithm, 277 one-to-one rule, 292 power rule, 283 product rule, 283 properties of, 274 quotient rule, 283 regression equation, 145 solving equations, 293 transformations of, 278–279 Logarithmic re-expression of data, 298–300 Logic biconditional, 808 chain rule, 812 compound statement, 803 conclusion, 807 conditionals (implications), 807 conjunction, 803 direct reasoning, 811 disjunction, 803 existential quantifiers, 802 hypothesis, 807 implications (conditionals), 807 indirect reasoning (modus tollens), 811 law of detachment (modus ponens), 811 logically equivalent, 804 modus ponens (law of detachment), 811 negation, 801 quantifiers, 801 statement, 801 tautology, 809 truth table, 803 universal quantifiers, 801 valid reasoning, 809–810 Logically equivalent, 804. See also Logic Logistic curve, 258

Logistic function, 258–259 Logistic growth and decay function, 258 Logistic regression, 145 Long division of polynomials, 197 Long-range navigation (LORAN), 608 LORAN (long-range navigation) system, 608 Lower bound of function, 89 for real zeros, 202 test for real zeros, 202 LRAM (left rectangular approximation method), 750 Lurking variable, 728

M Magnitude of real number. See Absolute value of vector, 456, 458 Main diagonal of matrix, 534 Major axis of ellipse, 592 Mapping, 80 Mathematical induction, 687, 690 extended principle of, 692 inductive hypothesis, 687 principle of, 688 Mathematical model, 64, 76 Mathematical modeling, 64 Matrix, 530. See also Solving systems of equations addition and subtraction of, 530–531 determinant of, 535–536 elementary row operations, 546–548 equation, 550–551 invertible, 550 multiplication of, 532–534 multiplicative identity, 534 multiplicative inverse, 534 nonsingular, 534 properties of, 537 reduced row echelon form, 548 row echelon form, 547, 548 scalar multiple of, 531 singular, 534 square, 530 symmetric, 541 transpose of, 534 Maximum, local, 90 Maximum r-value, 496 Maximum sustainable population, 268 Mean, 704–705 Measure of angle, 338 Measure of center, 704–705 Measure of spread, 707 Median, 704–705 Midpoint formula in Cartesian space, 630 coordinate plane, 15 number line, 14 Minimum, local, 90 Minor axis of ellipse, 592 Minor of a matrix, 535 Minute (of angle), 320 Mirror method (inverse), 123 Mode, 704, 706 Modeling data, 144–145

6965_SE_Index_pp939-949.qxd

1/25/10

10:54 AM

Page 945

INDEX

with cubic functions, 145 with exponential functions, 145 line of best fit, 144 with logarithmic functions, 145 with logistic functions, 145 with power functions, 145 with quadratic functions, 145 with quartic functions, 145 with sinusoidal functions, 145 Modified boxplot, 710 Modulus of complex number, 504 Modus popens (law of detachment), 811 Modus tollens (indirect reasoning), 811. See also Logic Monomial function, 175–176 Monomials, 784 Motion harmonic, 390–391 projectile, 57, 479–480 vertical free-fall, 167 Multiplication principle of counting, 643 Multiplication property of equality, 21 of inequality, 23 Multiplicative identity of algebraic expressions, 6 of complex number, 51 Multiplicative identity matrix, 534 Multiplicative inverse of complex number, 51 of matrix, 534 of real number, 6 Multiplicity of zero of function, 189

N n factorial, 644 n-set, 644, 647 NAPIER, JOHN (1550–1617), 274 Nappe of a cone, 580 Natural base e, 256–258 Natural exponential function, 255 Natural logarithm, 277 Natural logarithmic function, 278 Natural number, 2 Nautical mile, 324 Navigation, 321 nCr, 646 NDER ƒ1a2, 766 Negation, 801. See also Logic Negative angle, 338 Negative assocation, 717, 718 Negative correlation, 717, 718 Negative linear correlation, 161 Negative number, 3 NEUMANN, JOHN VON (1903–1957), 658 NEWTON, ISAAC (1642–1727), 167, 732, 737, 749, 755 Newton-meter, 471 Newton’s Law of Cooling, 296 NINT1ƒ1x2, x, a, b2, 767 Non-identity, disproving, 416–417 Nonlinear system of equations, 521–522 Nonrigid transformation, 129 Nonsingular matrix, 534 Normal curve, 711

Normal distribution, 711 nPr, 645 nth power, 7 nth root, 779 of complex number, 508 of unity, 508 Number line, real, 3 Number system, real. See Real number Numerator, 791 Numerical derivative, 766 Numerical derivative of ƒ at a, 766 Numerical integral, 767 Numerical model, 64

O Objective function, 567 Observational study, 720 Obtuse triangle, 435 Octants, 629 Odd degree, polynomial function of, 214 Odd-even identity, 406–407 Odd function, 91, 406–407 One-sided limit, 759 One-to-one function, 122 exponential, 292 logarithmic, 292 trigonometric, 378 Open interval, 5 Operations for equivalent equations, 22 Opposite, algebraic, 6 Order of magnitude, 294 Order of matrix, 530 Ordered pair and inverse relation, 121–122 of real number, 12 and relation, 115 solution of equation, 21 Ordered set, 3 Ordinary annuity, 308 Origin coordinate plane, 12 number line, 3 space, 629 Orthogonal vectors, 469 Outcomes, equally likely, 658 Outliers, 709

P Parabola, 99, 477, 581, 621 axis of, 164, 582 directrix, 581 focal chord of, 590 focal length, 583 focal width, 583 focus of, 581, 583 latus rectum of, 590 reflective property of, 586–587 standard form of, 583 transformations of, 586 translations of, 584–586 with vertex, 583, 584 Paraboloid of revolution, 586 Parallel lines, 31–32 Parallelogram, 458 Parameter, 110, 475, 704

945

Parameter interval, 475 Parametric curve, 475 Parametric equation, 475 for circle, 475 eliminating the parameter, 476–477 graphing, 475–476 and inverse relations, 121–122 for a line in Cartesian space, 475–476 for lines and line segments, 477–478 motion along a line, 478–479 motion in the plane, 479–481 motion of Ferris Wheels, 481 Parametrically defined relation, 119 Parametrization of curve, 475 Partial fraction, 557 decomposition, 557 with quadratic factors, 558–560 Partial sums, sequence of, 682 PASCAL, BLAISE (1623–1662), 653, 658 Pascal’s triangle, 653–654 Perfect square trinomial, 787 Perihelion, 595 Period, 345 of sinusoid, 353 of tangent function, 361 Periodic function, 345 Permutations, 644–646 Perpendicular lines, 31–32 pH, 295 Phase shift, 352 Picture graph, 693 Pie chart, 693 Piecewise-defined function, 104 Placebo, 721 Plane in space, 632 Plane trigonometry. See Trigonometric function Planetary orbit, 624–625 Platonic solid, 446 Point of discontinuity, 85 Point of intersection, 45 Point-plotting method, 31 Point-slope form, 29, 30 Polar axis, 487 Polar coordinate system, 487 coordinate conversion equations, 490 finding distance, 491 graphing, 487 Polar coordinates, 487 Polar distance formula, 493 Polar equation cardioid, 498 and conics, 621–622 equivalent, 494 graph of, 495–496 limaçon curve, 497–499 rose curve, 496–497 standard form, 622 symmetry, 494–495 Polar form of a complex number, 504 Pole, 487 PÓLYA, GEORGE (1887–1985), 70 Polynomial, 784. See also Rational function adding and subtracting, 784 binomial products, 784 characteristic, 543

6965_SE_Index_pp939-949.qxd

946

1/25/10

10:54 AM

Page 946

INDEX

Polynomial (continued) degree of, 784 factoring, 214, 785–788 inequality, 236 multiplying, 784 prime, 785 standard form, 784 terms of, 185, 784 Polynomial function, 158 combining with trigonometric function, 369 complex conjugates, 51 cubic, 185, 187 degree of, 158 division algorithm, 197 end behavior, 187–189 Factor Theorem, 199 Fundamental Theorem of Algebra, 210 higher-degree, 187–189 Intermediate Value Theorem, 190 interpolation, 192 leading coefficient and term, 185, 784 Linear Factorization Theorem, 210 linear factors, 210 long division, 197 of odd degree, 214 quadratic, 164–166 quartic, 185 quotient, 197 with real coefficients, 214 regression, 145 remainder, 197 Remainder Theorem, 198 synthetic division, 199–201 zeros of, 201, 210–211 Polynomial interpolation, 192 Population, 711 Population growth, 260–261 Position vector, 456 Positive angle, 338 Positive association, 717 Positive correlation, strong, 717 Positive linear correlation, 161 Positive number, 3 Power function, 174 identifying the graph, 178 regression equation, 145 Power-reducing identity, 428 Power rule for logarithms, 283 Present value of annuity, 309 Prime polynomial, 785 Principal nth root, 779 Principle of mathematical induction, 688 Probability, 659 binomial, 665 conditional, 663 distribution, 659 of event, 658 expected value, 668 function, 659 independent events, 661 sample space, 658 strategy, 660 Probability simulations, 724–725 Problem-solving process, 70–71 linear regression line, 145

modeling data, 144 Product binomial, 785 of complex numbers, 50, 505 of functions, 110 of scalar and vector, 458 of a sum and difference, 786 of two matrices, 532 Product rule for logarithms, 283 Projectile motion, 57, 479–480 Projection of u onto v, 470 Proof, 132, 469 Properties of absolute value, 13 of additive inverse, 6 of algebraic expressions, 6 of dot product, 467 of equality, 21 of exponents, 7 of inequalities, 23 of limits, 756–757 of logarithms, 274, 276, 277, 283 of matrices, 537 of radicals, 780 Proving an identity disproving non-identity, 416–417 strategies, 413–417 Pseuo-random numbers, 723 Pythagorean identities, 405 Pythagorean relation ellipse, 591, 594 hyperbola, 603, 605 Pythagorean Theorem, 14

Q Quadrantal angle, 342, 346 Quadrants of plane, 12 measure of angle, 339 trigonometric functions, 339 Quadratic equation, 41, 51 Quadratic (second-degree) equation in two variables, 581, 612 Quadratic factors, irreducible, 214 Quadratic formula, 42 Quadratic function, 164–166 graphing, 164–166 irreducible over the reals, 51, 214 line of symmetry, 164 modeling data, 145 nature of, 166 opening upward or downward, 164 regression equation, 145 transformations of, 164–166 vertex form of, 164–165 Quadratic inequality, 55 Quadric surfaces, 633 Quantifiers, 801. See also Logic Quantitative variable, 693 Quartic polynomial function, 185, 187 regression, 145 Quartile, 707 Quotient of complex numbers, 51, 505 of functions, 110

Quotient identity, 404 Quotient polynomial, 197, 557 Quotient rule for logarithms, 283

R Radar tracking system, 491 Radian, 321 arc length formula, 322–323 degree-radian conversion, 322 in navigation, 321 Radical, 779 properties of, 780 rational exponents, 781 rationalizing the denominator, 781 simplifying, 780, 782 Radical inequality, 240 Radicand, 779 Radio signal, 608 Radioactive decay, 266 Radius of circle, 15 of sphere, 631 Random number tables, 723 Randomization, 721 Randomness, 719–720, 723–724 Range of data, 707 of function, 80, 81 interquartile, 707 inverse function, 122 Rational equation, 228 extraneous solution of, 229 solving, 228 Rational exponent, 781 Rational expression, 791. See also Algebraic expression; Real number adding, 784 compound, 793 domain of, 791 multiplying, 792 reducing, 791 Rational function, 218. See also Polynomial end behavior asymptote, 221 horizontal asymptote, 221 and partial fractions, 557 reciprocal function, 219 slant asymptote, 221 transformation of, 219 vertical asymptote, 221 x- and y-intercepts, 221 Rational inequality, 239 Rational number, 2 Rational zeros of polynomial, 201 Rationalizing the denominator, 781 Re-expression of data, 287, 298–300 Real axis of complex plane, 503 Real number, 2. See also Algebraic expression; Rational expression absolute value, 13 bounded intervals, 4 coefficients of polynomial, 185 distance and midpoint formulas, 13–15 inequalities, 3 integers, 2 intervals of, 4

6965_SE_Index_pp939-949.qxd

2/2/10

12:30 PM

Page 947

INDEX

irrational numbers, 2 natural and whole numbers, 2 order of, 3 positive and negative, 3 trigonometric functions of, 344 Real number line, 3 Real part of complex number, 49 Real zeros of polynomial, 201. See also Zero of a function lower bound test, 202 rational and irrational zeros, 201 upper bound test, 202 Reciprocal, algebraic, 6 Reciprocal function, 219 Reciprocal identities, 404 Reciprocal transformation, 219 Rectangular coordinate system. See Cartesian coordinate system Recursive formula, 254 Recursively defined sequence, 670 Reduced form of fraction, 791 Reduced row echelon form, 548 Reduction formula, 424 Reference triangle, 341, 346 Reflecting telescope, 607 Reflections across axes, 131–133 of quadratic function, 164 through a line, 131 Reflective property of a hyperbola, 607 Reflective property of a parabola, 586–587 Reflective property of an ellipse, 597–598 Reflexive property, 21 Regression analysis, 163 cubic, 145 exponential, 145 line, 144 linear, 145 logistic, 145 natural logarithmic, 145 power, 145 quadratic, 145 quartic, 145 selecting a model, 144 sinusoidal, 145 Relation, 115 inverse, 121–122 parametrically defined, 119 Relative humidity, 224 Relative maximum and minimum. See Local (relative) maximum and minimum Relevant domain, 82 Remainder polynomial, 197 Remainder Theorem, 198 Removable discontinuity, 84–85 Repeated zeros, 190 Replication, 722 Residual, 173 Resistant measure of center, 705 Resolving the vector, 460 Response bias, 721 Response variable, 642 Restricted growth, 252

Richter scale, 290, 295 RIEMANN, GEORG BERNHARD (1826–1866), 750 Riemann sum, 750, 770 Right circular cone, 580 Right-hand limit, 758 Right-handed coordinate frame, 629 Right rectangular approximation method (RRAM), 750 Right triangle, 329. See also Solving triangles; Trigonometric function; Trigonometric identity determining, 333, 388 isosceles, 330 solving, 333, 388 Rigid transformation, 129 Root, 70. See also Radical; Real zeros of polynomial; Zero of a function of complex number, 508–510 cube root, 779 nth root, 508, 779 of unity, 508 Root mean square deviation, 710 Rose curve, 496–497 Rotation formulas, 538, 614 Rotation method (inverse), 124 Rotation of axes, 614 Rotation of conic, 615–616 Row echelon form, 547, 548 Row operations, elementary, 546–548 Row subscript (matrix), 530 RRAM (right rectangular approximation method), 750 Rule of Signs, Descartes’, 208

S Sample convenience, 721 random numbers in, 723–724 Sample space, 658 Sample survey, 720 Scalar, 458, 531 Scalar multiple of a matrix, 531 Scalar multiple of a vector, 458 Scatter plot, 12 Scientific notation, 8 Secant function, 329, 363 acute, 329 any angle, 340 graph of, 363 special angles, 330 Secant line, 740 Second (of angle), 320 Second-degree (quadratic) equation in two variables, 581, 612 Second-degree equation in x and y, 581, 637 Second quartile, 707 Segment, of circle, 450 Semiconjugate axis of a hyperbola, 603, 605 Semimajor axis of ellipse, 592, 594 Semiminor axis of ellipse, 592, 594 Semiperimeter of triangle, 445 Semitransverse axis of a hyperbola, 603, 605 Sequence, 670 arithmetic, 672–674

947

convergence/divergence of, 671 end behavior, 671 explicitly defined, 670 Fibonacci, 675 finite and infinite, 670 geometric, 672–674 limit of, 671 of partial sums, 682 Sequence (continued) recursively defined, 670 and series, 682 Series, 681–682 arithmetic, 679 geometric, 682 infinite geometric, 683 sum of, 682 summation notation, 678 Set builder notation, 2 Sign chart, 237, 240 Similar geometric figures, 329 Simple harmonic motion, 390–391 Simplifying radicals, 780, 782 Simulation, probability, 724–725 Sine function, 329, 350 acute angles, 329 any angle, 340 cofunction identities, 406 graph of, 350 harmonic motion, 390–391 inverse, 379 law of sines, 434–438 period of, 350 special angles, 330 of sum and difference identity, 423–424 Singular matrix, 534 Sinusoid, 352. See also Trigonometric function; Trigonometric identity 68-95-99.7 rule, 712 16-point unit circle, 346 amplitude of, 352 combining transformations of, 371–372 frequency of, 353 harmonic motion, 390–391 maximum and minimum, 352 period, 353 phase shift, 352, 353 regression, 145 sums and differences, 371–372 Skewed left (distribution), 709 Skewed right (distribution), 709 Slant asymptote, 221 Slant line, 159 Slope-intercept form, 30 Slope of a line, 28 Solution approximate. See Approximate solution of equation in x, 21, 31 of equation in x and y, 565 extraneous, 229 of inequality in x, 23 of inequality in x and y, 565 of system of equations, 520 of system of inequalities, 567 Solution set of an inequality, 23

6965_SE_Index_pp939-949.qxd

948

1/25/10

10:54 AM

Page 948

INDEX

Solving algebraically, graphically, numerically, 76–77 Solving equations, 40–45, 68–71 absolute value, 45 algebraically, 56–57 completing the square, 41 discriminant, 47 exponential, 292–293 extracting square roots, 41 factoring, 43 graphically, 40 intersections, 45 quadratic formula, 42 rational, 228 trigonometric, 408–409, 430–431 x-intercepts, 40 Solving inequalities, 23 absolute value, 54 double, 25 Solving systems of equations elimination, 522–524 Gaussian elimination, 544–545 graphically, 522 intersections, 522 inverse matrices, 550–551 matrices, 546–547 nonlinear, 521–522 substitution, 520 Solving triangles, 333, 435–437 ambiguous case, 435–437 areas, 444 law of cosines, 442–447 law of sines, 434–438 number of triangles, 442 right, 333, 388 Speed, 184, 461 Sphere, 631 Spiral of Archimedes, 499 Split-stem plot, 696 Spread of data, 707 Square of difference, 785 of sum, 785 Square matrix, 530, 535–536 Square root, 41, 779 Square root function, 100, 179 Square system of equations, 550 Square viewing window, 31 Squaring function, 172 Standard deviation, 710 Standard form of complex number, 49 of polar equation of conic, 622 of polynomial, 185, 784 of quadratic equation, 164 Standard form equation of circle, 15 of conic (polar), 622 of ellipse, 591, 594 of hyperbola, 602, 603, 605 of parabola, 583 of quadratic function, 164 of sphere, 631 Standard position of angle, 329, 338 Standard representation of a vector, 456

Standard unit vector, 460, 467, 633 Statement, 801. See also Logic Statistics, 704 boxplot, 709 categorical, 693 descriptive, 704 five-number summary, 707–708 histogram, 697 inferential, 704 mean, 704–705 median, 704–705 misuses of, 717–725 mode, 704, 706 split-stem plot, 696 stemplot, 694 time plot, 699 weighted mean, 707 Statute mile, 324 Stem, 694 Stem-and-leaf plot, 694 Stemplot, 694 Step function, 101 Stretch and shrink, 133–135 quadratic function, 164 sinusoid, 352 tangent function, 361 Strong association, 718 Strong positive correlation, 717 Subjects, 721 Subtraction, 6 Sum of functions, 110 identity, 421–425 perfect square trinomial, 787 of sinusoids, 371–372 of two cubes, 787 Sum of series, 682 finite arithmetic, 679 finite geometric, 680 infinite geometric, 683 partial sums, 682 Sum of vector, 458 Sum-to product formulas, 452 Summation notation, 678 Sungrazer, 600 Supply curve, 525 Symmetric difference quotient, 766 Symmetric distribution, 709 Symmetric matrix, 541 Symmetric property, 21 Symmetry about the origin, 90–91, 494 about x-axis, 90, 494 about y-axis, 91, 494 of polar graphs, 494–495 Synthetic division of polynomials, 199–201 System of equations, 520. See also Solving systems of equations System of inequalities, 566–567

T Tail-to-head representation, in vector operations, 458 Tangent function, 329, 361 acute angles, 329

any angle, 340 graph of, 361 inverse, 380–381 special angles, 330 Tangent line, 738–739 Tangent line of a function, 739 Tangent line of a parabola, 590 Tangent of a sum or difference of angles, 424 Tautology, 809. See also Logic Terminal point of a directed line segment, 457 Terminal point of a vector, 457 Terminal side of angle, 338 Terms of polynomial, 185 Terms of sequence, 670 Tetrahedron, 446 Third quartile, 707 Three-dimensional Cartesian coordinate system (Cartesian space), 629 Three-dimensional space, 629 Time plot, 699 Tower of Hanoi, 687–688 TRACE function, 101 Transcendental function, 252 Transformation, 129 of circles parametrically, 485 combining, 135–136 of ellipse, 594 of exponential functions, 255–256 of hyperbola, 605 of logarithmic function, 278–279 nonrigid, 129 of parabola, 584 of quadratic function, 164–166 of reciprocal function, 219 reflection, 131 rigid, 129 of sinusoid, 354–355 stretch and shrink, 133–135 translation, 129–131 Transitive property of equations, 21 of inequalities, 23 Translation of axes, 613–614 of parabola, 584–586 of quadratic function, 164–166 of sinusoid, 354 of tangent function, 361 Translation formulas, 614 Transpose of a matrix, 534 Transverse axis of hyperbola, 603 Treatments, 721 Tree diagram, 662 Triangle. See also Right triangle; Solving triangles acute, 435 area of, 444 obtuse, 435 Pascal’s, 653–654 reference, 341 Triangular form of linear system, 544–545 Triangular number, 656 Trichotomy property, 4

6965_SE_Index_pp939-949.qxd

2/2/10

2:48 PM

Page 949

INDEX

Trigonometric equation factoring, 409 using identities, 430–431 Trigonometric expression factoring, 407 simplifying, 407–408 sinusoid, 352 Trigonometric form of a complex number, 504 Trigonometric function, 329, 338, 342. See also Cosecant function; Cosine function; Cotangent function; Secant function; Sine function; Tangent function of acute angles, 329 amplitude, 352 of any angle, 340 combining with inverse, 382–383 combining with polynomial, 369 and complex numbers, 504 domain, 351, 361, 364 even and odd, 406–407 frequency, 352 inverse, 380–384 one-to-one, 380–381 phase shift, 352 properties of, 351, 361, 364 quadrantal angle, 342, 346 range, 351, 361, 364 of real numbers, 344 reference triangle, 341 and right triangles, 329 signs of, 341 sinusoids, 352, 354 of special angles, 330 using calculator, 331–332 and vectors, 460 Trigonometric identity, 404 in calculus, 417 cofunction, 406 domain of validity, 404 double-angle, 428 half-angle, 429–430 odd-even, 406–407 perfect square, 787 power-reducing, 428 proof strategies, 413–417 proving, 413–416 Pythagorean, 405 quotient, 404 reciprocal, 404 sum and difference, 421–425 Trinomials, 784 Truth table, 803. See also Logic Two-body problem, 624 Two-dimensional vector, 456–458 Two-sided limits, 759

U Unbounded interval, 5 Undercoverage bias, 721 Union of two sets, 55 Unit circle, 344 16-point, 346

roots of unity, 508 and sine function, 344 and trigonometric functions, 344 Unit vector, 459–460, 633 Universal quantifiers. See Logic Unrestricted growth, 252 Upper bound of function, 89 test for real zeros, 202

V Valid reasoning, 809–810. See also Logic Value of annuity, 308 Value of investment, 307 Variable, 5 categorical and quantitative, 693 confounding, 721 dependent and independent, 80 explanatory and response, 642 lurking, 728 Variance, 710 Vector, 456, 633 addition and scalar multiplication, 458, 633 and complex number, 504 component form of, 456, 633 components of, 456, 633 direction angle, 460 direction of, 456, 458 dot product, 467, 633 equal, 457, 633 head minus tail rule, 457 initial and terminal points of directed line segments, 457, 633 length/magnitude of, 456, 633 linear combination, 460 magnitude of, 458, 633 resolving, 460 trigonometric form of, 504 two-dimensional, 456–458 unit, 459, 633 zero, 456, 633 Vector form of equation for a line in Cartesian space, 457 Velocity, 461 average, 736 changing, 748–749 constant, 747 instantaneous, 738 VENN, JOHN (1834–1923), 662 Venn diagram, 661 Verbal description, functions from, 142–143 Vertex of an angle, 338 of ellipse, 591, 594 of hyperbola, 602, 605 of parabola, 164, 582 parabola with, 583, 584 of right circular cone, 580 Vertical asymptote, 92, 93, 221, 361 Vertical component of vector, 460

Vertical free-fall motion, 167 Vertical line test, 81 Vertical stretch and shrink, 133–135 of quadratic function, 164 of tangent function, 361 Vertical translation, 129–131 of quadratic function, 164 of sinusoid, 354 of tangent function, 361 Very weak negative association, 718 Viewing window, grapher, 31 slope, 36 square, 31 Voluntary response bias, 721 VON NEUMANN, JOHN (1903–1957), 658

W Weak negative correlation, 717 WEIERSTRASS, KARL (1815–1897), 755 Weight, 707 Weighted mean, 707 WESSEL, CASPAR (1745–1818), 503 Whispering gallery, 597 Whole numbers, 2 Work, 471 Wrapping function, 344

X x-axis, 12, 629 x-coordinate, 12, 629 x-intercept, 31, 69, 70, 221 xy-plane, 629 xz-plane, 629

Y y-axis, 12, 629 y-coordinate, 12, 629 y-intercept, 29, 221. See also Function yz-plane, 629

Z z-axis, 629 z-coordinate, 629 z-intercept, 629 ZENO OF ELEA (c 490–425 B.C.E.), 748 Zero factor property, 41, 69 Zero factorial, 645 Zero matrix, 532 Zero of a function, 69–70 complex conjugate, 51, 211–212 finding, 202, 212 higher-degree polynomials, 201, 212 multiplicity of, 189 rational and irrational, 201 real, 201 repeated, 190 Zero polynomial, 158–159 Zero vector, 456, 633 Zoom out, 188

949

Implementing the Common Core State Standards with Precalculus: Graphical, Numerical, Algebraic

Contents About This Guide .......................................................................................................2 Common Core State Standards For Mathematics............................................................3 Transitioning to the Common Core State Standards ........................................................4 Additional Content for Complete Coverage of the (+) Standards ......................................5 Expanding Section 6.6: Closeness and Betweenness in a Complex World ....................5 Expanding Section 9.3: Random Variables and Expected Value...................................5 The Standards for Mathematical Practices .....................................................................6 Common Core State Standards and Precalculus: Graphical, Numerical, Algebraic, 8th Edition ...............................................................................................11 Closeness and Betweenness in a Complex World ........................................................33 Random Variables and Expected Value.......................................................................37

About This Guide Pearson is pleased to offer this Guide to Implementing the Common Core State Standards as a complement to Precalculus: Graphical, Numerical, Algebraic, 8th edition. In this Guide, you will find information about the Common Core State Standards that can be useful as you look to implement the (+) standards in your precalculus course. On page 3, you’ll find an overview of the Common Core State Standards, including the recent history leading to their establishment, and a brief description of the Standards for Mathematical Content, with a focus on the (+) standards, and the Standards for Mathematical Practice. Author Dan Kennedy, Ph.D., contributed his thoughts (pages 4 and 5) on incorporating the Common Core State Standards into this precalculus course. He describes the additional content that has been added and explains why covering the (+) standards will make stronger mathematical thinkers. Kennedy also notes that he and the other authors have embedded in this textbook for many years the practices and habits of mind that the Standards for Mathematical Practice in the Common Core State Standards stress as essential to developing mathematical proficiency (pages 6–10). Pages 11 through 32 show Correlations of the (+) Standards for Mathematical Content and indicate where the standards are addressed throughout Precalculus: Graphical, Numerical, Algebraic. You’ll notice that in addition to all (+) standards, this course also covers many of the non (+) Standards for Mathematical Content, which are expected to be taught in Algebra 1, Geometry, and Algebra 2. Beginning on page 33 is the additional content written to ensure complete coverage of (+) standards in the course. Closeness and Betweenness in a Complex World is an extension of Section 6.6 written to inform the student on complex planes. Random Variables and Expected Value expands on Section 9.3 and the textbook’s coverage of probability. Each expanded section is followed by accompanying exercises. Answers to the exercises are on pages 47–49.

2

Common Core State Standards For Mathematics The Common Core State Standards Initiative is a state-led initiative coordinated by the National Governors Association Center for Best Practices (NGA Center) and the Council of Chief State School Officers (CCSSO), with a goal of developing a set of standards in mathematics and in English language arts that would be implemented in many, if not most states in the United States. The Common Core State Standards for Mathematics (CCSSM) were developed by mathematicians and math educators and reviewed by many professional groups and state department of education representatives of the 48 participating states. The members of the writing committee looked at state standards from high performing states in the United States and from high-performing countries around the world and developed standards that reflect the intent and content of these exemplars. The final draft was released in June 2010 after nearly 12 months of intense development, review, and revision. To date, over 40 states have adopted these new standards and are currently working to develop model curricula or curriculum frameworks based These standards identify on these standards.

the knowledge and skills students should gain throughout their K–12 careers so that upon graduation from high school, students will be college- or career-ready.

These standards identify the knowledge and skills students should gain throughout their K–12 careers so that upon graduation from high school, students will be college- or career-ready. The standards include rigorous content and application of knowledge through higher-order thinking skills.

The CCSSM consist of two interrelated sets of standards, the Standards for Mathematical Practice and the Standards for Mathematical Content. The Standards for Mathematical Practice describe the processes, practices, and dispositions that lead to mathematical proficiency. The eight standards are common to all grade levels, K–12, highlighting that these processes, practices, and dispositions are developed throughout one’s school career. A discussion of these standards is found on pages 6–9. The Standards for Mathematical Content are grade-specific for Kindergarten through Grade 8; at the high school level, the standards are not structured by course or grade; rather they are organized into six conceptual categories. Within each conceptual category are domains and clusters, which each consists of one or more standards. Most of the high school standards are meant to be covered by the end of three years of high school mathematics; a few standards, indicated with (+), represent more advanced topics and expand upon the content learned in the core high school curriculum. These are generally addressed in advanced high school mathematics courses, such as Precalculus, Advanced Statistics, or Discrete Mathematics. An overview of how the (+) standards are addressed in Precalculus: Graphical, Numerical, Algebraic is found on page 5.

3

Transitioning to the Common Core State Standards By Daniel Kennedy, Ph.D. Textbooks like this one evolve in many ways for many reasons, only some of which follow the design of the authors. The primary purpose of a book called precalculus, of course, is to prepare students for courses in calculus, and nine out of ten chapters of our book, Precalculus: Graphical, Numerical, Algberaic, are devoted to doing just that, in what we feel is the most pedagogically effective way. Chapter 9 on Discrete Mathematics, on the other hand, contains quite a bit of material that is not so important for studying calculus, but very important for coping with data in a quantitative world. In the last few decades, the attention of primary and secondary mathematics education has been shifting slowly but inexorably toward quantitative literacy and away from the classical focus on calculus and its attendant calculations. We have responded by adding more and more material to Chapter 9, all the time wondering whether teachers would have time to cover it in a typical school year. The need to meet every state’s individual standards requirements meant the inclusion of certain precalculus topics, and the exclusion of potentially more useful quantitative literacy material. That is why the adoption of Common Core State Standards seems to be a fine idea. Once we all agree on what ought to be taught, we can design our textbooks, our teacher development, and our assessments to ensure that students learn that content effectively. (The fact that they could already do this is one of the reasons for the remarkable success of the College Board's Advanced Placement courses over the years.) Eventually, we should be able to design more focused courses at every level and make the textbooks considerably smaller. This will, however, take time. For now, ironically, most textbooks will actually have to supplement their coverage –– as the country gets serious about quantitative literacy while remaining cautious about curricular reform. We were actually quite pleased that our book was already so consistent with the first set of Common Core State Standards; indeed, we only needed to expand the complex plane coverage in Section 6.6 and the treatment of probability theory in Section 9.3.

4

Additional Content for Complete Coverage of the (+) Standards Expanding Section 6.6: Closeness and Betweenness in a Complex World Most precalculus textbooks, ours included, have given scant attention to complex numbers, since they rarely come up in a first-year calculus course. They are algebraically important for understanding the Fundamental Theorem of Algebra, and they tie some important concepts together in DeMoivre's Theorem, but not much knowledge of the complex plane is required for those (primarily algebraic) applications. The Common Core State Standards, obviously looking ahead to courses beyond first-year calculus, have prescribed a somewhat deeper geometrical understanding of the complex plane for college-bound secondary students. This is easily provided with this brief supplement, which could be covered within one class period. As a nice bonus, students will understand why a "complex line" is impossible.

Expanding Section 9.3: Random Variables and Expected Value From the beginning, one of our strategies for keeping Chapter 9 from becoming too huge was to cover just enough probability theory to explain the statistical applications. (Remember: It's a precalculus book.) This led us to side-step the classical terminology of random variables and expected value, which the Common Core State Standards have now opted to include. Admittedly, it is much easier to explain some statistical concepts (like Bayesian decision-making) if one has access to all that terminology, so it is almost liberating to have it now available through this supplement. The Common Core vision is that many of the statistical concepts introduced in Chapter 9 will eventually be old news to precalculus students, making the more formal approach to probability theory an appropriate extension at this level. Note that the emphasis is still on the statistical applications, but the probability theory underlying them can be richer. How long it will take to teach this supplement will be heavily dependent on how much students already know about statistics. In fact, if students are lacking in statistical background, the teacher might want to incorporate examples from Sections 9.7 and 9.8 as needed.

5

The Standards for Mathematical Practices For us authors, the most gratifying part of the Common Core State Standards is the part labeled "Mathematical Practices." These standards give clear evidence that the authors of these standards wanted us to know that what we teach is only part of the goal. Equally important is how we teach it and why.

These standards give clear evidence that the authors of these standards wanted us to know that what we teach is only part of the goal. Equally important is how we teach it and why.

We would hope that anyone familiar with the layout of our book will take a look at the Mathematical Practices and demand to know how we got a copy of them ten years ago. In fact, these practices have been pillars of mathematics education reform for more than 20 years, and we very much wanted them to be reflected in our textbooks. In case you have not seen them, here are the eight practices that should be at the core of good mathematics education:

1

Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Our book is all about solving problems. We invoke George Polya's famous four-step process in the first section of the book. Our Examples model problem solving from every angle, and our Exercises range from the simple and computational to the rich and multi-representational.

6

2

Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Ideally, this is what it should mean to "do" mathematics, but many courses get bogged down in quantitative procedures and skills and miss the reasoning part. The Explorations in each section of our book get students to reason things out (abstractly and quantitatively) and thereby discover the procedures. Each set of Exercises includes problems (writing to learn) that reinforce the reasoning step, along with additional explorations and problems designed to extend the ideas.

3

Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Although we cannot guarantee that all teachers will use our books as we hope they will, our Explorations are designed to promote this kind of activity in every classroom. Indeed, we hope teachers will use this approach consistently in their teaching so that students will realize that the teacher is not the only one in the classroom who can initiate or sustain a learning experience.

7

4

Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

This is such an important goal of our book that we have chosen to begin and end our allimportant first chapter with sections on modeling. The very name of the book reflects the importance of understanding how to model the real world graphically, numerically, and algebraically. In each new edition of the book we update the numerical data so that students will see how the mathematics they are studying can be used to model the world they are living in right now.

5

Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

We doubt that any author team of any precalculus textbook anywhere has been more closely associated with educational technology, particularly graphing calculators, than we have. Not only have we been urging the use of technology in mathematics education for more than two decades, but we have also been vigilant about insisting on the wise use of technology while helpfully pointing out the pitfalls. The strategic use of appropriate tools has been a hallmark of our approach from the start.

8

6

Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

We suppose that any mathematical textbook would support this practice philosophically, but we feel strongly enough about it to invest some considerable ink into explaining it to the student. For example, the first section of the book discusses what it means to "solve" a problem and to "prove" that something is true mathematically, and we revisit those ideas in later Examples. It almost goes without saying that we address issues of calculator precision (and the limitations thereof), in context, throughout the book.

7

Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

We essentially used this practice as our guiding principle when reworking the order of topics in this book. Graphing calculators enabled us to expose students to the comparative study of functions before becoming immersed in their algebraic behavior, so we did. Students could thereby appreciate (and hopefully discuss) the structure of functions from the beginning of the course, thus raising to a new level the unifying principle of function first envisioned by Leonhard Euler in 1748. It is also a guiding principle of our book that students will understand function behavior in all its representations: graphical, numerical, algebraic, and verbal.

9

8

Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Every Example in our textbook is paired with an Exercise that allows students to affirm their understanding by employing this mathematical practice. Constructive learning is ideal when it works, but students can also learn by seeing examples, then looking for and expressing regularity by solving similar problems. This is also how homework plays a role, and we have much to say about homework in our book, including when calculator use is appropriate and when it is not. 

We hope that all of these Mathematical Practices will resonate strongly with teachers who have been using Precalculus: Graphical, Numerical, Algebraic. The more we can incorporate them into our daily teaching, the better it will be for our students. We expect some adjustments to be made to the Common Core State Standards in the near future regarding what topics should realistically be taught and when, but in our estimation the how and the why seem to be on pretty firm ground already.

10

Common Core State Standards

Precalculus: Graphical, Numerical, Algebraic, 8th Edition The following shows the alignment of Precalculus, Graphical, Numerical, Algebraic to the High School Standards for Mathematical Content in the Common Core State Standards. We have included all of the standards, both non (+) and (+), to help teachers understand the progression of concepts and skills in high school mathematics. However, the goal of Precalculus: Graphical, Numerical, Algebraic is to cover only those standards marked in red with a (+). You'll notice there are coverage gaps in some of the non (+) standards as they are meant to be studied before taking this course.

Key N-RN.1

N-CN.3 ( +) ★ TE TK ‘See related concepts and skills.’

non (+) standards that students study in Algebra 1, Geometry, and Algebra 2 (+) standards, the main focus of study in fourth-year courses, like precalculus modeling standards page references in the annotated Teacher’s Edition page references in the additional lessons of this Transition Kit indicates there is no direct instruction, but there is a lesson(s) that can be used to create instruction

You'll notice Geometry (+) standard G.GMD.2, which requires informal arguments using Cavelieri’s principle with respect to formulas for volume of a sphere and other solid figures, is not covered in this course. The authors believe formulas for volume should be taught in geometry courses and therefore this standard is covered in Pearson’s Geometry text.

Number and Quantity The Real Number System

N-RN

Extend the properties of exponents to rational exponents. N-RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

TE: 7, 781–783

N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

TE: 780–783

Use properties of rational and irrational numbers. N-RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

11

See related concepts and skills. TE: 2

Quantities

N-Q

Reason quantitatively and use units to solve problems. N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

TE: 66–67, 76 (#25–#28), 107 (#56), 126 (#34), 183 (#53), 193–194(#66–#68), 233 (#31), 263 (#58), 318, 397 (#36), 402, 517, 640, 733

N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.

TE: 250, 318, 402, 517, 640, 733

N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

TE: 250, 318, 402, 517, 640, 733

The Complex Number System

N-CN

Perform arithmetic operations with complex numbers. N-CN.1 Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.

TE: 49

N-CN.2 Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

TE: 49–50, 51, 52, 53, 62, 505–506, 511

N-CN.3 (+) Find the conjugate of a complex number; use conjugates

TE: 51, 53 (#33–#40), 62 (#80), 504–506, 511

to find moduli and quotients of complex numbers.

Represent complex numbers and their operations on the complex plane. N-CN.4 (+) Represent complex numbers on the complex plane in

TE: 503–505, 511, 515

rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

N-CN.5 (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

N-CN.6 ( +) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

12

TE: 49, 50, 51, 52, 53, 62, 505, 506, 511, 515 TK: 6.6.1 Also see related concepts and skills. TE: 49, 52, 503, 504, 511

Use complex numbers in polynomial identities and equations. N-CN.7 Solve quadratic equations with real coefficients that have complex solutions.

TE: 213, 215 (#27–#36), 247 (#49–#52)

N-CN.8 ( +) Extend polynomial identities to the complex numbers.

TE: 49–51

N-CN.9 (+) Know the Fundamental Theorem of Algebra; show that it

TE: 210, 211 215 (#21)

is true for quadratic polynomials.

Vector and Matrix Quantities

N-VM

Represent and model with vector quantities. N-VM.1 (+) Recognize vector quantities as having both magnitude

TE: 456–458, 633

and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

N-VM.2 (+) Find the components of a vector by subtracting the

TE: 456, 461, 464 (#29–#32)

coordinates of an initial point from the coordinates of a terminal point.

N-VM.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.

TE: 461–463, 465, 515 (#74, #75), 516 (#76, #77)

Perform operations on vectors. N-VM.4 (+) Add and subtract vectors.

N-VM.4.a Add vectors end-to-end, componentwise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

TE: 458–459, 464 (#13, #14)

N-VM.4.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

TE: 460–462, 465 (#49, #50)

N-VM.4.c Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

TE: 464 (#15, #18–#20), 514 (#1, #2)

13

N-VM.5 (+) Multiply a vector by a scalar.

N-VM.5.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).

TE: 459, 464 (#16)

N-VM.5.b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

TE: 458–459

Perform operations on matrices and use matrices in applications. N-VM.6 (+) Use matrices to represent and manipulate data, e.g., to

TE: 531, 533–534, 541

represent payoffs or incidence relationships in a network.

N-VM.7 (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

TE: 531, 540 (#11c–#16c), 573 (#1c, #2c)

dimensions.

TE: 530–534, 540 (#11– #28), 573 (#1–#8)

N-VM.9 (+) Understand that, unlike multiplication of numbers, matrix

TE: 537

N-VM.8 (+) Add, subtract, and multiply matrices of appropriate

multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

N-VM.10 ( +) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

N-VM.11 ( +) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

N-VM.12 ( +) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

14

TE: 532, 534–535, 537, 540, 541

See related concepts and skills. TE: 532, 540 (#15, #16, #23, #24) See related concepts and skills. TE: 535, 537–538

Algebra Seeing Structure in Expressions

A-SSE

Interpret the structure of expressions A-SSE.1 Interpret expressions that represent a quantity in terms of its context.★

A-SSE.1.a Interpret parts of an expression, such as terms, factors, and coefficients.

TE: 779–783, 784–790, 791–795

A-SSE.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity.

TE: 785–790

A-SSE.2 Use the structure of an expression to identify ways to rewrite it.

TE: 780–783, 784–790, 791–795

Write expressions in equivalent forms to solve problems A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★

A-SSE.3.a Factor a quadratic expression to reveal the zeros of the function it defines.

TE: 40, 46 (#1–#6), 785– 786, 789

A-SSE.3.b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

TE: 41–42, 46 (#13–#18), 166, 169 (#33–#38)

A-SSE.3.c Use the properties of exponents to transform expressions for exponential functions.

TE: 270, 305, 306, 310

A-SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.★

TE: 680–681, 685 (#37, #38), 731 (#75, #76)

Arithmetic with Polynomials and Rational Expressions

A-APR

Perform arithmetic operations on polynomials A-APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

TE: 784–785, 789 (#9–#18)

Understand the relationship between zeros and factors of polynomials A-APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

15

TE: 198–199, 204 (#13– #18), 246 (#29, #30)

A-APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

TE: 189–190 (#33–#42, #49–#52), 246 (#5–#8)

Use polynomial identities to solve problems A-APR.4 Prove polynomial identities and use them to describe numerical relationships.

TE: 785–788, 789, 790, 796

A-APR.5 (+ ) Know and apply the Binomial Theorem for the expansion

TE: 652–655, 656–657, 729 (#19–#26)

of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.

Rewrite rational expressions A-APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

TE: 791–792, 793–794, 795 (#33–#44, #63–#72)

A-APR.7 ( +) Understand that rational expressions form a system

TE: 792–793, 795 (#45–#62)

analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Creating Equations ★

A-CED

Create equations that describe numbers or relationships A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

TE: 47 (#59–#61), 71, 77 (#47), 230–232, 233, 234, 260–261, 263 (#55–#58)

A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

TE: 67, 71, 75 (#15), 76 (#24), 79 (#63), 92–93, 144, 147, 151 (#50, #51), 163, 167, 170 (#53), 171 (#58, #59, #62, #63), 172 (#67, #68), 220–223, 230, 268, 269, 297, 302 (#51, #52), 567–570, 571 (#31–#37), 572 (#38), 603–604

A-CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

TE: 92–93, 220–232, 332– 333 (#13–#22), 241–242, 260–261, 266–267, 268– 269, 567–570, 571 (#31– #37), 572 (#38), 603–604

A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

TE: 27 (#70–#73), 66 (Exploration 1), 76 (#22), 114, 140, 141, 305, 306

16

Reasoning with Equations and Inequalities

A-REI

Understand solving equations as a process of reasoning and explain the reasoning A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

TE: 22, 23, 40, 41, 228, 229, 230, 292, 293, 413, 414, 415, 416

A-REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

TE: 228, 229, 230, 231, 232, 233, 235

Solve equations and inequalities in one variable A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

TE: 22, 23, 24, 25, 26, 27, 61

A-REI.4 Solve quadratic equations in one variable.

A-REI.4.a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.

TE: 41–42, 46, 47 (#68)

A-REI.4.b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

TE: 40, 41, 42, 43, 46, 47

Solve systems of equations A-REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

See related concepts and skills. TE: 544–545

A-REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

TE: 520–521, 523, 524–535, 526, 527, 528

A-REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

TE: 526 (#2, #11–#14, #17, #18), 527 (#27, #28, #38, #39)

A-REI.8 (+ ) Represent a system of linear equations as a single matrix

TE: 546–547, 553 (#25–#28)

equation in a vector variable.

A-RE I.9 ( +) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

17

TE: 550–552, 554 (#49–#52)

Represent and solve equations and inequalities graphically A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

TE: 31, 36 (#27–#30), #37 (#51, #52), 69, 83, 87–88, 89, 91–92, 93–94, 95 (#21– #24, #29–#34), 96 (#63– #66), 99–101, 105, 106 (#1– #12), 107 (#53–#56), 108 (#64), 164–165, 169, (#13– #18), 176–179, 182 (#37– #42), 185, 193 (#9–#12), 219–220, 225 (#5–#10), 226 (#31–#36), 255, 256, 258, 259, 262 (#15–#30), 278, 279, 281 (#37–#58), 350, 351, 357 (#13–#28), 361– 362, 363–364, 365 (#1– #12), 366 (#13–#16)

A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★

TE: 520–522, 523, 524, 525, 526 (#13–#18), 527 (#35–#42), 529 (#65, #66)

A-REI.12 Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

TE: 565–566, 567 570, 571, 574, 575

18

Functions Interpreting Functions

F-IF

Understand the concept of a function and use function notation F-IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

TE: 80–84, 94, 95, 96, 97, 99–105, 106, 107, 152, 153

F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

TE: 83–84. 95 (#9–#16, #17–#20), 97 (#78), 107 (#56), 140–147, 148 (#15– #20), 150 (#49), 153 (#11– #18), 154 (#59–#64)

F-IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

TE: 670–675, 676, 677, 730 (#47–#62)

Interpret functions that arise in applications in terms of the context F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★

TE: 159–163, 164–168, 170, 171, 172, 176–181, 183, 184, 185–192, 194, 195, 218–224, 227, 248, 249, 250, 252–261, 262, 263, 265–270, 271, 272, 273, 277–280, 281, 282, 290, 291, 350–356, 358, 359, 360, 361–364, 367

F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.★

TE: 81–82, 95, 99–102, 105, 106, 140–141, 176–181, 219, 224, 255, 259, 266, 267, 268, 269, 270, 278, 350, 356, 361, 362, 363, 364

F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★

TE: 160–161, 170 (#53), 171 (60), 172 (#67), 173 (#78)

19

Analyze functions using different representations F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★

F-IF.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima.

TE: 66–68, 99, 103–104, 106, 159–163, 164–168, 169, 170, 171, 172, 173, 246, 247, 248, 249, 250

F-IF.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

TE: 100, 101, 103, 104– 105, 106, 107, 108, 179

F-IF.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

TE: 158–168, 169, 170, 171, 172, 173, 185–192, 193, 194, 195, 196, 246, 247, 248, 249, 250

F-IF.7.d (+) Graph rational functions, identifying

TE: 218–224, 225, 226, 227, 247

zeros and asymptotes when suitable factorizations are available, and showing end behavior.

F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

F-IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

TE: 252–258, 260, 262, 266–268, 274, 277–279, 281, 314, 315, 318

F-IF.8.a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

TE: 40–43, 47 (#1–#6, #13– #18), 145, 164–168, 169, 171 (#61–#65), 246, 248, 249, 250

F-IF.8.b Use the properties of exponents to interpret expressions for exponential functions.

TE: 7, 253, 254, 260, 262 (#39, #40), 265–267, 270 (#1–#6), 271 (#29–#34), 272, 273

F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

20

TE: 90–92, 103–104, 105, 176, 179, 219, 255, 256, 259, 350, 351, 361, 362, 363, 364

Building Functions

F-BF

Build a function that models a relationship between two quantities F-BF.1 Write a function that describes a relationship between two quantities.★

F-BF.1.a Determine an explicit expression, a recursive process, or steps for calculation from a context.

TE: 254, 261 (#11, #12), 266, 267, 271 (#58, #59, #62, #63), 191, 248 (#83, #85, #86), 249 (#90, #93– #95), 395 (#27, #28), 396 (#29, #32), 670–671, 672– 673, 674–675, 676 (#1–#10, #21–#31),

F-BF.1.b Combine standard function types using arithmetic operations.

TE: 110, 116 (#1–#8), 117 (#9, #10)

F-BF.1.c (+) Compose functions.

TE: 111–114, 117, 118, 154

F-BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★

TE: 670–675, 676, 677

Build new functions from existing functions F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

TE: 90–92, 95 (#47–#54), 103–104, 105, 129–136, 137, 138, 139, 153 (#37– #40), 176, 179, 185, 219, 220, 255, 256, 258, 259, 350, 351, 352, 353, 355, 361

F-BF.4 Find inverse functions.

F-BF.4.a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.

TE: 121–125, 126, 127, 128

F-BF.4.b ( +) Verify by composition that one

TE: 124, 129 (#27–#32)

function is the inverse of another.

F-BF.4.c (+) Read values of an inverse function

TE: 123–124, 126 (#23–#26)

from a graph or a table, given that the function has an inverse.

F-BF.4.d ( +) Produce an invertible function from

TE: 123–124, 126 (#23–#26)

a non-invertible function by restricting the domain.

F-BF.5 (+ ) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

21

TE: 274–277, 281, 283, 285, 288, 293, 297, 298

Linear, Quadratic, and Exponential Models★

F-LE

Construct and compare linear, quadratic, and exponential models and solve problems F-LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.

F-LE.1.a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

See related concepts and skills. TE: 33 (Figure P.29(b)), 160– 161, 170 (#51, #52), 254– 255, 261 (#11, #12)

F-LE.1.b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

TE: 160–163, 170 (#53), 172 (#67, #68)

F-LE.1.c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

TE: 260–261, 262 (#51), 263 (#52–#58), 265–270, 271, 272, 273

F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

TE: 33–34, 37 (#45, #51), 163, 170 (#53), 172 (#67, #68), 266–268, 271 (#33, #34), 272

F-LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

TE: 258–261, 262 (#51–#56)

F-LE.4 For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

TE: 296–297 (Example 7), 301 (#49, #50)

Interpret expressions for functions in terms of the situation they model F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context.

22

TE: 258–261, 263 (#56), 269–270, 272 (#46), 273 (#58), 316 (#76, #94)

Trigonometric Functions

F-TF

Extend the domain of trigonometric functions using the unit circle F-TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

TE: 320–323, 325, 327, 399

F-TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

TE: 338–346, 347–349, 400 (#17–#28)

F-TF.3 (+) Use special triangles to determine geometrically the values

TE: 341–342, 343, 344, 346, 347 (#25–#36), 400 (#29–#32)

of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.

F-TF.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

TE: 345–346, 348 (#49– #52), 350, 351, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 400 (#61–#66)

Model periodic phenomena with trigonometric functions F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.★

TE: 354, 355, 358 (#57–#60)

F-TF.6 (+) Understand that restricting a trigonometric function to a

TE: 378–384, 385

domain on which it is always increasing or always decreasing allows its inverse to be constructed.

F-TF.7 (+) Use inverse functions to solve trigonometric equations that

TE: 378–384, 386 (#55)

arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.★

Prove and apply trigonometric identities F-TF.8 Prove the Pythagorean identity sin2( θ) + cos2( θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

TE: 345, 405–406, 410

F-TF.9 ( +) Prove the addition and subtraction formulas for sine,

TE: 421–425, 426, 427, 428, 450 (#5)

cosine, and tangent and use them to solve problems.

23

Geometry Congruence

G-CO

Experiment with transformations in the plane G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

TE: 13, 15, 31, 32, 338, 816, 817, 819, 825, 826

G-CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G-CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G-CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G-CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Understand congruence in terms of rigid motions G-CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G-CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

See related concepts and skills. TE: 329

G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

See related concepts and skills. TE: 329

24

Prove geometric theorems G-CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

See related concepts and skills. TE: 19 (#37, #54, #55, #59), 20 (#65), 39 (#71)

G-CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

TE: 16–17

Make geometric constructions G-CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

TE: 513 (#78), 589 (#71, #72), 611 (#69)

G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Similarity, Right Triangles, and Trigonometry

G-SRT

Understand similarity in terms of similarity transformations G-SRT.1 Verify

experimentally the properties of dilations given by a center and a scale factor:

G-SRT.1.a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. G-SRT.1.b The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

25

G-SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

See related concepts and skills. TE: 196 (#85), 329

G-SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

See related concepts and skills. TE: 329

Prove theorems involving similarity G-SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

TE: 19 (#54, #55, #59)

G-SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

TE: 196 (#85)

Define trigonometric ratios and solve problems involving right triangles G-SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

TE: 329–331, 355

G-SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.

TE: 331 (Exploration 2)

G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★

TE: 334, 336 (#61–#65)

Apply trigonometry to general triangles G-SRT.9 (+) Derive the formula A = 1/2 ab sin(C) for the

TE: 337 (#78)

area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

G-SRT.10 ( +) Prove the Laws of Sines and Cosines and use

TE: 434–438, 442

them to solve problems.

G-SRT.11 ( +) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

Circles

TE: 437–438, 439, 440, 445–446, 448, 449

G-C

Understand and apply theorems about circles G-C.1 Prove that all circles are similar.

See prerequisite concepts and skills. TE: 15

26

G-C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

TE: 320

G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

G-C.4 (+) Construct a tangent line from a point outside a given circle to the circle.

See related concepts and skills. TE: 39 (#70)

Find arc lengths and areas of sectors of circles G-C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

TE: 328 (#71)

Expressing Geometric Properties with Equations

G-GPE

Translate between the geometric description and the equation for a conic section G-GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

TE: 15–16, 19 (#41–#48), 799

G-GPE.2 Derive the equation of a parabola given a focus and directrix.

TE: 584–585, 588 (#15, #16, #23, #24), 799

G-GPE.3 (+) Derive the equations of ellipses and hyperbolas

TE: 592–593, 594, 599 (#23, #24, #33. #34), 604–605, 609 (#23–#25, #35, #36), 799

given the foci, using the fact that the sum or difference of distances from the foci is constant.

Use coordinates to prove simple geometric theorems algebraically G-GPE.4 Use coordinates to prove simple geometric theorems algebraically.

TE: 13–17, 19 (#37–#40, #45–#48, #53–#55, #59), 20 (#65–#70)

G-GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

TE: 31–32, 37 (#41–#43)

G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

See related concepts and skills. TE: 14–15, 17 (#23–#28)

G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★

TE: 14, 20 (#66), 797

27

Geometric Measurement and Dimension

G-GMD

Explain volume formulas and use them to solve problems G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

See related concepts and skills. TE: 27 (#72), 142–143, 153 (#67), 231, 241, 243 (#60), 248 (#87), 797

G-GMD.2 (+) Give an informal argument using Cavalieri’s

See page 14 for comments.

principle for the formulas for the volume of a sphere and other solid figures. G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★

TE: 27 (#72), 142-143, 149 (#33), 155 (#67), 191, 194-195, 231, 241, 243 (#59, #60), 248 (#87), 797

Visualize relationships between two-dimensional and three dimensional objects G-GMD.4 Identify the shapes of two-dimensional crosssections of three-dimensional objects, and identify threedimensional objects generated by rotations of twodimensional objects.

TE: 580-581, 591, 602

Modeling with Geometry

G-MG

Apply geometric concepts in modeling situations G-MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★

TE: 47 (#61), 114, 140, 142, 143, 191, 195, 207, 231, 243, 248, 324, 328, 542, 598

G-MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).★

TE: 207

G-MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★

TE: 59 (#46), 140–143, 191, 195, 231, 243

28

Statistics and Probability Interpreting Categorical and Quantitative Data

S-ID

Summarize, represent, and interpret data on a single count or measurement variable S-ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

TE: 697–698, 701 (#9–#11), 703 (#31)

S-ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

TE: 704–712, 713–716

S-ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

TE: 698, 703 (#31), 704–712, 713– 716

S-ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

TE: 711–712, 715 (#41, #42)

Summarize, represent, and interpret data on two categorical and quantitative variables S-ID.5 Summarize categorical data for two categories in twoway frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies).

TE: 697, 698, 701 (#7, #8), 702 (#23, #24)

Recognize possible associations and trends in the data. S-ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

S-ID.6.a Fit a function to the data;

TE: 33–34, 37, 67, 75, 76, 79, 82, 144, 147, 151, 163, 167–168, 170, 172, 179–181, 183

S-ID.6.b Informally assess the fit of a function by plotting and analyzing residuals.

TE: 33–34, 37, 67, 75, 76, 79, 82, 144, 147, 151, 163, 167–168, 170, 172, 179–181, 183

S-ID.6.c Fit a linear function for a scatter plot that suggests a linear association.

TE: 33–34, 37, 67, 75, 76, 79, 82, 144, 163, 170, 172

use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

29

Interpret linear models S-ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

TE: 29–30, 33, 34, 37, 38, 67, 75, 79, 163, 170, 172 (#67)

S-ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.

TE: 146, 162, 163, 167, 179, 717– 719, 725

S-ID.9 Distinguish between correlation and causation.

TE: 162, 728 (#53)

Making Inferences and Justifying Conclusions

S-IC

Understand and evaluate random processes underlying statistical experiments S-IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

TE: 704, 719–720, 723–724, 726, 727, 728, 732

S-IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation

TE: 719–728, 730, 731, 732

Make inferences and justify conclusions from sample surveys, experiments, and observational studies S-IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

TE: 719–728, 730, 731, 732

S-IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

TE: 720–721, 726 (#7–#12), 727 (#33, #34), 732 (#113, #114)

S-IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

TE: 721–723, 724–725, 726, 727, 728

S-IC.6 Evaluate reports based on data.

TE: 695, 696, 697, 699–700, 701, 702, 703, 707, 708, 709, 710, 712, 733

30

Conditional Probability and the Rules of Probability

S-CP

Understand independence and conditional probability and use them to interpret data S-CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

TE: 661–662, 667 (#27, #28)

S-CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

TE: 661, 667 (#33, #34)

S-CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

TE: 663–664, 667 (#31, #32)

S-CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

Probability concepts are taught in Section 9.3. TE: 658–669

S-CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

TE: 661–663, 666, 667

Use the rules of probability to compute probabilities of compound events in a uniform probability model S-CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

Conditional probability is introduced in Section 9.3. TE: 663

S-CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

TE: 662

S-CP.8 ( +) Apply the general Multiplication Rule in a

TE: 663, 667 (#31, #32)

uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

S-CP.9 ( +) Use permutations and combinations to compute probabilities of compound events and solve problems.

31

TE: 658, 661, 666 (#5, #6), 667 (#33, #34)

Using Probability to Make Decisions

S-MD

Calculate expected values and use them to solve problems S-MD.1 ( +) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

TE: 723–724, 727 (#33–#36) TK: 9.3.1, 9.3.2, 9.3.3, 9.3.7, 9.3.8

TE: 668–669 (#61, #62) TK: 9.3.3, 9.3.4, 9.3.5, 9.3.6, 9.3.7, 9.3.8

S-MD.2 ( +) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

S-MD.3 ( +) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

S-MD.4 ( +) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

TE: 659–660, 668 (#57, #61), 669 (#62) TK: 9.3.2, 9.3.3

TE: 659–660, 668 (#57, #61), 669 (#62) TK: 9.3.6, 9.3.7, 9.3.8

Use probability to evaluate outcomes of decisions S-MD.5 ( +) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

S-MD.5.a Find the expected payoff for a game of chance.

TE: 668–669 (#61, #62) TK: 9.3.3, 9.3.4, 9.3.5

S-MD.5.b. Evaluate and compare strategies on the basis of expected values.

TE: 668–669 (#61, #62) TK: 9.3.6, 9.3.7

S-MD.6 ( +) Use probabilities to make fair decisions (e.g.,

TE: 723–724, 727 (#33, #34)

drawing by lots, using a random number generator).

S-MD.7 ( +) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

32

TE: 664, 667 (#34, #35, #38), 727– 725, 732 (#113, #114) TK: 9.3.3, 9.3.4, 9.3.5, 9.3.6, 9.3.7

Closeness and Betweenness in a Complex World An extension of Section 6.6 The real number line makes it easy to visualize closeness and betweenness in the world of real numbers, partly because the line represents an ordering of the real numbers, increasing from left to right along the line. The complex number system has some algebraic advantages (for example, we can factor any quadratic polynomial), but it has the disadvantage of not being orderable along a "complex number line." EXAMPLE 1 Proving that a Complex Number Line is Impossible Show that all three of these inequalities lead to contradictions according to the algebraic properties of order: i = 0, i > 0, i < 0. SOLUTION Suppose i = 0. Multiply both sides by i to get –1 = 0, a contradiction. Suppose i > 0. Multiply both sides by i to get –1 > 0, a contradiction. Suppose i < 0. Multiply both sides by i (remembering to switch the inequality because this time we are multiplying by a negative number) to get –1 > 0, a contradiction! This shows that you can't even put i on the same line with the real numbers and preserve the algebraic properties of order, so there can be no complex number line. Fortunately, we can still use the complex plane to understand closeness, as we hope you will discover in the following exploration. EXPLORATION 1 Measuring Closeness in the Complex Plane Which number is closer to 3 + 2i : 2 + 3i or 3 + 4i ? 1. Graph the three numbers in the complex plane and answer the question graphically. 2. For real numbers, we measure the distance from x to y by x ! y . Does this appear to work for complex numbers? (Be sure to use the definition of absolute value in Section 6.6.) We hope you concluded in the exploration above that the distance between two complex numbers x and y in the complex plane is, conveniently, x ! y , just as it is for distance between two real numbers on the real line. In fact, we can make this a definition. DEFINITION Distance Between Two Complex Numbers The distance between the complex numbers x and y is x ! y . That is, if x = a + bi and y = c + di , then the distance between them is

(a ! c) + (b ! d)i = (a ! c)2 + (b ! d)2 .

33

The fact that the concept of "closeness" can be extended to the (unordered) complex numbers is more important than it might seem to you now. Calculus is explained by limits, and limits are explained by closeness. The main intent of a course called "precalculus" is to prepare you to study the calculus of real-valued functions, but if you go on to study the calculus of complex-valued functions you will revisit the connections you have made here. The concept of "betweenness" does not extend as nicely, because the very notion of betweenness should imply that the numbers are, in fact, lined up. One related concept that does extend naturally, however, is that of the mean of two numbers. Notice that the mean of two real numbers is represented on the real line by the midpoint of the segment between them:

x

y

x+y 2

Conveniently, the mean of two complex numbers is also represented in the complex plane by the midpoint of the segment between them:

y x+ y 2

x This follows from the midpoint formula for two points in the plane. That is, if x = a + bi and

y = c + di , then

x + y (a + c) + (b + d)i ! a + c " ! b + d " = =# $+# $i . 2 2 % 2 & % 2 &

Although the picture is now two-dimensional, we would still say that the points on the segment connecting x and y "lie between" x and y. We can also extend this idea to other points in the complex plane, as the following example will show.

34

EXAMPLE 2 Extending the Concept of Betweenness a. Use an absolute value inequality to describe the set of real numbers that lie between 3 and 15. b. If the same absolute value inequality were to be used in the complex plane, what numbers would lie between 3+ 2i and 9 + 10i ? SOLUTION a. The mean of 3 and 15 is 9, which is 6 units away from each number. The set of numbers between 3 and 15 is described by x ! 9 < 6 . b. The mean of 3 + 2i and 9 + 10i is 6 + 6i, which is

(6! 3)2 + (6! 2)2 = 5 units away

from each number. The inequality z ! (6+ 6i ) < 5 describes the interior of a circle of radius 5 in the complex plane around the point 6 + 6i.

This extends the algebraic and geometric concepts of betweenness in a natural way, but how should we describe it? It would be misleading to say that every point in the circle "lies between" 3 + 2i and 9 + 10i, as that should really describe the points on the connecting segment, even in higher dimensions. A more appropriate connection can be made using the concept of distance. The inequality in (a) describes the set of numbers on the real line whose distance is less than 6 units from the mean. Geometrically, their graph is an open line segment centered at the mean. The inequality in (b) describes the set of numbers in the complex plane whose distance is less than 6 units from the mean. Geometrically, their graph is the interior of a circle centered at the mean. We say that the points inside the circle form a "neighborhood" around the complex number 6 + 6i, just as the points in the open segment form a "neighborhood" around the real number 9. Neighborhoods will play an important role when you encounter limits of complex functions and limits of functions of several variables in future mathematics courses.

35

Exercises for “Closeness and Betweenness in a Complex World,” an extension of Section 6.6 1. Find the distance between 4+ 3i and 1+ 5i . 2. Which is closer to 8+ 3i , 2+ i or 12! 2i ? 3. a. Accurately plot the complex numbers A = !2! 4i , B = 1+ 1i and C = 3+ 4i . b. Does B appear to lie on the line segment connecting A and C? c. Find the distance between each pair of complex numbers in part (a). d. What has to be true about the distances to ensure that B lies on the line segment between A and C? 4. Find a such that the distance between P = a ! 7i and Q = !4! 9i is 6 units. 5. Find the midpoint between !8+ 10i and 4+ 3i . 6. a. Write an absolute value inequality to express the neighborhood of all complex numbers z that are closer to 6! i than is the number !2+ 14i . b. Write a brief geometric description of the neighbor expressed in 6a.

36

Random Variables and Expected Value An extension of Section 9.3 EXPLORATION 1 The Game Show Audience Suppose a TV game show host offers everyone in the audience the choice of three deals: 1. Pick an envelope from five in his hand. The envelopes contain a ten dollar bill, a twenty dollar bill, a fifty dollar bill, a hundred dollar bill, and a check for five thousand dollars. 2. Choose one of three suitcases. Two are empty and the third contains 240 twentydollar bills. 3. Take $500 with no strings attached. Which deal would you take? Which deal do the show sponsors hope you will take? If you are an optimist, you might try for the biggest payoff and hope for the best. An optimist would probably choose deal #1 and hope to win $5000. If you are a pessimist, you might choose deal #3, which guarantees a better payoff ($500) than the worst-case scenarios in the other two deals. The show sponsors, however, who need to pay off a large number of choosers, should consider the probabilities of the various payoffs. On the average:

•

they will lose (1/5)($10) + (1/5)($20) + (1/5)($50) + (1/5)($100) + (1/5)($5000)= $1036 per person for those who choose deal 1; they will lose (2/3)($0) + (1/3)($4800) = $1600 per person for those who

•

choose deal 2; they will lose $500 per person for those who choose deal 3.

•

So they should hope for an audience of pessimists who will play it safe and take deal #3. They also should hope that people will not opt for deal #2, which has the largest expected payoff. In fact, the audience would come out ahead if they all secretly conspired to pick deal #2 and split their total earnings equally! Not many of us will be lucky enough to face an offer like the one above, but companies face these kinds of decisions all the time. The Bayesian strategy uses probabilities to predict the optimal choice in the long run. The Bayesian choice in Exploration 1, for example, would be deal #2. To set a context for studying these Bayesian choices, we need some definitions. DEFINITION Random Variable A random variable is a function that assigns a real number value to every outcome in a sample space. 37

It is convenient to denote random variables with capital letters and the individual values assumed by a random variable with subscripted lower-case letters. Thus X might be a random variable that takes on values {x1, x2, K , xn} and Y might be a random variable that takes on values

{y1,

y2, K , yn}. We can then use the notation P (X ) to denote a

probability function (Section 8.3) that assigns probabilities to the values of a random variable. We can describe the probability distribution with a table, as in Section 8.3. EXAMPLE 1 Probability Distribution of a Random Variable Let X be the random variable that gives the sum of the numbers on the top faces when two fair, 6-sided dice are rolled. Make a table showing the probability distribution for X. SOLUTION The values of X come from the sample space of all possible sums: {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. The probability distribution is shown in Table 1. X P(X)

2

3

4

5

6

7

8

9

10

11

12

1/36

2/36

3/36

4/36

5/36

6/36

5/36

4/36

3/36

2/36

1/36

Table 1: Probability distribution for the total on two fair dice.

Since probability functions of random variables are real-valued functions of real numbers, we can graph them in the coordinate plane. The most useful visualization is given by a histogram (Section 9.7) in which the heights of the rectangles are probabilities rather than frequencies. A histogram for the probability function in Example 1 is shown below. Note that we put the values of X in list L1 and the corresponding probabilities in list L2 .

38

MARGIN NOTE

Seeing the Mean If you look at the histogram of the probability distribution, you should be able to see why the mean would be 7. You can estimate the mean of any probability distribution by visualizing the balancing point of its histogram along the horizontal axis, and that is easy to do with a symmetric distribution like this.

Like frequency distributions, probability functions have means and standard deviations. We can find them on a calculator as shown below:

In the case of a probability distribution, we give the mean a special name to suggest its significance in probability modeling. DEFINITION Expected Value of a Random Variable If X is a random variable with probability function P(X), the mean of the probability distribution is also called the mean of X (denoted µ X ) or the expected value of X (denoted E(X)). Computationally, the expected value is the sum of the values of X n

times their respective probabilities: E (X ) = µ X = ! xk P (xk ) . k=1

EXAMPLE 2 Finding Expected Value For $5, a carnival huckster will let you draw a card at random from a fair deck of 52. He will pay you a dollar times the value of your card (ace = 1), but zero for a face card. What is the expected value of the payoff? Would you pay $5 to play this game? SOLUTION

1 1 1 1 1 1 1 1 1 1 3 !1+ ! 2+ ! 3+ ! 4+ ! 5+ ! 6+ ! 7+ ! 8+ ! 9+ !10+ ! 0 13 13 13 13 13 13 13 13 13 13 13 55 = " 4.23 13

E (X ) =

The expected value might not influence your decision to play or not play, but it is very important to the huckster, who can expect to make an average of 77 cents per game in the long run by charging $5 to play it You might get lucky playing once or twice, but the expected value shows that if you play this game for a long time at $5 per play, you can expect to lose an average of 77 cents per game. An important random variable in mathematical modeling is the random variable that counts the number of successes in n independent trials of an experiment with two possible outcomes. This random variable has a binomial probability distribution (as discussed in Section 9.3).

39

EXAMPLE 3 Expected Value and Binomial Distributions A fair coin is flipped five times. What is the expected number of heads? Interpret the expected value in the context of this example. SOLUTION If X is the random variable that counts the number of heads on five tosses, then X takes on the values {0, 1, 2, 3, 4, 5} . Recall that the probabilities can be found by expanding the polynomial ( p + q)5 , where p is the probability of heads and q is the probability of tails (in this case, p = q = 1/2 ).

( p + q)5 = 1p5 + 5p4q + 10p3q2 + 10p2q3 + 5pq4 + 1q5 1 5 10 10 5 1 = + + + + + 32 32 32 32 32 32 The expected value is E (X ) =

1 5 10 10 5 1 ! 0+ !1+ ! 2+ ! 3+ ! 4+ ! 5 = 2.5. 32 32 32 32 32 32

This does not mean that we "expect" 2.5 heads when we toss the coin 5 times, as that is impossible. It does mean that if we perform the 5-toss experiment many times, the mean (or average) number of heads will be around 2.5. It is no coincidence that the expected value turned out to be half the number of tosses; in fact, that is the value you probably expected (pun unavoidable)! What number of heads would you expect on five tosses if the coin were weighted with the probability of heads equal to 3/5? If you would expect three, you would be right. We can state the general case as a theorem.

THEOREM

Expected Value for a Binomial Distribution If X counts the number of successes on n independent trials of a binomial experiment with probability p of success, then E (X ) = np .

EXAMPLE 4 A Chimpanzee Takes Organic Chemistry A chimpanzee is shown 100 questions from a multiple-choice test in Organic Chemistry. As each question appears on a screen, the chimpanzee presses one of four buttons to indicate the selected answer (A, B, C, or D). What is the chimpanzee's expected score? SOLUTION If we assume that the chimpanzee is unfamiliar with the material, the test amounts to 100 independent trials of a binomial experiment with probability ¼ of success. The expected number of correct answers is 100(1/4) = 25. Remember also that the expected value is really a mean, so a large class of chimpanzees would probably produce some scores higher than 25 (and, of course, some lower). An actual student taking this test should not conclude from a score of 25 that "Well, at least I showed that I knew 25% of the test." (See margin note.)

40

MARGIN NOTE The Guessing Penalty You may know that some standardized multiple-choice tests are scored with a "guessing penalty" that would bring the average chimpanzee's score down to zero. For a test with four options, as in Example 4, the formula would be R ! 13W for R right answers and W wrong answers. The chimp with the expected score of 25 would get an adjusted score of 25! 13 (75) = 0. Note that, under this grading system, a student who gets 25 right and leaves the other 75 blank would get an adjusted score of 25 and could justifiably claim to have known 25% of the test.

EXAMPLE 5 Playing the Lottery The payoff in your state lottery is up to 20 million dollars. To win, you must choose six numbers between 01 and 46 correctly. Optimistically, you purchase ten different lottery tickets at $2 each. What is the expected value of the game for you? SOLUTION The game rules will probably inform you that there are 9,366,819 possible lottery tickets of this type. You have ten of them, so your expected payoff in dollars is:

0!

9366809 10 + 20,000,000! " 21.35. 9366819 9366819

Unfortunately, you paid $20 for the tickets, bringing your expected profit down to $1.35. Probabilities in real life are often estimated rather than computed, but expected values can still guide Bayesian decision-making. EXAMPLE 6 The Warranty Extension For an extra $49 you can buy extended warranty protection on your new HDTV for three years. If the set requires maintenance during that time, it will be free. You estimate that there is a 5% chance that you might need a $300 repair within three years and a 1% chance that you will need a $500 repair, but the chances are 94% that you will need no repair at all. Based on expected values, should you purchase the extended coverage? SOLUTION If you DO purchase the coverage, your expected value (in dollars) is !49+ 0(0.94) + 0(0.05) + 0(0.01) = !49. If you DO NOT purchase the coverage, your expected value (in dollars) is 0+ 0(0.94) + (!300)(0.05) + (!500)(0.01) = !20. The Bayesian strategy would be to decline the extended warranty and hope for the best. (The optimist would agree, while the pessimist would buy the coverage and avoid the possibility of those potential big losses.)

41

In most statistical applications, the probabilities used for real-world random variables are empirical probabilities (inferred from data) rather than theoretical probabilities (determined by the laws of probability). Expected values are determined using the same formula in either case and can be equally useful in decision-making. Looking at Example 6 from the other direction, the company offering the extended warranty must estimate those repair probabilities as accurately as they can so that they will know what to charge for the protection. They can track the repair bills incurred by a large number of customers and use a frequency table to compute the probabilities empirically, as in Example 7. EXAMPLE 7 A sample of 500 customers nationwide shows that they incurred repair bills in the first three years as recorded in Table 2: Repair Bill 0 50 100 200 300 500

Frequency 460 20 8 5 5 2

Table 2: A frequency table of 500 repair bills.

If they charge each customer $29 for the extended warranty, what is their expected profit? SOLUTION We first approximate the probability distribution for the repair bills by dividing the frequencies by the sample size (500). The relative frequency table is shown in Table 3: Repair Bill 0 50 100 200 300 500

Probability 0.92 0.04 0.016 0.01 0.01 0.004

Table 3: A relative frequency table of 500 repair bills.

The expected value in dollars is:

0(0.92) + 50(0.04) + 100(0.016) + 200(0.01) + 300(0.01) + 500(0.004) = 10.60

If they charge $29 for the coverage, the expected profit per customer opting for the coverage is $29 – $10.60 = $18.40.

42

EXAMPLE 8 The Barber's Chair Bill the barber keeps a careful record of how long it takes him to cut the hair of 100 random customers during a typical week. The frequencies are recorded in Table 4: Minutes in Chair 8 9 10 11 12

Number of Customers 32 25 22 15 6

Table 4: Frequency table for the barber.

What is the mean number of minutes a customer spends in Bill's chair? If there are four people waiting when he opens his shop, what is the expected time it will take for Bill to give all four of them haircuts? SOLUTION We begin by dividing the frequencies by the total number of customers, thus creating the empirical probability distribution shown in Table 5: Minutes in Chair 8 9 10 11 12

Probability 0.32 0.25 0.22 0.15 0.06

Table 5: Empirical probability table for the barber.

Then E (X ) = 0.32(8) + 0.25(9) + 0.22(10) + 0.15(11) + 0.06(12)

= 9.38 minutes Since it takes (on average) 9.38 minutes per customer, the expected length of time for Bill to cut the hair of four customers is 4(9.38) = 37.52 minutes.

43

Exercises for “Random Variables and Expected Value,” an extension of Section 9.3 From the reading: 1. Make up an example of a random variable and specifically identify how it assigns numbers to outcomes in the sample space. 2. What is a probability distribution? 3. Describe two ways to display a probability distribution. 4. What is another name for the mean of a probability distribution? 5. What is a simple way to calculate the mean of a binomial probability distribution? 6. Describe the difference between empirical probabilities and theoretical probabilities. Applying the concepts: 7. A bag contains twenty numbered balls. Five of the balls have the number 1 on them. Three balls have the number 2 on them. Seven balls have the number 3 on them. Four balls are marked with a 4. One ball has a 5 on it. Let Y be the random variable that gives the number on a ball chosen at random from the bag. a. Show the probability distribution for Y in a table. b. Calculate the expected value for Y. 8. One hundred people were asked how many televisions are in their home. The histogram below shows the empirical probability distribution of the results of the survey. The top of each bar corresponds to an integer multiple of 0.05. Calculate the expected value. 0.25 0.15 0.05 1

2

3

4

5 6

7

9. Not all dice are six-sided. If two ten-sided dice with faces numbered one through ten are rolled, what is the expected value for the sum of the dice?

44

10. At a carnival, people pay The Great Flubini $2 each to try to guess their weight within five pounds. If he guesses successfully, no prize is given. If he is off from five to ten pounds, participants get a prize worth $5. If he is off by more than ten pounds, participants get a $10 prize. The Great Fubini is correct 72% of the time, off from five to ten pounds 21% of the time and off by more than ten pounds 7% of the time. If The Great Fubini guesses the weights of 300 people per day, on average, is the carnival making money or losing money? How much per day? 11. A circular spinner consists of eight identical sectors. The spinner is equally likely to land in any of the eight sectors. Four of the eight yield no payout at all if the spinner lands there. Three of the eight pay $6 if the spinner lands there. One sector pays $10. It costs $5 to spin the spinner once. Use expected value to determine if it is a wise idea to play this game. 12. A game of chance consists of paying $4 to randomly draw a card from a deck of 50. The table below shows your chances of each net outcome. For example, 40% of the cards say, “You lose.” and so your net result is losing the $4 you paid to play. Complete the table below so that the game is a fair game. This means the expected value is 0. X

–4

–3

?

2

4

P(X)

0.4

0.1

0.2

0.1

0.2

13. An unfair coin is one that is weighted so that the probability of getting heads does not equal the probability of getting tails. Suppose you have a coin weighted so it is likely to land on tails 60% of the time. Let X be the random variable that counts the number of tails obtained on four tosses. 4 a. Expand (t + h) .

b. If t is the probability of tails and h is the probability of heads calculate the value of the term 4t3h . What does the value represent? c. Find the expected number of tails, E(X), using the polynomial. d. Calculate the expected value using E (X ) = np . 14. Most people have dreamed of showing up to a final exam for a class they never attended. Marcel actually did this for his Mandarin language final exam, so he had to randomly guess on the 80-question multiple choice exam. Each question had 5 choices, A, B, C, D, or E, with only one answer being correct. Each question was worth one point. a. What was the expected score for Marcel? b. Explain why his actual score may have differed from his expected score.

45

15. From problem 14, suppose Marcel’s professor had imposed a “guessing penalty” of

1 point off the final score for each incorrect answer. Would the adjusted score 4

have been a more accurate representation of his lack of Mandarin knowledge? Explain.

16. A multiple choice test consists of 150 questions each with n choices, where n is an integer greater than or equal to two. Each correct answer earns one point. Find the appropriate “guessing penalty” in terms of n so that the expected score for randomly guessing on all questions is zero. 17. A three year extended warranty on a laptop computer is offered for $79. Consumer reports show that typically, 4% of owners incur a $200 repair in that time and 1% of owners incur a $300 repair in that time. Does the extended warranty have a positive expected value for the consumer who purchases it? Show the work that leads to your conclusion. 18. The records of 1000 randomly selected customers of the XTREM car insurance company showed the following repair payouts by the company in the last fiscal year. What was the expected value of a payout? Repair Payout (Dollars)

Number

0

726

1000

83

2000

46

3000

52

4000

63

5000

17

6000

13

19. From problem 18, if the average annual premium payment collected from a customer was $1600, what was the company’s expected income per customer for extended warranties that year? 20. Make up a game of chance involving payment to play and payouts, and compute the expected value of the game for the player.

46

ANSWER SETS Answers to the exercises for “Closeness and Betweenness in a Complex World” 1.

13 40 < 41.

2. 2+ i is closer because 3. a.

4

i

C

3 2 1

B R

!6

!5

!4

!3

!2

!1

1

2

3

4

5

6

!1 !2 !3

A

!4

b. Yes, B appears to lie on the line segment between A and C. c. AB = 34 , BC = 13 , AC = 89 d. AB + BC = AC (So B is not on the line segment between A and C.) 4. a = !4+ 4 2 or a = !4! 4 2 5. !2+

13 i 2

6. a. z ! (6! i ) < 17 b. The neighborhood includes all points within a circle of radius the complex number 6! i .

47

17 around

Answers to the exercises for “Random Variables and Expected Value” From the reading: 1. Answers will vary. One example: The number cars entering a parking garage every five minutes between 8am and 9am. The random variable is the set of 12 whole numbers you get corresponding to cars entering from 8 to 8:05, 8:05 to 8:10, etc. 2. A probability distribution is a pairing of each element in the set of values of the random variable with the probability of that value occurring. 3. A probability distribution may be displayed as a table of values or a histogram. 4. Another name for the mean of a probability distribution is the expected value. 5. The mean of a binomial probability distribution is found by multiplying the number of independent trials by the probability of success. 6. Empirical probabilities are probabilities inferred from collected data and theoretical probabilities are those determined by the laws of probability. Applying the concepts: 7. a. Y

1

2

3

4

5

P(Y)

5 20

3 20

7 20

4 20

1 20

b.

53 or 2.65 20

8. 3.8 9. 11 10. The expected value is !0.31 dollars which means, on average, the carnival is losing 31 cents per person who plays the game. The carnival is losing about $93 per day. 11. It is not wise to play the game. The expected value for the player is –1.5 dollars which means, on average, a player loses $1.50 per turn. 12. The missing value is 4.5. 13. a. t4 + 4t3h + 6t2h2 + 4th3 + h4 b. 0.3456 is the theoretical probability of getting exactly three tails and one head in four flips of the coin. c. E(X) = 2.4 d. E(X) = 2.4

48

14. a. 16 b. The expected value of 16 is the expected average score of all scores if the test was repeated many times with random guessing. Any one trial may result in a variety of different scores. 15. Yes. The penalty results in an expected value of zero. Since Marcel has never been to class, he probably knows no Mandarin at all.

1 points for each wrong answer. Expected number correct is 150 and n !1 n 150 150 ! " 150 . Solve #1$ %150$ number wrong is 150! & # (a) = 0 for a. n n ( ' n

16. Take off

17. If you purchase the coverage your expected value is –79 dollars. If you do not purchase the coverage your expected value is –11 dollars. It seems financially wise to not purchase the coverage. 18. $746 19. $854 per customer per year 20. Answers will vary.

49