College Algebra: A Graphing Approach

805 Pages·2004·15.62 MB·English

by  Ron (Ron Larson) Larson

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LIBRARY OF FUNCTIONS SUMMARY
Linear Function Absolute Value Function Square Root Function
f(cid:1)x(cid:2)(cid:3)mx(cid:4)b f(cid:1)x(cid:2)(cid:3)(cid:9)x(cid:9)(cid:3)(cid:10)x, x ≥ 0 f(cid:1)x(cid:2)(cid:3)(cid:8)x
(cid:2)x, x < 0
y y y
2 4
3 f(x) = x
(0, b) 1
f(x) = x
2
x x
(− b , 0( (− b , 0( −2 −1 (0, 0) 2 1
m m
−1
x
f(x) = mx + b, f(x) = mx + b, −1 (0, 0) 2 3 4
m > 0 m < 0 −2 −1
Domain:(cid:1)(cid:2)(cid:1), (cid:1)(cid:2) Domain:(cid:1)(cid:2)(cid:1), (cid:1)(cid:2) Domain:(cid:4)0, (cid:1)(cid:2)
Range:(cid:1)(cid:2)(cid:1), (cid:1)(cid:2) Range:(cid:4)0, (cid:1)(cid:2) Range:(cid:4)0, (cid:1)(cid:2)
x-intercept:(cid:1)(cid:2)b(cid:7)m, 0(cid:2) Intercept:(cid:1)0, 0(cid:2) Intercept:(cid:1)0, 0(cid:2)
y-intercept:(cid:1)0, b(cid:2) Decreasing on (cid:1)(cid:2)(cid:1), 0(cid:2) Increasing on (cid:1)0, (cid:1)(cid:2)
Increasing when m > 0 Increasing on (cid:1)0, (cid:1)(cid:2)
Decreasing when m < 0 Even function
y-axis symmetry
Greatest Integer Function Quadratic (Squaring) Function Cubic Function
f(cid:1)x(cid:2)(cid:3)(cid:5)x(cid:6) f(cid:1)x(cid:2)(cid:3)ax2 f(cid:1)x(cid:2)(cid:3)x3
y y y
f(x) = [ [ x]]
3 3 3
2 2 2
1 1 f(x) =ax2,a>0
(0, 0)
x x x
−3 −2 −1 1 2 3 −2 −1 1 2 3 4 −3 −2 1 2 3
−1 f(x) =ax2,a<0 −1 f(x) = x3
−2 −2
−3 −3 −3
Domain:(cid:1)(cid:2)(cid:1), (cid:1)(cid:2) Domain:(cid:1)(cid:2)(cid:1), (cid:1)(cid:2) Domain:(cid:1)(cid:2)(cid:1), (cid:1)(cid:2)
Range:the set of integers Range (cid:1)a > 0(cid:2):(cid:4)0, (cid:1)(cid:2) Range:(cid:1)(cid:2)(cid:1), (cid:1)(cid:2)
x-intercepts:in the interval (cid:4)0, 1(cid:2) Range (cid:1)a < 0(cid:2):(cid:1)(cid:2)(cid:1), 0(cid:3) Intercept:(cid:1)0, 0(cid:2)
y-intercept:(cid:1)0, 0(cid:2) Intercept:(cid:1)0, 0(cid:2) Increasing on (cid:1)(cid:2)(cid:1), (cid:1)(cid:2)
Constant between each pair of Decreasing on (cid:1)(cid:2)(cid:1), 0(cid:2)for a > 0 Odd function
consecutive integers Increasing on (cid:1)0, (cid:1)(cid:2)for a > 0 Origin symmetry
Jumps vertically one unit at Increasing on (cid:1)(cid:2)(cid:1), 0(cid:2)for a < 0
each integer value Decreasing on (cid:1)0, (cid:1)(cid:2)for a < 0
Even function
y-axis symmetry
Relative minimum (cid:1)a > 0(cid:2),
relative maximum (cid:1)a < 0(cid:2),
or vertex:(cid:1)0, 0(cid:2)
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Rational (Reciprocal) Function Exponential Function Logarithmic Function
1
f(cid:1)x(cid:2)(cid:3) f(cid:1)x(cid:2)(cid:3)ax, a > 0, a(cid:5)1 f(cid:1)x(cid:2)(cid:3)log x, a > 0, a(cid:5)1
x a
y y y
3 1
f(x) = f(x) = log x
x 1 a
2
1 f(x) = ax f(x) = a−x
(0, 1) (1, 0)
x x
−1 1 2 3 1 2
x
−1
Domain:(cid:1)(cid:2)(cid:1), 0(cid:2)(cid:1)(cid:1)0, (cid:1)) Domain:(cid:1)(cid:2)(cid:1), (cid:1)(cid:2) Domain:(cid:1)0, (cid:1)(cid:2)
Range:(cid:1)(cid:2)(cid:1), 0(cid:2)(cid:1)(cid:1)0, (cid:1)) Range:(cid:1)0, (cid:1)(cid:2) Range:(cid:1)(cid:2)(cid:1), (cid:1)(cid:2)
No intercepts Intercept:(cid:1)0, 1(cid:2) Intercept:(cid:1)1, 0(cid:2)
Decreasing on (cid:1)(cid:2)(cid:1), 0(cid:2) and (cid:1)0, (cid:1)(cid:2) Increasing on (cid:1)(cid:2)(cid:1), (cid:1)(cid:2) Increasing on (cid:1)0, (cid:1)(cid:2)
Odd function for f(cid:1)x(cid:2)(cid:3)ax y-axis is a vertical asymptote
Origin symmetry Decreasing on (cid:1)(cid:2)(cid:1), (cid:1)(cid:2) Continuous
Vertical asymptote:y-axis for f(cid:1)x(cid:2)(cid:3)a(cid:2)x Reflection of graph of f(cid:1)x(cid:2)(cid:3)ax
Horizontal asymptote:x-axis x-axis is a horizontal asymptote in the line y(cid:3)x
Continuous
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College Algebra
A Graphing Approach
Fourth Edition
Ron Larson
Robert P. Hostetler
The Pennsylvania State University
The Behrend College
Bruce H. Edwards
The University of Florida
With the assistance of David C. Falvo
The Pennsylvania State University
The Behrend College
Houghton Mifflin Company Boston New York
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Vice President and Publisher:Jack Shira
Associate Sponsoring Editor:Cathy Cantin
Development Manager:Maureen Ross
Assistant Editor:Lisa Pettinato
Assistant Editor:James Cohen
Supervising Editor:Karen Carter
Senior Project Editor:Patty Bergin
Editorial Assistant:Allison Seymour
Production Technology Supervisor:Gary Crespo
Executive Marketing Manager:Michael Busnach
Senior Marketing Manager:Danielle Potvin
Marketing Associate:Nicole Mollica
Senior Manufacturing Coordinator:Priscilla Bailey
Composition and Art:Meridian Creative Group
Cover Design Manager:Diana Coe
Cover photograph ©Lucidio Studio,Inc./SuperStock
We have included examples and exercises that use real-life data as well as technology output
from a variety of software. This would not have been possible without the help of many people
and organizations. Our wholehearted thanks go to all their time and effort.
Copyright ©2005 by Houghton Mifflin Company. All rights reserved.
No part of this work may be reproduced or transmitted in any form or by any means,electronic
or mechanical,including photocopying and recording,or by any information storage or retrieval
system,without the prior written permission of Houghton Mifflin Company unless such copy-
ing is expressly permitted by federal copyright law. Address inquiries to College Permissions,
Houghton MifflinCompany,222 Berkeley Street,Boston,MA 02116-3764.
Printed in the U.S.A.
Library of Congress Catalog Card Number:2003113989
ISBN:0-618-39437-0
123456789–DOW–08 07 06 05 04
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Contents iii
Contents
A Word from the Authors (Preface) vi
C
O
Features Highlights x
N
T
E
Chapter P Prerequisites 1 N
T
P.1 Real Numbers 2 S
P.2 Exponents and Radicals 12
P.3 Polynomials and Factoring 24
P.4 Rational Expressions 37
P.5 The Cartesian Plane 47
P.6 Exploring Data: Representing Data Graphically 58
Chapter Summary 67 Review Exercises 68
Chapter Test 72
Chapter 1 Functions and Their Graphs 73
Introduction to Library of Functions 74
1.1 Graphs of Equations 75
1.2 Lines in the Plane 86
1.3 Functions 99
1.4 Graphs of Functions 113
1.5 Shifting, Reflecting, and Stretching Graphs 125
1.6 Combinations of Functions 134
1.7 Inverse Functions 145
Chapter Summary 155 Review Exercises 156
Chapter Test 160
Chapter 2 Solving Equations and Inequalities 161
2.1 Linear Equations and Problem Solving 162
2.2 Solving Equations Graphically 172
2.3 Complex Numbers 183
2.4 Solving Equations Algebraically 191
2.5 Solving Inequalities Algebraically and Graphically 210
2.6 Exploring Data: Linear Models and Scatter Plots 222
Chapter Summary 231 Review Exercises 232
Chapter Test 236
Cumulative Test: Chapters P–2 237
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iv Contents
Chapter 3 Polynomial and Rational Functions 239
3.1 Quadratic Functions 240
3.2 Polynomial Functions of Higher Degree 251
3.3 Real Zeros of Polynomial Functions 264
3.4 The Fundamental Theorem of Algebra 279
3.5 Rational Functions and Asymptotes 286
3.6 Graphs of Rational Functions 296
3.7 Exploring Data: Quadratic Models 305
Chapter Summary 312 Review Exercises 313
Chapter Test 318
Chapter 4 Exponential and Logarithmic Functions 319
4.1 Exponential Functions and Their Graphs 320
4.2 Logarithmic Functions and Their Graphs 332
4.3 Properties of Logarithms 343
4.4 Solving Exponential and Logarithmic Equations 350
4.5 Exponential and Logarithmic Models 361
4.6 Exploring Data: Nonlinear Models 373
Chapter Summary 382 Review Exercises 383
Chapter Test 388
Chapter 5 Linear Systems and Matrices 389
5.1 Solving Systems of Equations 390
5.2 Systems of Linear Equations in Two Variables 401
5.3 Multivariable Linear Systems 411
5.4 Matrices and Systems of Equations 427
5.5 Operations with Matrices 442
5.6 The Inverse of a Square Matrix 457
5.7 The Determinant of a Square Matrix 466
5.8 Applications of Matrices and Determinants 474
Chapter Summary 484 Review Exercises 486
Chapter Test 492
Cumulative Test: Chapters 3–5 493
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Contents v
Chapter 6 Sequences, Series, and Probability 495
6.1 Sequences and Series 496
6.2 Arithmetic Sequences and Partial Sums 507
C
6.3 Geometric Sequences and Series 516
O
N
6.4 Mathematical Induction 526
T
E
6.5 The Binomial Theorem 534 N
T
6.6 Counting Principles 542 S
6.7 Probability 552
Chapter Summary 565 Review Exercises 566
Chapter Test 570
Chapter 7 Conics and Parametric Equations 571
7.1 Conics 572
7.2 Translations of Conics 586
7.3 Parametric Equations 595
Chapter Summary 603 Review Exercises 604
Chapter Test 608
Cumulative Test: Chapters 6–7 609
Appendices Appendix A Technology Support Guide A1
Appendix B Proofs of Selected Theorems A25
Appendix C Concepts in Statistics A31
C.1 Measures of Central Tendency and Dispersion A31
C.2 Least Squares Regression A40
Appendix D Solving Linear Equations and Inequalities A42
Appendix E Systems of Inequalities A45
E.1 Solving Systems of Inequalities A45
E.2 Linear Programming A55
Answers to Odd-Numbered Exercises and Tests A65
Index of Applications A163
Index A167
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vi A Word from the Authors
A Word from the Authors
Welcome to College Algebra: A Graphing Approach, Fourth Edition. We are
pleased to present this new edition of our textbook in which we focus on
making the mathematics accessible, supporting student success, and offering
instructors flexibility in how the course can be taught.
Accessible to Students
Over the years we have taken care to write this text with the student in mind.
Paying careful attention to the presentation, we use precise mathematical
language and a clear writing style to develop an effective learning tool. We
believe that every student can learn mathematics, and we are committed to
providing a text that makes the mathematics of the college algebra course
accessible to all students. For the Fourth Edition,we have revised and improved
many text features designed for this purpose.
Throughout the text, we present solutions to many examples from multiple
perspectives—algebraic, graphic, and numeric. The side-by-side format of this
pedagogical feature helps students to see that a problem can be solved in
more than one way and to see that different methods yield the same result. The
side-by-side format also addresses many different learning styles.
We have found that many college algebra students grasp mathematical concepts
more easily when they work with them in the context of real-life situations.
Students have numerous opportunities to do this throughout the Fourth Edition,
in examples and exercises, including developing models to fit current real data.
To reinforce the concept of functions, we have compiled all the elementary
functions as a Library of Functions. Each function is introduced at the first point
of use in the text with a definition and description of basic characteristics; all
elementary functions are also presented in a summary on the front endpapers of
the text for convenient reference.
We have carefully written and designed each page to make the book more
readable and accessible to students. For example, to avoid unnecessary page
turning and disruptions to students’ thought processes, each example and
corresponding solution begins and ends on the same page.
Supports Student Success
During more than thirty years of teaching and writing, we have learned many
things about the teaching and learning of mathematics. We have found that
students are most successful when they know what they are expected to learn and
why it is important to learn it. With that in mind,we have enhanced the thematic
study thread throughout the Fourth Edition.
Each chapter begins with a list of section references and a study guide,What You
Should Learn,which is a comprehensive overview of the chapter concepts. This
study guide helps students prepare to study and learn the material in the chapter.
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A Word from the Authors vii
Using the same pedagogical theme, each section begins with a set of section
learning objectives—What You Should Learn. These are followed by an engaging
real-life application—Why You Should Learn It—that motivates students and
illustrates an area where the mathematical concepts will be applied in an exam-
ple or exercise in the section. The Chapter Summary—What Did You Learn?—at
the end of each chapter is a section-by-section overview that ties the learning
objectives from the chapter to sets of Review Exercisesat the end of each chapter.
Throughout the text, other features further improve accessibility. Study Tips are
provided throughout the text at point-of-use to reinforce concepts and to help
students learn how to study mathematics. Explorations have been expanded in
order to reinforce mathematical concepts. Each Example with worked-out
solution is followed by a Checkpoint,which directs the student to work a similar
exercise from the exercise set. The Section Exercises now begin with a
Vocabulary Check, which gives the students an opportunity to test their P
understanding of the important terms in the section. Synthesis Exercises check R
E
students’ conceptual understanding of the topics in each section and Review F
A
Exercisesprovide additional practice with the concepts in the chapter or previous C
E
chapters. Chapter Tests,at the end of each chapter,and periodic Cumulative Tests
offer students frequent opportunities for self-assessment and to develop strong
study- and test-taking skills.
The use of technology also supports students with different learning styles, and
graphing calculators are fully integrated into the text presentation. In the Fourth
Edition,a robust Technology Support Appendixhas been added to make it easier
for students to use technology. Technology Supportnotes are provided throughout
the text at point-of-use. These notes guide students to the Technology Support
Appendix,where they can learn how to use specific graphing calculator features
to enhance their understanding of the concepts presented in the text. These notes
also direct students to the Graphing Technology Guide,on the textbook website,
for keystroke support that is available for numerous calculator models.
Technology Tips are provided in the text at point-of-use to call attention to the
strengths and weaknesses of graphing technology, as well as to offer alternative
methods for solving or checking a problem using technology. Because students
are often misled by the limitations of graphing calculators, we have, where
appropriate, used color to enhance the graphing calculator displays in the
textbook. This enables students to visualize the mathematical concepts clearly
and accurately and avoid common misunderstandings.
Numerous additional text-specific resources are available to help students
succeed in the college algebra course. These include “live” online tutoring,
instructional DVDs and videos,and a variety of other resources,such as tutorial
support and self-assessment, which are available on CD-ROM and the Web. In
addition,the Student Success Organizeris a note-taking guide that helps students
organize their class notes and create an effective study and review tool.
Flexible Options for Instructors
From the time we first began writing textbooks in the early 1970s, we have
always considered it a critical part of our role as authors to provide instructors
with flexible programs. In addition to addressing a variety of learning styles,
the optional features within the text allow instructors to design their courses to
meet their instructional needs and the needs of their students. For example, the