College algebra graphs 3e raymond barnett

Preface vii Application Index xvi and Models 11-1 Using Graphing Calculators 2 1-2 Functions 21 1-3 Functions: Graphs and Properties 46 1-4 Functions: Graphs and Transformations 67 1-5 O

COLLEGE ALGEBRA: GRAPHS AND MODELS, THIRD EDITION

Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020 Copyright © 2009 by The McGraw-Hill Companies, Inc All rights reserved Previous editions © 2005, 2000 No part of this publication may be reproduced or distributed in any form or

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Library of Congress Cataloging-in-Publication Data

College algebra : graphs and models –– 3rd ed / Raymond A Barnett [et al.].

p cm.––(Barnett, Ziegler, and Byleen's precalculus series)

Includes index.

ISBN 978–0–07–305195–6 — ISBN 0–07–305195–0 (hard copy : alk paper)

1 Algebra––Textbooks 2 Algebra––Graphic methods––Textbooks I Barnett, Raymond A QA152.3.B37 2009

512.9––dc22

2007042212

www.mhhe.com

Raymond A Barnett, a native of and educated in California, received his B.A in

mathematical statistics from the University of California at Berkeley and his M.A inmathematics from the University of Southern California He has been a member of theMerritt College Mathematics Department and was chairman of the department for fouryears Associated with four different publishers, Raymond Barnett has authored or co-authored 18 textbooks in mathematics, most of which are still in use In addition tointernational English editions, a number of the books have been translated into Span-ish Co-authors include Michael Ziegler, Marquette University; Thomas Kearns, North-ern Kentucky University; Charles Burke, City College of San Francisco; John Fujii,Merritt College; and Karl Byleen, Marquette University

Michael R Ziegler received his B.S from Shippensburg State College and his M.S.

and Ph.D from the University of Delaware After completing postdoctoral work at theUniversity of Kentucky, he was appointed to the faculty of Marquette University where

he currently holds the rank of Professor in the Department of Mathematics, Statistics,and Computer Science Dr Ziegler has published more than a dozen research articles

in complex analysis and has co-authored more than a dozen undergraduate ics textbooks with Raymond Barnett and Karl Byleen

mathemat-Karl E Byleen received his B.S., M.A., and Ph.D degrees in mathematics from the

University of Nebraska He is currently an Associate Professor in the Department ofMathematics, Statistics, and Computer Science of Marquette University He has pub-lished a dozen research articles on the algebraic theory of semigroups and co-authoredmore than a dozen undergraduate mathematics textbooks with Raymond Barnett andMichael Ziegler

Dave Sobecki earned a B.A in math education from Bowling Green State University,

then went on to earn an M.A and a Ph.D in mathematics from Bowling Green He is

an Associate Professor in the Department of Mathematics and Statistics at Miami versity in Hamilton, Ohio He has written or co-authored five journal articles, elevenbooks and five interactive CD-ROMs Dave lives in Fairfield, Ohio, with his wife (Cat)and dogs (Large Coney and Macleod) His passions include Ohio State football, ClevelandIndians baseball, heavy metal music, travel, and home improvement projects

Uni-About the Authors

Barnett, Ziegler, Byleen and Sobecki’s Precalculus Series

College Algebra, Eighth Edition

This book is the same as Precalculus without the three chapters on trigonometry.

ISBN 0-07-286738-8, ISBN 978-0-07-286738-1

Precalculus, Sixth Edition

This book is the same as College Algebra with three chapters of trigonometry added.

The trigonometry functions are introduced by a unit circle approach

ISBN 0-07-286739-6, ISBN 978-0-07-286739-8

College Algebra with Trigonometry, Eighth Edition

This book differs from Precalculus in that College Algebra with Trigonometry uses

right triangle trigonometry to introduce the trigonometric functions

ISBN 0-07-331264-9, ISBN 978-0-07-331264-4

College Algebra: Graphs and Models, Third Edition

This book is the same as Precalculus: Graphs and Models without the three chapters

on trigonometry This text assumes the use of a graphing calculator

ISBN 0-07-305195-0, ISBN 978-0-07-305195-6

Precalculus: Graphs and Models, Third Edition

This book is the same as College Algebra: Graphs and Models with three additional

chapters on trigonometry The trigonometric functions are introduced by a unit circleapproach This text assumes the use of a graphing calculator

ISBN 0-07-305196-9, ISBN 978-0-07-305-196-3

College Algebra with Trigonometry: Graphs and Models

This book is the same as Precalculus: Graphs and Models except that the

trigono-metric functions are introduced by right triangle trigonometry This text assumes theuse of a graphing calculator

ISBN 0-07-291699-0, ISBN 978-0-07-291699-7

Preface vii Application Index xvi

and Models 11-1 Using Graphing Calculators 2

1-2 Functions 21

1-3 Functions: Graphs and Properties 46

1-4 Functions: Graphs and Transformations 67

1-5 Operations on Functions; Composition 84

1-6 Inverse Functions 99

Chapter 1 Review 119Chapter 1 Group Activity: Mathematical Modeling:

Choosing a Cell Phone Provider 126

Quadratic Functions 1272-1 Linear Functions 128

2-2 Linear Equations and Models 151

2-3 Quadratic Functions 172

2-4 Complex Numbers 190

2-5 Quadratic Equations and Models 206

2-6 Additional Equation-Solving Techniques 226

2-7 Solving Inequalities 241

Chapter 2 Review 258Chapter 2 Group Activity: MathematicalModeling in Population Studies 265

Cumulative Review Exercises

Chapters 1–2 267

Functions 2733-1 Polynomial Functions and Models 274

3-2 Polynomial Division 291

3-3 Real Zeros and Polynomial Inequalities 303

3-4 Complex Zeros and Rational Zeros of

Polynomials 3203-5 Rational Functions and Inequalities 336

3-6 Variation and Modeling 361

Chapter 3 Review 371Chapter 3 Group Activity: InterpolatingPolynomials 378

Functions 3814-1 Exponential Functions 3824-2 Exponential Models 3994-3 Logarithmic Functions 4164-4 Logarithmic Models 4304-5 Exponential and Logarithmic Equations 440Chapter 4 Review 462

Chapter 4 Group Activity: Comparing RegressionModels 456

Cumulative Review Exercises Chapters 3–4 457

Equations and Inequalities 4635-1 Systems of Linear Equations in Two

Variables 4645-2 Systems of Linear Equations in Three Variables 482

5-3 Systems of Linear Inequalities 4955-4 Linear Programming 510

Chapter 5 Review 522Chapter 5 Group Activity: Heat Conduction 526

6-1 Systems of Linear Equations: Gauss–JordanElimination 528

6-2 Matrix Operations 5476-3 Inverse of a Square Matrix 5656-4 Matrix Equations and Systems of LinearEquations 578

6-5 Determinants 5886-6 Properties of Determinants 5976-7 Determinants and Cramer’s Rule 604Chapter 6 Review 610

Chapter 6 Group Activity: Using Matrices toFind Cost, Revenue, and Profit 616

Cumulative Review Exercises

CHAPTER 7 Sequences, Induction, and

Probability 6217-1 Sequences and Series 622

7-2 Mathematical Induction 633

7-3 Arithmetic and Geometric Sequences 644

7-4 The Multiplication Principle, Permutations, and

Combinations 659

7-5 Sample Spaces and Probability 676

7-6 The Binomial Formula 696

8-2 Ellipse 723

8-3 Hyperbola 735

8-4 Systems of Nonlinear Equations 750

Chapter 8 Review 761

Chapter 8 Group Activity: Focal Chords 765

Cumulative Review Exercises

Chapters 7–8 766

VI

A-1 Algebra and Real Numbers A-2A-2 Exponents A-13

A-3 Radicals A-27A-4 Polynomials: Basic Operations A-36A-5 Polynomials: Factoring A-47A-6 Rational Expressions: Basic Operations A-58Appendix A Review *

Appendix A Group Activity: Rational and Irrational Numbers *

*Available online at www.mhhe.com/barnett

Graphing A-69B-1 Linear Equations and Inequalities A-70B-2 Cartesian Coordinate System A-82B-3 Basic Formulas in Analytic Geometry A-91

C-1 Significant Digits A-106C-2 Partial Fractions A-109C-3 Descartes’ Rule of Signs A-118C-4 Parametric Equations *

*Available online at www.mhhe.com/barnett

Student Answers SA-1

Enhancing a Tradition of Success

We take great satisfaction from the fact that more than 100,000 students have learned lege algebra from a Barnett Series textbook Ray Barnett is one of the masters of collegetextbook writing His central approach is proven and remains effective for today’s students

col-The third edition of College Algebra: Graphs and Models has benefited greatly

from the numerous contributions of new coauthor Dave Sobecki of Miami UniversityHamilton Dave brings a fresh approach to the material and many good suggestions

for improving student accessibility Every aspect of the revision focuses on making

the text more relevant to students, while retaining the precise presentation of the ematics for which the Barnett name is renowned

math-Specifically we concentrated on the areas of writing, worked examples, exercises,technology, and design Based on numerous reviews, advice from expert consultants,and direct correspondence with many users of previous editions, we feel that this edi-tion is more relevant than ever before We hope you will agree

Writing Without sacrificing breadth or depth or coverage, we have rewritten nations to make them clearer and more direct As in previous editions, the text empha-sizes computational skills, real-world data analysis and modeling, and problem solv-ing rather than theory

expla-Examples In the new edition, even more solved examples in the book provide ical solutions side-by-side with algebraic solutions By seeing the same answer result

graph-from their symbol manipulations and graph-from graphical approaches, students gain insight

into the power of algebra and make important conceptual and visual connections.Likewise, we added expanded color annotations to many examples, explaining thesolution steps in words Each example is then followed by a similar matched problemfor the student to solve Answers to the matched problems are located at the end ofeach section for easy reference This active involvement in learning while reading helpsstudents develop a more thorough understanding of concepts and processes

Exercises With an eye to improving student performance and to make the book

more useful for instructors, we have extensively revised the exercise sets We addedhundreds of new writing questions as well as exercises at the easy to moderate leveland expanded the variety of problem types to ensure a gradual increase in difficultylevel throughout each exercise set

Technology Although technology is employed throughout, we strive to balancealgebraic skill development with the use of technology as an aid to learning and prob-lem solving We assume that students using the book will have access to one of thevarious graphing calculators or computer programs that are available to perform thefollowing operations:

• Simultaneously display multiple graphs in a user-selected viewing window

• Explore graphs using trace and zoom

Preface

• Approximate roots and intersection points

• Approximate maxima and minima

• Plot data sets and find associated regression equations

• Perform basic matrix operations, including row reduction and inversionMost popular graphing calculators perform all of these operations The majority of thegraphing calculator images in this book are “screen shots” from a Texas InstrumentsTI-84 Plus graphing calculator Students not using that TI calculator should be able toproduce similar results on any calculator or software meeting the requirements listed.The proper use of such calculators is covered in Section 1-1

A Central Theme

In the Barnett series, the function concept serves as a unifying theme A brief look

at the table of contents reveals this emphasis A major objective of this book is thedevelopment of a library of elementary functions, including their important proper-ties and uses Employing this library as a basic working tool, students will be able toproceed through this book with greater confidence and understanding

Design: A New Book with a New Look

The third edition of College Algebra: Graphs and Models presents the subject in the

precise and straightforward way that users have come to rely on in the Barnett books, now updated for students in the twenty-first century The changes to the textare manifested visually in a new design We think the pages of this edition offer amore contemporary and inviting visual backdrop for the mathematics We are confi-dent that the design and other changes to the text described here will help to improveyour students’ enjoyment of and success in the course

text-Features

New to the Third Edition

An extensive reworking of the narrative throughout the chapters has made the

lan-guage less formal and more engaging for students

A new full-color design gives the book a more contemporary feel and will appeal

to students who are accustomed to high production values in books, magazines, andnonprint media

Even more examples now feature side-by-side solutions integrating algebraic and

graphical solution methods This format encourages students to investigate matical principles and processes graphically and numerically, as well as algebraically

mathe-The increased use of annotated algebraic steps, in small colored type, to walk

students through each critical step in the problem-solving process helps students low the authors’ reasoning and improve their own problem-solving strategies

fol-An fol-Annotated Instructor’s Edition is now available and contains answers to

exercises in the text, including answers to section, chapter review, and cumulativereview exercises These answers are printed in a second color, adjacent to correspon-ding exercises, for ease of use by the instructor

More balanced exercise sets give instructors maximum flexibility in assigning

homework We added exercises at the easy to moderate level and expanded the

ety of problem types to ensure a gradual increase in difficulty level throughout eachexercise set The division of exercise sets into A (routine, easy mechanics), B (moredifficult mechanics), and C (difficult mechanics and some theory) is no longer explicit

in the student edition of the text: the letter designations appear only in the AnnotatedInstructor’s Edition This change was made in order to avoid fueling students’ anxi-ety about challenging exercises As in previous editions, students at all levels can bechallenged by the exercises in this text

Hundreds of new writing questions encourage students to think about the

impor-tant concepts of the section before solving computational problems Problem numbers

that appear in blue indicate problems that require students to apply their reasoningand writing skills to the solution of the problem

Features Retained

Examples and matched problems introduce concepts and demonstrate problem-solving

techniques using side-by-side algebraic and graphical solution methods Each fully solved example is followed by a similar Matched Problem for the student to workthrough while reading the material Answers to the matched problems are located atthe end of each section, for easy reference This active involvement in the learningprocess helps students develop a more thorough understanding of algebraic and graph-ical concepts and processes

care-Graphing calculator technology is integrated throughout the text for

visualiza-tion, investigavisualiza-tion, and verification Graphing calculator screens displayed in the textare actual output Although technology is employed throughout, the authors strive tobalance algebraic skill development with the use of technology as an aid to learningand problem solving

Annotated steps of examples and developments are found throughout the text to

help students through the critical stages of problem-solving Think Boxes (colordashed boxes) are used to enclose steps that, with some experience, many studentswill be able to perform mentally

Applications throughout the third edition give the student substantial experience

in modeling and solving real-world problems, fulfilling a primary objective of the text.Over 500 application exercises help convince even the most skeptical student thatmathematics is relevant to everyday life Chapter Openers are written to highlightinteresting applications and an Applications Index is included to help locate applica-tions from particular fields

Explore-discuss boxes are interspersed throughout each section They foster

con-ceptual understanding by asking students to think about a relationship or processbefore a result is stated Verbalization of mathematical concepts, processes, and results

is strongly encouraged in these investigations and activities

Group activities at the end of each chapter involve multiple concepts discussed

in the chapter These activities strongly encourage the verbalization of mathematicalconcepts, results, and processes All of these special activities are highlighted toemphasize their importance

Foundations for calculus icons are used to mark concepts that are especially

per-tinent to a student’s future study of calculus

Interpretation of graphs icons are used to mark exercises that ask students to

make determinations about equations or functions based on graphs

Key Content Changes

Chapter 1, Functions, Graphs, and Models Functions are now introduced interms of relations, providing more flexibility in discussing correspondences betweensets of objects

Chapter 2, Modeling with Linear and Quadratic Functions Quadratic ities are now covered using a more general test point method, so that all nonlinearinequalities are solved using the same method

inequal-Chapter 3, Polynomial and Rational Functions Section 3-1 (Polynomial tions and Models) has been split into two sections, so Section 3-2 is now entirelydevoted to division of polynomials, including remainder and factor theorems crucial

Func-to the study of zeros of polynomials later in the chapter The chapter also features anew section (3-6) on direct, inverse, joint, and combined variation This new sectionsupports the book’s emphasis on mathematical modeling

Chapter 5, Modeling with Systems of Equations and Inequalities A new tion 5-2 on Systems of Linear Equations in Three Variables was added, while Section5-1 on Systems of Linear Equations in Two Variables was reorganized

Sec-Chapter 6 Matrices and Determinants The material on matrix solutions to ear systems is now found in Section 6-1

lin-Supplements

MathZone McGraw-Hill’s MathZone is a complete online tutorial and homeworkmanagement system for mathematics and statistics, designed for greater ease of usethan any other system available Instructors have the flexibility to create and sharecourses and assignments with colleagues, adjunct faculty, and teaching assistants withonly a few clicks of the mouse All algorithmic exercises, online tutoring, and a vari-ety of video and animations are directly tied to text-specific materials

MathZone is completely customizable to suit individual instructor and student needs.Exercises can be easily edited, multimedia is assignable, importing additional content iseasy, and instructors can even control the level of help available to students while doingtheir homework Students have the added benefit of full access to the study tools to indi-vidually improve their success without having to be part of a MathZone course.MathZone has automatic grading and reporting of easy-to-assign algorithmicallygenerated problem types for homework, quizzes and tests Grades are readily accessi-ble through a fully integrated grade book that can be exported in one click to MicrosoftExcel, WebCT, or BlackBoard

• Assessment capabilities, powered through ALEKS, which provide students andinstructors with the diagnostics to offer a detailed knowledge base through advancedreporting and remediation tools.

• Faculty with the ability to create and share courses and assignments with colleaguesand adjuncts, or to build a course from one of the provided course libraries

• An Assignment Builder that provides the ability to select algorithmically generatedexercises from any McGraw-Hill math textbook, edit content, as well as assign avariety of MathZone material including an ALEKS Assessment

• Accessibility from multiple operating systems and Internet browsers

ALEKS ALEKS (Assessment and LEarning in Knowledge Spaces) is a dynamic

online learning system for mathematics education, available over the Web 24/7.ALEKS assesses students, accurately determines their knowledge, and then guidesthem to the material that they are most ready to learn With a variety of reports, Text-book Integration Plus, quizzes, and homework assignment capabilities, ALEKS offersflexibility and ease of use for instructors

• ALEKS uses artificial intelligence to determine exactly what each student knows and

is ready to learn ALEKS remediates student gaps and provides highly efficient ing and improved learning outcomes

learn-• ALEKS is a comprehensive curriculum that aligns with syllabi or specified books Used in conjunction with a McGraw-Hill text, students also receive links totext-specific videos, multimedia tutorials, and textbook pages

text-• Textbook Integration Plus enables ALEKS to be automatically aligned with syllabi

or specified McGraw-Hill textbooks with instructor chosen dates, chapter goals,homework, and quizzes

• ALEKS with AI-2 gives instructors increased control over the scope and sequence

of student learning Students using ALEKS demonstrate a steadily increasing tery of the content of the course

mas-• ALEKS offers a dynamic classroom management system that enables instructors tomonitor and direct student progress toward mastery of course objectives

• See www.aleks.com

Student’s Solutions Manual Prepared by Dave Sobecki, the Student’s Solutions

Manual provides comprehensive, worked-out solutions to all of the odd-numbered

exercises from the text The steps shown in the solutions match the style of solvedexamples in the textbook

Video Lectures on Digital Video Disk (DVD) In the videos, J D Herdlick of

St Louis Community College at Meramec introduces essential definitions, theorems,formulas, and problem-solving procedures and then works through selected exercisesfrom the textbook, following the solution methodology employed in the text In addi-tion, new instructional videos on graphing calculator operations help students masterthe most essential calculator skills used in the college algebra course The video series

is available on DVD or online as an assignable element of MathZone The DVDs are

with Disabilities Act Standards for Accessible Design Instructors may use them asresources in a learning center, for online courses, and/or to provide extra help to stu-dents who require extra practice.

NetTutor Available through MathZone, NetTutor is a revolutionary system thatenables students to interact with a live tutor over the World Wide Web NetTutor’sWeb-based, graphical chat capabilities enable students and tutors to use mathematicalnotation and even to draw graphs as they work through a problem together Studentscan also submit questions and receive answers, browse previously answered questions,and view previous live-chat sessions Tutors are familiar with the textbook’s objec-tives and problem-solving styles

CTB (Computerized Test Bank) Online Available through MathZone, this puterized test bank, utilizing algorithm-based testing software, enables users to createcustomized exams quickly This user-friendly program enables instructors to searchfor questions by topic, format, or difficulty level; to edit existing questions or to addnew ones; and to scramble questions and answer keys for multiple versions of thesame test Hundreds of text-specific open-ended and multiple-choice questions areincluded in the question bank Sample chapter tests and final exams in MicrosoftWord®and PDF formats are also provided

com-Instructor’s Solutions Manual Prepared by Dave Sobecki, and available on

Math-Zone, the Instructor’s Solutions Manual provides comprehensive, worked-out solutions

to all exercises in the text The methods used to solve the problems in the manual arethe same as those used to solve the examples in the textbook

You Can Customize this Text with McGraw-Hill/Primis Online A digital base offers you the flexibility to customize your course including material from thelargest online collection of textbooks, readings, and cases Primis leads the way incustomized eBooks with hundreds of titles available at prices that save your studentsover 20% off bookstore prices Additional information is available at 800-228-0634

In addition to the authors, many others are involved in

the successful publication of a book We wish to thank

personally the following people who reviewed the third

edition manuscript and offered invaluable advice for

improvements:

Laurie Boudreaux, Nicholls State University

Emma Borynski, Durham Technical Community College

Barbara Burke, Hawaii Pacific University

Sarah Cook, Washburn University

Donna Densmore, Bossier Parish Community College

Alvio Dominguez, Miami-Dade College (Wolfson

Cam-pus)

Joseph R Ediger, Portland State University

Angela Everett, Chattanooga State Technical Community

College

Mike Everett, Santa Ana College

Toni Fountain, Chattanooga State Technical Community

College

Scott Garten, Northwest Missouri State University

Perry Gillespie, Fayetteville State University

Dana Goodwin, University of Central Arkansas

Judy Hayes, Lake-Sumter Community College

James Hilsenbeck, University of Texas–Brownsville

Lynda Hollingsworth, Northwest Missouri State University

Michelle Hollis, Bowling Green Community College of

WKU

Linda Horner, Columbia State Community College

Tracey Hoy, College of Lake County

Byron D Hunter, College of Lake County

Michelle Jackson, Bowling Green Community College of

WKU

Tony Lerma, University of Texas–Brownsville

Austin Lovenstein, Pulaski Technical College

Jay A Malmstrom, Oklahoma City Community College

Lois Martin, Massasoit Community College

Mikal McDowell, Cedar Valley College

Rudy Meangru, LaGuardia Community College

Dennis Monbrod, South Suburban College

Sanjay Mundkar, Kennesaw State University

Elaine A Nye, Alfred State College

Dale Oliver, Humboldt State University

Jorge A Perez, LaGuardia Community College

Susan Pfeifer, Butler Community College

Dennis Reissig, Suffolk County Community College

Brunilda Santiago, Indian River Community CollegeNicole Sifford, Rivers Community College

James Smith, Columbia State Community CollegeJoyce Smith, Chattanooga State Technical CommunityCollege

Shawn Smith, Nicholls State UniversityMargaret Stevenson, Massasoit Community CollegePam Stogsdill, Bossier Parish Community CollegeJohn Verzani, College of Staten Island

Deanna Voehl, Indian River Community CollegeIanna West, Nicholls State University

Fred Worth, Henderson State University

We also wish to thankHossein Hamedani for providing a careful and thoroughcheck of all the mathematical calculations in the book(a tedious but extremely important job)

Dave Sobecki for developing the supplemental manualsthat are so important to the success of a text

Jeanne Wallace for accurately and efficiently producingmost of the manuals that supplement the text

Mitchel Levy for scrutinizing our exercises in the uscript and making recommendations that helped us tobuild balanced exercise sets

man-Tony Palermino for providing excellent guidance inmaking the writing more direct and accessible to stu-dents

Pat Steele for carefully editing and correcting the uscript

man-Jay Miller for his careful technical proofread of the firstpages

Katie White for organizing the revision process, mining the objectives for the new edition, and supervis-ing the preparation of the manuscript

deter-Sheila Frank for guiding the book smoothly through allpublication details

All the people at McGraw-Hill who contributed theirefforts to the production of this book, especially DawnBercier

Producing this new edition with the help of all theseextremely competent people has been a most satisfyingexperience

from their symbol manipulations and from

graphical approaches, students gain insightinto the power of algebra and make importantconceptual and visual connections

Examples and Matched Problems

Integrated throughout the text, completely

worked examples and practice problems are

used to introduce concepts and demonstrate

problem-solving techniques—algebraic,

graph-ical, and numerical Each Example is followed

by a similar Matched Problem for the student

to work through while reading the material

Answers to the matched problems are located

at the end of each section, for easy reference

This active involvement in the learning

process helps students develop a thorough

understanding of algebraic concepts and

processes

Using Exponential Function Properties

Find all solutions to 4x⫺3 ⫽ 8.

Divide both sides by 2.

10

Table 3 Home Ownership Rates

Home Ownership Year Rate (%)

EXAMPLE 5 Home Ownership Rates

The U.S Census Bureau published the data in Table 3 on home ownership rates.

(A) Let x represent time in years with representing 1900, and let y represent

the corresponding home ownership rate Use regression analysis on a graphing calculator to find a logarithmic function of the form that models

the data (Round the constants a and b to three significant digits.)

(B) Use your logarithmic function to predict the home ownership rate in 2010.

SOLUTIONS

(A) Figure 1 shows the details of constructing the model on a graphing calculator (B) The year 2010 corresponds to Evaluating at predicts a home ownership rate of 71.4% in 2010.

x⫽ 110 x⫽ 110. y1⫽ ⫺36.7 ⫹ 23.0 ln x

y ⫽ a ⫹ b ln x

x⫽ 0

0 0 100

Refer to Example 5 The home ownership rate in 1995 was 64.7%.

(A) Find a logarithmic regression equation for the expanded data set.

(B) Predict the home ownership rate in 2010.



ANSWERS TO MATCHED PROBLEMS

1.95.05 decibels 2.7.80 3.2.67 4.1 kilometer per second less

5.(A) (B) ⫺31.5 ⫹ 21.7 ln x 70.5%

Balanced Exercise Sets

College Algebra: Graphs and Models, third

edition, contains more than 4,000 problems

Each Exercise set is designed so that an age or below-average student will experiencesuccess and a very capable student will bechallenged Exercise sets are found at the end

aver-of each section in the text The AnnotatedInstructor’s Edition features A (routine, easymechanics), B (more difficult mechanics), and

C (difficult mechanics and some theory) ignations to denote these levels and helpinstructors in the assignment building process.Problem numbers that appear in blue indicateproblems that require students to apply theirreasoning and writing skills to the solution ofthe problem

des-In Problems 1–8, determine the validity of each statement If a statement is false, explain why.

15 Explain why squaring both sides of an equation sometimes

introduces extraneous solutions.

16 Would raising both sides of an equation to the third power

ever introduce extraneous solutions? Why or why not?

17 Write an example of a false statement that becomes true

when you square both sides What would every possible ample have in common?

ex-18 How can you recognize when an equation is of quadratic type?

In Problems 19–32, solve algebraically and confirm cally, if possible.

1 4x⫺ 3 ⫽ 2

1 3x⫹ 5 ⫽ 3

110x ⫹ 1 ⫹ 8 ⫽ 0 15x ⫹ 6 ⫹ 6 ⫽ 0

14 ⫺ x ⫽ 4 14x ⫺ 7 ⫽ 5

3x3/2⫺ 5x1/2 ⫹ 12 ⫽ 0 10

57.Explain why the following “solution” is incorrect:

58.Explain why the following “solution” is incorrect.

In Problems 59–62, solve algebraically and confirm cally, if possible.

x⫹ 3 ⫹ 25 ⫽ 144

1x⫹ 3 ⫹ 5 ⫽ 12 (x⫺ 3) 4⫹ 3(x ⫺ 3)2 ⫽ 4

13t ⫹ 4 ⫹ 1t ⫽ ⫺3 1u ⫺ 2 ⫽ 2 ⫹ 12u ⫹ 3

(x2⫹ 2x)2⫺ (x2⫹ 2x) ⫽ 6 (m2⫺ m)2⫺ 4(m2⫺ m) ⫽ 12

100.NAVIGATION A speedboat takes 1 hour longer to go 24 miles up

a river than to return If the boat cruises at 10 miles per hour in still water, what is the rate of the current?

101.CONSTRUCTION A gardener has a 30 foot by 20 foot lar plot of ground She wants to build a brick walkway of uniform

rectangu-to have 400 square feet of ground left for planting, how wide (rectangu-to two decimal places) should she build the walkway?

102.CONSTRUCTION Refer to Problem 101 The gardener buys enough bricks to build 160 square feet of walkway Is this sufficient (to two decimal places) can she build the walkway with these bricks?

103.CONSTRUCTION A 1,200 square foot rectangular garden is closed with 150 feet of fencing Find the dimensions of the garden to the nearest tenth of a foot.

en-104.CONSTRUCTION The intramural fields at a small college will cover a total area of 140,000 square feet, and the administration has the dimensions of the field.

105.ARCHITECTURE A developer wants to erect a rectangular ing on a triangular-shaped piece of property that is 200 feet wide and

build-400 feet long (see the figure).

Property Line

FIRST STREET

Property A

Proposed Building

(B) Building codes require that this building have a footprint of that will satisfy the building codes?

(C) Can the developer construct a building with a footprint of

of a building constructed in this manner?

106.ARCHITECTURE An architect is designing a small A-frame cottage for a resort area A cross-section of the cottage is an ters The front wall of the cottage must accommodate a sliding door positioned as shown in the figure.

(A) Express the area A(w) of the door as a function of the width

lem 105.]

(B) A provision of the building code requires that doorways

of the doorways that satisfy this provision.

(C) A second provision of the building code requires all ways to be at least 2 meters high Discuss the effect of this re- quirement on the answer to part B.

door-107.TRANSPORTATION A delivery truck leaves a warehouse

and travels north to factory A From factory A the truck travels

the figure on the next page) The driver recorded the truck’s

the end of the trip and also at factory B, but forgot to record it at that it was farther from the warehouse to factory A than it was

DOOR DETAIL Page 1 of 4

5 meters

h

w

4 meters

*Euclid’s theorem: If two triangles are similar, their corresponding

sides are proportional:

Applications

One of the primary objectives of this book is

to give the student substantial experience in

modeling and showing real-world problems

More than 500 application exercises help

con-vince even the most skeptical student that

mathematics is relevant to everyday life The

most difficult application problems are marked

with two stars (* *), the moderately difficult

application problems with one star (*), and

easier application problems are not marked

An Application Index is included immediately

preceding Chapter 1 to locate particular

appli-cations

Boiling point of water, 44

Boy-girl composition of families, 684

Braking distance and speed, 370

Cell phone charges, 125, 126

Cell phone subscribers, 490, 492

Combined area, 183 Combined outcomes, 660–661 Combined variation, 367–368 Committee selection, 687 Communications, 764 Competitive rowing, 169 Completion of years of college, 319 Compound interest, 391–395, 442, 450,

460, 461, 709 Computer design, 413–414 Computer-generated tests, 662 Computer science, 60–61, 65, 125, 270, A–26

Construction, 65, 82, 183, 189, 223, 224,

240, 263, 269, 270, 290, 314–315,

319, 335, 360, 377, 438, 760, A–58, A–73, A–109

Consumer debt, 38–40 Continuous compound interest, 393–395 Cost analysis, 150, 156–157, 169, 263, 563

Credit union debt, 40 Cryptography, 574–575, 577–578, 616 Data analysis, 38–41, 125–126, 490–492, A–91–92, A–96

Delivery charges, 66 Demand, 94, 111, 270 Demographics, 150 Depreciation, 150, 263, 269, 415 Depth of a well, 234–236, 240 Design, 215–216, 236–237, 236–238,

240, 263, 264, 734, 760 Diamond prices, 161–164 Diet, 475–476, 522, 525, 588, 615, 620 Distance-rate-time, 157–159

Distance-rate-time problems, 158–159 Divorce, 291

Dominance relation, 564–565 Drawing cards, 686–687 Drug use, 264

Earthquake intensity, 433–435, 448 Earthquakes, 170, 438, 451, 455, 461 Earth science, 170, 256, 414, 482, A–26

Ecology, 439 Economics, 94–95, 657, 709, 767, A–26, A–73

Economy stimulation, 654–655 Efficiency, 460

Electrical circuit, A–73 Electricity, 370 Empirical probabilities for an insurance company, 682–683

Employee training, 359, 415 Engineering, 189, 370, 371, 658, 722,

734, 764, 768, A–109 Environmental science, 98 Epidemics, 408–409 Estimating weight, 285 Evaporation, 83, 99 Explosive energy, 438 Fabrication, 335 Falling objects, 44, 184, 189, 262, 363,

364, 481, 482, 632, 658 Finance, 398, 399, 481, 494, 620, 658 Fish weight, 285

Fixed costs, 251, 253 Flight ground speed, 170 Flight navigation, 150 Fluid flow, 83, 98–99 Food chain, 658 Force of stretched spring, 150, 362 Gaming, 413

Gas mileage, 188–189 Genealogy, 658 Genetics, 370 Geology, 149 Geometry, 240, 262, 318, 371, 377, 460,

494, 525, 546, 587, 659, 760, 767, A–46

Global warming, 149 Half-life, 403, 404 Health care, 290–291 Heat conduction, 526 Height of bungee jumper, 124 History of technology, 414 Home ownership, 319, 437 Horsepower and speed, 370 Hydroelectric power consumption, 286

APPLICATION INDEX XVII

Net cash flow, 6–7

Newton’s law of cooling, 414, 415, 451

Nuclear power, 416, 750

Numbers, 760

Nutrition, 481, 509–510, 525–526, 547,

Officer selection, 668–669 Olympic games, 125, 169, 171 Optimal speed, 219–220, 220, 225, 265, 271

Ozone levels, A–91 – A–92 Packaging, 335, A–47 Parabolic reflector, 719–720 Pendulum, A–35

Photic zone, 414, 451 Photography, 370, 415, 451, 659 Physics, 44, 45, 124, 149, 150, 370, 376,

460, 658, A–35, A–95, A–96 Physiology, 360

Plant nutrition, 509, 522 Political science, 264, 768 Politics, 149, 564 Pollution, 521–522 Population growth, 266, 400–401, 413,

414, 450, 455, 460, 495, 657–658 Position of moving object, 124 Present value, 398, 399, 455 Price and demand, 20, 94, 111, 118, 124,

189, 190, 251–253, 270, 478 Price and revenue, 21

Price and supply, 118, 478, A–96

Pricing, 263, 270 Prize money, 650–651 Production costs, 82 Production scheduling, 481, 488–490,

494, 504–506, 510–513, 546–547,

555, 587 Profit, 19, 64–65, 94–95, 188, 251–253,

256, 290, 318, 376 Profit analysis, 263, 460 Profit and loss analysis, 269 Projectile motion, 184–185, 189, 250,

256, A–132 – 134 Propagation of a rumor, 409 Psychology, 360, 371, 510, 521 Purchasing, 521, 542–543, 620 Puzzle, 546, 615, 658–659 Quality control, 709 Radioactive decay, 403–404 Radioactive tracers, 414 Rate of change, 142–144, 144 Rate-time, 480

Relativistic mass, A–35 Rental charges, 57–58, 156, 157, 219, 220 Replacement time, 360

Research and development analysis, 46 Resource allocation, 494–495, 509, 521,

525, 587, 615

Revenue, 64, 111–112, 118, 125, 188,

189, 190, 290 Revenue analysis, 45, 610 Richter scale, 433, 434, 435 Rocket flight, 435–436 Rolling two dice, 679–680, 686 Safety research, 83

Salary increment, 632 Sales analysis, 45 Sales commissions, 66, 169, 552–553 Selecting officers, 668–669

Selecting subcommittees, 671–672 Serial number counting, 673 Service charges, 66 Shipping, 270, 319, 460 Signal light, 722 Simple interest, 376 Sociology, 510, 522 Solid waste disposal, 216–218 Sound detection, 170 Sound intensity, 431–433, 438, 455, 461 Space vehicles, 438

Sports, 125, 169, A–108 Sports medicine, 263 Stock prices, 67, 149 Stopping distance, 225, 271 Storage, 335

Subcommittee selection, 671–672 Supply and demand, 164–167, 171, 264–265, 477–478, 481 Telephone charges, 66, 125, 126 Temperature, 19, 149, 150, 170, A–96 Timber harvesting, 82–83

Time and speed, 370 Time measurement with atomic clock, A–23 Time spent studying, 262

Tire mileage, 66 Transportation, 65, 223–224, 521, 709, 760 Underwater pressure, 144

Variable costs, 251, 253 Vibration of air in pipe, 365, 371 Volume of cylindrical shell, A–44 Weather balloon, 257

Weight and speed, 370 Weight estimates, 285 Well depth, 234–235 Wildlife management, 415, 455 Women in the workforce, 377 Work, 376

World population, 450

1-6 Inverse FunctionsChapter 1 ReviewChapter 1 Group Activity:Mathematical Modeling:Choosing a Cell Phone Provider

OUTLINE

Functions, Graphs,

and Models

THE function concept is one of the most important ideas in

mathematics To study math beyond the elementary level, you

absolutely need to have a solid understanding of functions and

their graphs In this chapter, you’ll learn the fundamentals of

what functions are all about, and how to use them In

subse-quent chapters, this will pay off as you study particular types

of functions in depth In the first section of this chapter, we

dis-cuss the techniques involved in using an electronic graphing

device like a graphing calculator In the remaining sections, we

introduce the concept of functions and discuss general

prop-erties of functions and their graphs Everything you learn in

this chapter will increase your chance of success in this

course, and in almost any other course you may take that

in-volves mathematics

C

2 C H A P T E R 1 FUNCTIONS, GRAPHS, AND MODELS

The use of technology to aid in drawing and analyzing graphs is revolutionizing ematics education and is the primary motivation for this book Your ability to interpretmathematical concepts and to discover patterns of behavior will be greatly increased

math-as you become proficient with an electronic graphing device In this section we duce some of the basic features of electronic graphing devices Additional features will

intro-be introduced as the need arises If you have already used an electronic graphing device

in a previous course, you can use this section to quickly review basic concepts If youneed to refresh your memory about a particular feature, consult the Technology Index

at the end of this book to locate the textbook discussion of that particular feature

Z Using Graphing Calculators

We will begin with the use of electronic graphing devices to graph equations We will

refer to any electronic device capable of displaying graphs as a graphing utility The

two most common graphing utilities are handheld graphing calculators and ers with appropriate software It’s essential that you have such a device handy as youproceed through this book

comput-Since many different brands and models exist, we will discuss graphing tors only in general terms Refer to your manual for specific details relative to yourown graphing calculator.*

calcula-An image on the screen of a graphing calculator is made up of darkened rectangles

called pixels (Fig 1) The pixel rectangles are the same size, and don’t change in size

dur-ing any application Graphdur-ing calculators use pixel-by-pixel plottdur-ing to produce graphs

Z Using Graphing Calculators

Z Understanding Screen Coordinates

Z Using the Trace, Zoom, and Intersect Commands

Z Mathematical Modeling

(a) Image on

a graphing calculator.

(b) Magnification to show pixels.

Z Figure 1 Pixel-by-pixel plotting

on a graphing calculator.

*Manuals for most brands of graphing calculators are readily available on the Internet.

The accuracy of the graph depends on the resolution of the graphing calculator.Most graphing calculators have screen resolutions of between 50 and 75 pixels perinch, which results in fairly rough but very useful graphs Some computer systemscan print very high quality graphs with resolutions greater than 1,000 pixels per inch.Most graphing calculator screens are rectangular The graphing screen on a graph-ing calculator represents a portion of the plane in the rectangular coordinate system.But this representation is an approximation, because pixels are not really points, as isclearly shown in Figure 1 Points are geometric objects without dimensions (you canthink of them as “infinitely small”), whereas a pixel has dimensions The coordinates

of a pixel are usually taken at the center of the pixel and represent all the infinitelymany geometric points within the pixel Fortunately, this does not cause much of aproblem, as we will see

The portion of a rectangular coordinate system displayed on the graphing screen

is called a viewing window and is determined by assigning values to six window

vari-ables: the lower limit, upper limit, and scale for the x axis and the lower limit, upper

limit, and scale for the y axis Figure 2(a) illustrates the names and values of standard

window variables, and Figure 2(b) shows the resulting standard viewing window.

The names Xmin, Xmax, Xscl, Ymin, Ymax, and Yscl will be used for the six

window variables Xscl and Yscl determine the distance between tick marks on the x

and y axes, respectively Xres is a seventh window variable on some graphing

calcu-lators that controls the screen resolution; we will always leave this variable set to thedefault value 1 The window variables may be displayed slightly differently by yourgraphing calculator In this book, when a viewing window of a graphing calculator ispictured in a figure, the values of Xmin, Xmax, Ymin, and Ymax are indicated bylabels to make the graph easier to read [see Fig 2(b)] These labels are always cen-tered on the sides of the viewing window, regardless of the location of the axes

REMARK: We think it’s important that actual output from existing graphing calculators

be used in this book The majority of the graphing calculator images in this book arescreen dumps from a Texas Instruments TI-84 graphing calculator Occasionally weuse screen dumps from a TI-86 graphing calculator, which has a wider screen Youmay not always be able to produce an exact replica of a figure on your graphing cal-culator, but the differences will be relatively minor

We now turn to the use of a graphing calculator to graph equations that can bewritten in the form

(1)Graphing an equation of the type shown in equation (1) using a graphing calculator

is a simple three-step process:

y  (some expression in x)

S E C T I O N 1–1 Using Graphing Calculators 3

Z Figure 2 A standard viewing

window and its dimensions.

10

10

10

10

(a) Standard window variable values

(b) Standard viewing window

Z GRAPHING EQUATIONS USING A GRAPHING CALCULATOR

Step 1.Enter the equation

Step 2.Enter values for the window variables (A good rule of thumb forchoosing Xscl and Yscl, unless there are reasons to the contrary, is

to choose each about one-tenth the width of the corresponding able range.)

vari-The following example illustrates this procedure for graphing the equation

(See Example 1 of Appendix B, Section B-2 for a hand-drawn sketch ofthis equation.)

y  x2 4

Use a graphing calculator to graph for and

SOLUTION

Press the key to display the equation editor and enter the equation [Fig 3(a)] PressWINDOW to display the window variables and enter the given values for these variables[Fig 3(b)] Press GRAPH to obtain the graph in Figure 3(c) (The form of the screens inFigure 3 may differ slightly, depending on the graphing calculator used.) 

(a) Enter equation.

(c) Press the graph command (b) Enter window variables.

Often, it is helpful to think about an appropriate viewing window before ing to graph an equation This can help save time, as well as increase your odds ofseeing the whole graph

start-*Answers to matched problems in a given section are found near the end of the section, before the exercise set.

S E C T I O N 1–1 Using Graphing Calculators 5

Find an appropriate viewing window in which to graph the equation with a graphing calculator

y  1x  13

SOLUTIONS

Algebraic Solution

We begin by thinking about reasonable x values for this

equation Since y values are determined by an

expres-sion under a root, only x values that result in

being nonnegative will have an associated y value So we

write and solve the inequality

This tells us that there will be points on the graph only

for x values 13 or greater.

Next, we make a table of values for selected x ues to see what y values are appropriate Note that we

val-chose x values that make it easy to compute y.

To clearly display all of these points and leave some space

around the edges, we choose

val-values greater than or equal to 13

To clearly display all of these points and leavesome space around the edges, we choose

Find an appropriate viewing window in which to graph the equation

with a graphing calculator



y  2  1x  15

*Many graphing calculators can construct a table of values like the one in Figure 5 using the TBLSET and TABLE commands.

The next example illustrates how a graphing calculator can be used as an aid tosketching the graph of an equation by hand The example illustrates the use of alge-braic, numeric, and graphic approaches; an understanding of all three approaches will

be a big help in problem solving

to Hand Graphing—Net Cash Flow

The net cash flow y in millions of dollars of a small high-tech company from

1998–2006 is given approximately by the equation

(2)

where x represents the number of years before or after 2002, when the board of

direc-tors appointed a new CEO

(A) Construct a table of values for equation (2) for each year starting with 1998 and

ending with 2006 Compute y to one decimal place.

(B) Obtain a graph of equation (2) in the viewing window of your graphing tor Plot the table values from part A by hand on graph paper, then join thesepoints with a smooth curve using the graph in the viewing window as an aid

Recall that x represents years before or after 2002, and y represents cash flow

calcula-S E C T I O N 1–1 Using Graphing Calculators 7

Z Figure 8 Net cash flow.

1 In applied problems, it’s a great idea to begin by writing down what each variable

in the problem represents

2 Table 1 is useful for analyzing this data since it gives us specific detail; the tion and graph are also useful because they give us a quick overview of cash flow.Each viewpoint has its specific value 

equa-MATCHED PROBLEM 3

The company in Example 3 is in competition with another company whose net cash

flow y in millions of dollars from 1998 to 2006 is given approximately by the equation

(3)

where x represents the number of years before or after 2002.

(A) Construct a table of values for equation (3) for each year starting with 1998 and

ending with 2006 Compute y to one decimal place.

(B) Plot the points corresponding to the table by hand, then hand sketch the graph ofthe equation with the help of a graphing calculator



4  x  4

y  1  1.9x  0.2x3

ZZZEXPLORE-DISCUSS 1

The choice of the viewing window has a pronounced effect on the shape of

a graph Graph in each of the following viewing windows:

(A) (B) (C)

Which window gives the best view of the graph of this equation, and why?

100  x  100, 100  y  100

10  x  10, 10  y  10

1  x  1, 1  y  1

y  x3 2x

Z Understanding Screen Coordinates

We now take a closer look at screen coordinates of pixels Earlier we indicated that the

coordinates of the center point of a pixel are usually used as the screen coordinates of

the pixel, and these coordinates represent all points within the pixel As you might

expect, screen coordinates of pixels change as you change values of window variables

To find screen coordinates of various pixels, move a cursor around the viewing

window and observe the coordinates displayed on the screen A cursor is a special

symbol, such as a plus or times sign, that locates one pixel on the screen at

a time As the cursor is moved around the screen, it moves from pixel to pixel To see

()()

of the four pixels that are closest to (2, 2) The coordinates displayed on your screenmay vary slightly from these, depending on the graphing calculator used.

Z Figure 9 Screen coordinates

Z Using the Trace, Zoom, and Intersect Commands

When analyzing the graph of an equation, it’s often useful to find the coordinates of

certain points on the graph Using the TRACE command on a graphing calculator is

one way to accomplish this The trace feature places a cursor directly on the graphand only permits movement left and right along the graph The coordinates displayedduring the tracing movement are coordinates of points that satisfy the equation Inmost cases, these coordinates are not the same as the pixel screen coordinates dis-played using the unrestricted cursor movement that we discussed earlier Instead, they

are the exact coordinates of points on the graph.

7  y  7.

7  x  7

ZZZEXPLORE-DISCUSS 2

Graph the equation in a standard viewing window

(A) Without selecting the TRACE command, move the cursor to a point on the

screen that appears to lie on the graph of and is as close to (5, 5) as sible Record these coordinates Do these coordinates satisfy the equation

pos-(B) Now select the TRACE command and move the cursor along the graph

of to a point that has the same x coordinate found in part A Is the y

coordinate of this point the same as you found in part A? Do the coordinates

of the point using trace satisfy the equation

(C) Explain the difference in using trace along a curve and trying to use

unrestricted movement of a cursor along a curve

S E C T I O N 1–1 Using Graphing Calculators 9

ZZZEXPLORE-DISCUSS 3

Figure 11 shows the ZOOM menu on a TI-84.* We want to explore the effects

of some of these options on the graph of Enter this equation in theequation editor and select ZStandard from the ZOOM menu What are the win-dow variables? In each of the following, position the cursor at the origin andselect the indicated zoom option Observe the changes in the window variablesand examine the coordinates displayed by tracing along the curve

(A) ZSquare (B) ZDecimal (C) ZInteger (D) ZoomFit

y  x.

Zoom out

Original graph

Original graph

Zoom in

Z Figure 10 The zoom

operation.

Z Figure 11 The ZOOM menu

on a TI-84.

Most graphing calculators have a ZOOM command In general, zooming in on

a graph reduces the window variables and magnifies the portion of the graph visible

in the viewing window [Fig 10(a)] Zooming out enlarges the window variables so

that more of the graph is visible in the viewing window [Fig 10(b)]

*The ZOOM menu on other graphing calculators may look quite different from the one on the TI-84.

† On the TI-84, INTERSECT is found on the CALC (2nd-TRACE) menu.

Another command found on most graphing calculators is INTERSECT† or

ISECT This command enables the user to find the point(s) where two curves

inter-sect without using trace or zoom

The use of trace, zoom, and intersect is best illustrated by examples

10 C H A P T E R 1 FUNCTIONS, GRAPHS, AND MODELS

Let

(A) Use the TRACE command to find y when (B) Use the TRACE and ZOOM commands to find x when (C) Use the INTERSECT command to find x when

Round answers to two decimal places

it only allows us to input x values Instead we press the ZOOM command and

select Zoom In to obtain more accuracy (Fig 18) Then we select the TRACEcommand and move the cursor as close to y 5 as possible (Fig 19)

Now we see that when This is an improvement, but we can

do better Repeating the Zoom In command and tracing along the curve (Fig 20),

we see that x 7.37when y 5.00

5.37

7.76

(C) Enter in the graphing calculator and graph and in the standard ing window (Fig 21) Now there are two curves displayed on the graph The hor-izontal line is the graph of , and the other curve is the familiar graph ofThe coordinates of the intersection point of the two curves must satisfy both

view-equations Clearly, the y coordinate of this intersection point is 5 The x

coordi-nate is the value we are looking for We can use the INTERSECT command tofind the coordinates of the intersection point in Figure 21 When we select theINTERSECT command, we are asked to make three choices: the first curve, thesecond curve, and a guess When the desired equation is displayed at the top ofthe screen, press ENTER to select it (Figs 22 and 23) It doesn’t matter which

of the two graphs is designated as the first curve (If there are more than twocurves, use the up and down arrows to select the desired equation, then pressENTER.)

12 C H A P T E R 1 FUNCTIONS, GRAPHS, AND MODELS

To enter a guess, move the cursor close to the intersection point (Fig 24) and

press ENTER (Fig 25) On many models, you can enter an x value close to the

intersection rather than moving the cursor (We will see a more important use

of entering a guess in Example 5.) Examining Figure 25, we see that

Use a graphing calculator to solve the equation

Round answers to two decimal places

SOLUTION

We will solve this equation by graphing both sides in the same viewing window andfinding the intersection points First we enter and inthe graphing calculator (Fig 26) and graph in the standard viewing window (Fig 27)

A lot of students will always start with the standard viewing window, but in this

case it is a poor choice Because we are seeking the values of x that make the left

side of the equation equal to 200, we need a value for Ymax that is larger than 200.Changing Ymax to 300 and Yscl to 30 produces a new graph (Fig 28) The two curves

Examining Figure 29, we see that the values of are getting very large It’s unlikelythat there will be additional solutions to the equation for larger values of x.

Examining Figure 30, we see that there are values of that are on both sides of 200,

so there’s a good chance that there will be additional solutions for more negative

val-ues of x Based on the table valval-ues in Figure 30, we make the following changes in the

window variables:

This produces the graph in Figure 31 Examining the values of for values of x to

the left of this window (Fig 32), we conclude that there are no other intersection points

Now we use the INTERSECT command to find the x coordinates of the three

inter-section points in Figure 31 We select the two equations as before (Figs 33 and 34).This time, the guess is very important We have to specify which intersection point

to find To do this, we make a guess that is close to the desired point We first selectthe leftmost intersection point by moving the cursor to that point (Fig 35) and press-ing ENTER

intersect twice in this window The x coordinates of these points are the solutions we

are looking for But first, could there be other solutions that are not visible in thiswindow? To find out, we must investigate the behavior outside this window A table

is the most convenient way to do this (Figs 29 and 30)

14 C H A P T E R 1 FUNCTIONS, GRAPHS, AND MODELS

The coordinates of the leftmost intersection point are displayed at the bottom of thescreen (Fig 36) To find the other two intersection points, we repeat the entire process.When we get to the screen that asks for a guess, we place the cursor near the point

we are looking for and press ENTER (Figs 37 and 38)

Thus, we see that the solutions to are

intu-we can be more definitive in our reasoning For example, in Chapter 3 intu-we will showthat any equation like the one in Example 5 can have no more than three solutions

Examples 1 through 5 dealt with a variety of methods for finding the value of y that corresponds to a given value of x and the value(s) of x that correspond to a given value of y These methods are summarized in the following box.

Z FINDING SOLUTIONS TO AN EQUATION

Assume the equation is entered in a graphing calculator as (expression

Method 2. Set TBLSTART to a and display the table.

Method 3. Graph select the TRACE command, and enter a.

To find solutions (x, y) given use either of the ing methods:

follow-Method 1. Graph and use TRACE and ZOOM

Method 2. Graph y1and y2 b and use INTERSECT

Z Mathematical Modeling

Now that we are able to solve equations on a graphing calculator, there are many

appli-cations that we can investigate The term mathematical modeling refers to the process

of using an equation or equations to describe data from the real world The next ple develops a mathematical model for manufacturing a box to certain specifications

exam-S E C T I O N 1–1 Using Graphing Calculators 15

A packaging company plans to manufacture open boxes from 11- by 17-inch sheets

of cardboard by cutting x- by x-inch squares out of the corners and folding up the

sides, as shown in Figure 39

(A) Write an equation for the volume y of the resulting box in terms of the length x

of the sides of the squares that are cut out Indicate appropriate restrictions on x (B) Graph the equation for appropriate values of x Adjust the window variables for

y to include the entire graph of interest.

(C) Find the smallest square that can be cut out to produce a box with a volume of

(A) The dimensions of the box are expressed in terms of x in Figure 40(a), and the

box is shown in Figure 40(b) with the sides folded up and dimensions added

From this figure we can write an equation of the volume in terms of x and lish restrictions on x.

Note that the three dimensions of the box are x, and Of course,all of these dimensions have to be positive, so and

Now we solve the two latter inequalities:

Add 2x to each side.

Divide both sides by 2.

If x  5.5, then it is also less than 8.5.

We find that x has to be greater than zero and less than 5.5 (Inequalities are

reviewed in Appendix B, Section B-1.)The volume of a rectangular box is the product of its three dimensions, sothe volume of the box is given by

(4)(B) Entering this equation in a graphing calculator (it doesn’t need to be multipliedout) and evaluating it for several integers between 0 and 5 (Fig 41), it appears

that a good choice for the window dimensions for y is This choicecan easily be changed if there is too much space above the graph or if part ofthe graph we are interested in is out of the viewing window Figure 42 shows thegraph of equation (4) in the selected viewing window

0 y  200.

0 6 x 6 5.5

y  x(17  2x)(11  2x)

8.5 7 x 5.5 7 x

200

5.5

0 0

200

5.5

(C) We want to find the smallest value of x for which That is, we want tosolve the equation

We will solve this equation using the INTERSECT command Entering

in the graphing calculator and pressing GRAPH produces the two curves shown inFigure 43 The curves intersect twice Because we were asked for the smallest value

of x that satisfies the equation, we want the intersection point on the left (Fig 44) From Figure 44, we see that y is 150 when x is 1.19, so the smallest square that

can be cut out to produce a box with a volume of 150 cubic inches is 1.19 inches

y2 150

x(17  2x)(11  2x)  150

y 150

S E C T I O N 1–1 Using Graphing Calculators 17

18 C H A P T E R 1 FUNCTIONS, GRAPHS, AND MODELS

In Problems 1–6, determine if the indicated point lies in the

viewing window defined by

(A) Find the smallest rectangle in a Cartesian coordinate

system that will contain all the points in the table State your answer in terms of the window variables Xmin, Xmax, Ymin, and Ymax.

(B) Enter the window variables you determined in part A

and display the corresponding viewing window Can you use the cursor to display the coordinates of the points in the table on the graphing calculator screen?

Discuss the differences between the rectangle in the plane and the pixels displayed on the screen.

8 Repeat Problem 7 for the following table.

9 Explain the significance of Xmin, Xmax, and Xscl when

using a graphing calculator.

10 Explain the significance of Ymin, Ymax, and Yscl when

us-ing a graphus-ing calculator.

In Problems 11–16, graph each equation in a standard viewing

window.

In Problems 17–20, find an appropriate viewing window in

which to graph the given equation with a graphing calculator.

( 3, 5)

Xmin  7, Xmax  9, Ymin  4, Ymax  11

For each equation in Problems 21–26, use the TABLE command

on a graphing calculator to construct a table of values over the indicated interval, computing y values to the nearest tenth of a unit Plot these points on graph paper, then with the aid of a graph

on a graphing calculator, complete the hand sketch of the graph.

21. (Use even integers for the table.)

22. (Use odd integers for the table.)

23. (Use odd integers for the table.)

24. (Use even integers for the table.)

for the table.)

inte-gers for the table.)

In Problems 27–30, graph the equation in a standard viewing window Approximate to two decimal places the x coordinates of the points in this window that are on the graph of the equation and have the indicated y coordinates First use TRACE and ZOOM, then INTERSECT.

standard viewing window of a graphing calculator.

How do these graphs compare to the graph you drew in part A?

(C) Apply each of the following ZOOM options to the graphs in part B and determine which options produce a curve that looks like the curve you drew in part A:

ZDecimal, ZSquare, ZoomFit

the curve.

standard viewing window of a graphing calculator.

How do these graphs compare to the graph you drew in part A?

(C) Apply each of the following ZOOM options to the graphs in part B and determine which options produce a curve that looks like the curve you drew in part A:

ZDecimal, ZSquare, ZoomFit

In Problems 43–46, use the INTERSECT command on a

graphing calculator to solve each equation for the indicated

values of b Round answers to two decimal places.

In Problems 47–50, use the INTERSECT command on a

graphing calculator to solve each equation for the indicated

values of b Round answers to two decimal places.

ZOOM to approximate to four decimal places pare your result with the direct calculator evaluation of

ZOOM to approximate to four decimal places Compare your result with the direct calculator evaluation of

59.In a few sentences, discuss the difference between the mathematical coordinates of a point and the screen coordi- nates of a pixel.

60.In a few sentences, discuss the difference between the dinates displayed during unrestricted cursor movement and those displayed during the trace procedure.

coor-APPLICATIONS

61.PROFIT The monthly profit in dollars for a small consulting firm can be modeled by the equation

where x is the number of new clients acquired

during that month For tax purposes, the owner asks the manager

to try to make the December profit as close as possible to

$11,000 How many new clients should the manager try bring in?

62.METEOROLOGY The average monthly high temperature in grees Fahrenheit in Cincinnati for the first 10 months of the year can be modeled very accurately by the equation

de-where x is the month, with responding to January A wedding planner in Cincinnati is asked

cor-to set a date in a month where the high temperature is most likely

to be close to 75 degrees What month would be the best choice?

y  x2

( 12, 2)

3x 29  51 3

5  2x 0.05(x 4) 3 4  17  x

63.MANUFACTURING A rectangular open-top box is to be

con-structed out of an 8.5-inch by 11-inch sheet of thin cardboard by

cutting x-inch squares out of each corner and bending the sides

up, as in Figures 39 and 40 in Example 6 What size squares to

two decimal places should be cut out to produce a box with a

volume of 55 cubic inches? Give the dimensions to two decimal

places of all possible boxes with the given volume.

64.MANUFACTURING A rectangular open-top box is to be

con-structed out of a 9-inch by 12-inch sheet of thin cardboard by

cutting x-inch squares out of each corner and bending the sides

up as shown in Figures 39 and 40 in Example 6 What size

squares to two decimal places should be cut out to produce a box

with a volume of 72 cubic inches? Give the dimensions to two

decimal places of all possible boxes with the given volume.

65.MANUFACTURING A box with a lid is to be cut out of a

12-inch by 24-12-inch sheet of thin cardboard by cutting out six x-12-inch

squares and folding as indicated in the figure What are the

di-mensions to two decimal places of all possible boxes that will

have a volume of 100 cubic inches?

68.MANUFACTURING A drinking container in the shape of a right circular cone* has a volume of 50 cubic inches If the ra- dius plus the height of the cone is 8 inches, find the radius and the height to two decimal places.

69.PRICE AND DEMAND A nationwide office supply company

sells high-grade paper for laser printers The price per case y (in dollars) and the weekly demand x for this paper are related

approximately by the equation

(A) Complete the following table Approximate each value of x

to the nearest hundred cases.

(A) Complete the following table Approximate each value of x

to the nearest hundred cases.

de-71. PRICE AND REVENUE Refer to Problem 69 The revenue

from the sale of x cases of paper at $y per case is given by the

product (A) Use the results from Problem 69 to complete the following table of revenues.

66.MANUFACTURING A box with a lid is to be cut out of a

10-inch by 20-inch sheet of thin cardboard by cutting out six

x-inch squares and folding as indicated in the figure What are

the dimensions to two decimal places of all possible boxes that

will have a volume of 75 cubic inches? (Refer to the figure for

Problem 65.)

67 MANUFACTURING An oil tank in the shape of a right

circu-lar cylinder* has a volume of 40,000 cubic feet If regulations

for such tanks require that the radius plus the height must be

50 feet, find the radius and the height to two decimal places.

*Geometric formulas can be found in Appendix C.

72. PRICE AND REVENUE Refer to Problem 70 The revenue

from the sale of x cases of paper at $y per case is given by the

Z Defining Relations and Functions

Z Defining Functions by Equations

Z Finding the Domain of a Function

Z Using Function Notation

Z Modeling and Data Analysis

Z A Brief History of the Function Concept

The idea of correspondence plays a really important role in understanding the cept of functions, which is almost certainly the most important idea in this course.The good news is that you have already had years of experience with correspondences

con-in everyday life For example,For every person, there is a corresponding age

For every item in a store, there is a corresponding price

For every season, there is a corresponding Super Bowl champion

For every circle, there is a corresponding area

For every number, there is a corresponding cube

One of the most basic and important ways that math can be applied to other areas

of study is the establishment of correspondences among various types of phenomena

In many cases, once a correspondence is known, it can be used to make importantdecisions and predictions An engineer can use a formula to predict the weight capac-ity of a stadium grandstand A political operative decides how many resources to allo-cate to a race given current polling results A computer scientist can use formulas tocompare the efficiency of algorithms for sorting data stored on a computer An econ-omist would like to be able to predict interest rates, given the rate of change of themoney supply And the list goes on and on