College Algebra: Graphs and Models

805 Pages·2012·23.021 MB·English

by  Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen, Judith A. Penna

College Algebra
GRAPHS AND MODELS
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E D I T I O N
5
College Algebra
GRA PHS AND M ODELS
MARVIN L. BITTINGER
Indiana University Purdue University Indianapolis
JUDITH A. BEECHER
Indiana University Purdue University Indianapolis
DAVID J. ELLENBOGEN
Community College of Vermont
JUDITH A. PENNA
Indiana University Purdue University Indianapolis
Boston Columbus Indianapolis New York San Francisco Upper Saddle River
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Library of Congress Cataloging-in-Publication Data
College algebra : graphs and models/Marvin L. Bittinger [… et al.].
—5th ed.
p. cm.
Includes index.
ISBN-13: 978-0-321-78395-0
ISBN-10: 0-321-78395-6
1. Algebra—Textbooks. 2. Algebra—Graphic methods—Textbooks.
3. Function algebras—Textbooks. I. Bittinger, Marvin L.
QA154.3.C65 2013
512—dc22 2011011992
Copyright © 2013, 2009, 2006, 2001 Pearson Education, Inc.
All rights reserved. No part of this publication may be reproduced, stored in a r etrieval system, or transmitted, in any form or
by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the pub-
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MA 02116, fax your request to 617-671-3447, or e-mail at http://www.pearsoned.com/legal/permissions.htm.
1 2 3 4 5 6 7 8 9 10—DOW—15 14 13 12 11
ISBN-10: 0-321-78395-6
www.pearsonhi ghered.com ISBN-13: 978-0-321-78395-0
Contents
Preface xi
To the Student xix
Interactive Figures Index xx
R
Basic Concepts of Algebra 1
R.1 The Real-Number System 2
Real Numbers / Interval Notation / Properties of the Real
Numbers / Absolute Value
R.2 Integer Exponents, Scientific Notation, and Order of Operations 8
Integers as Exponents / Scientific Notation / Order of Operations
R.3 Addition, Subtraction, and Multiplication of Polynomials 17
Polynomials / Addition and Subtraction / Multiplication
R.4 Factoring 22
Terms with Common Factors / Factoring by Grouping / Trinomials of
the Type x2 + bx + c / Trinomials of the Type ax2 + bx + c, a (cid:2) 1 /
Special Factorizations
R.5 The Basics of Equation Solving 30
Linear Equations and Quadratic Equations / Formulas
R.6 Rational Expressions 35
The Domain of a Rational Expression / Simplifying, Multiplying, and Dividing
Rational Expressions / Adding and Subtracting Rational Expressions /
Complex Rational Expressions
R.7 Radical Notation and Rational Exponents 44
Simplifying Radical Expressions / An Application / Rationalizing
Denominators or Numerators / Rational Exponents
Study Guide 53 Review Exercises 54 Chapter Test 56
1
Graphs, Functions, and Models 57
1.1 Introduction to Graphing 58
Graphs / Solutions of Equations / Graphs of Equations / The Distance
Formula / Midpoints of Segments / Circles
VISUALIZING THE GRAPH 69
1.2 Functions and Graphs 74
Functions / Notation for Functions / Graphs of Functions /
Finding Domains of Functions / Visualizing Domain and Range /
Applications of Functions
v
vi CONTENTS
1.3 Linear Functions, Slope, and Applications 90
Linear Functions / The Linear Function f(x) = mx + b and Slope /
Applications of Slope / Slope–Intercept Equations of Lines / Graphing
f(x) = mx + b Using m and b / Applications of Linear Functions
VISUALIZING THE GRAPH 101
MID-CHAPTER MIXED REVIEW 106
1.4 Equations of Lines and Modeling 107
Slope–Intercept Equations of Lines / Point–Slope Equations of Lines /
Parallel Lines / Perpendicular Lines / Mathematical Models / Curve
Fitting / Linear Regression
1.5 Linear Equations, Functions, Zeros, and Applications 121
Linear Equations / Applications Using Linear Models / Zeros of
Linear Functions
1.6 Solving Linear Inequalities 139
Linear Inequalities / Compound Inequalities / An Application
Study Guide 144 Review Exercises 151 Chapter Test 155
2
More on Functions 159
2.1 Increasing, Decreasing, and Piecewise Functions; Applications 160
Increasing, Decreasing, and Constant Functions / Relative Maximum and
Minimum Values / Applications of Functions / Functions Defined Piecewise
2.2 The Algebra of Functions 175
The Algebra of Functions: Sums, Differences, Products, and Quotients /
Difference Quotients
2.3 The Composition of Functions 182
The Composition of Functions / Decomposing a Function as a Composition
MID-CHAPTER MIXED REVIEW 190
2.4 Symmetry 192
Symmetry / Even Functions and Odd Functions
2.5 Transformations 199
Transformations of Functions / Vertical Translations and Horizontal
Translations / Reflections / Vertical and Horizontal Stretchings and Shrinkings
VISUALIZING THE GRAPH 209
2.6 Variation and Applications 213
Direct Variation / Inverse Variation / Combined Variation
Study Guide 222 Review Exercises 229 Chapter Test 233
CONTENTS vii
3
Quadratic Functions and Equations; Inequalities 235
3.1 The Complex Numbers 236
The Complex-Number System / Addition and Subtraction /
Multiplication / Conjugates and Division
3.2 Quadratic Equations, Functions, Zeros, and Models 243
Quadratic Equations and Quadratic Functions / Completing the Square /
Using the Quadratic Formula / The Discriminant / Equations Reducible
to Quadratic / Applications
3.3 Analyzing Graphs of Quadratic Functions 259
Graphing Quadratic Functions of the Type f(x) = a (x - h)2 + k / Graphing
Quadratic Functions of the Type f(x) = ax2 + bx + c, a (cid:2) 0 / Applications
VISUALIZING THE GRAPH 269
MID-CHAPTER MIXED REVIEW 273
3.4 Solving Rational Equations and Radical Equations 275
Rational Equations / Radical Equations
3.5 Solving Equations and Inequalities with Absolute Value 283
Equations with Absolute Value / Inequalities with Absolute Value
Study Guide 287 Review Exercises 293 Chapter Test 295
4
Polynomial Functions and Rational Functions 297
4.1 Polynomial Functions and Modeling 298
The Leading-Term Test / Finding Zeros of Factored Polynomial
Functions / Finding Real Zeros on a Calculator / Polynomial Models
4.2 Graphing Polynomial Functions 315
Graphing Polynomial Functions / The Intermediate Value Theorem
VISUALIZING THE GRAPH 322
4.3 Polynomial Division; The Remainder Theorem and the Factor Theorem 325
Division and Factors / The Remainder Theorem and Synthetic Division /
Finding Factors of Polynomials
MID-CHAPTER MIXED REVIEW 333
4.4 Theorems about Zeros of Polynomial Functions 335
The Fundamental Theorem of Algebra / Finding Polynomials with Given Zeros /
Zeros of Polynomial Functions with Real Coefficients / Rational Coefficients /
Integer Coefficients and the Rational Zeros Theorem / Descartes’ Rule of Signs
4.5 Rational Functions 345
The Domain of a Rational Function / Asymptotes / Applications
VISUALIZING THE GRAPH 361
4.6 Polynomial Inequalities and Rational Inequalities 365
Polynomial Inequalities / Rational Inequalities
Study Guide 375 Review Exercises 385 Chapter Test 389
viii CONTENTS
5
Exponential Functions and Logarithmic Functions 391
5.1 Inverse Functions 392
Inverses / Inverses and One-to-One Functions / Finding Formulas for
Inverses / Inverse Functions and Composition / Restricting a Domain
5.2 Exponential Functions and Graphs 404
Graphing Exponential Functions / Applications / The Number e / Graphs of
Exponential Functions, Base e
5.3 Logarithmic Functions and Graphs 417
Logarithmic Functions / Finding Certain Logarithms / Converting Between
Exponential Equations and Logarithmic Equations / Finding Logarithms on a
Calculator / Natural Logarithms / Changing Logarithmic Bases / Graphs of
Logarithmic Functions / Applications
VISUALIZING THE GRAPH 429
MID-CHAPTER MIXED REVIEW 433
5.4 Properties of Logarithmic Functions 435
Logarithms of Products / Logarithms of Powers / Logarithms of Quotients /
Applying the Properties / Simplifying Expressions of the Type log ax and alogax
a
5.5 Solving Exponential Equations and Logarithmic Equations 443
Solving Exponential Equations / Solving Logarithmic Equations
5.6 Applications and Models: Growth and Decay; Compound Interest 453
Population Growth / Interest Compounded Continuously / Models of Limited
Growth / Exponential Decay / Exponential and Logarithmic Curve Fitting
Study Guide 470 Review Exercises 477 Chapter Test 481
6
Systems of Equations and Matrices 483
6.1 Systems of Equations in Two Variables 484
Solving Systems of Equations Graphically / The Substitution Method /
The Elimination Method / Applications
VISUALIZING THE GRAPH 494
6.2 Systems of Equations in Three Variables 500
Solving Systems of Equations in Three Variables / Applications /
Mathematical Models and Applications
6.3 Matrices and Systems of Equations 510
Matrices and Row-Equivalent Operations / Gaussian Elimination with
Matrices / Gauss–Jordan Elimination
6.4 Matrix Operations 518
Matrix Addition and Subtraction / Scalar Multiplication / Products of
Matrices / Matrix Equations
MID-CHAPTER MIXED REVIEW 529
CONTENTS ix
6.5 Inverses of Matrices 531
The Identity Matrix / The Inverse of a Matrix / Solving Systems of Equations
6.6 Determinants and Cramer’s Rule 538
Determinants of Square Matrices / Evaluating Determinants Using
Cofactors / Cramer’s Rule
6.7 Systems of Inequalities and Linear Programming 546
Graphs of Linear Inequalities / Systems of Linear Inequalities / Applications:
Linear Programming
6.8 Partial Fractions 558
Partial Fraction Decompositions
Study Guide 564 Review Exercises 570 Chapter Test 574
7
Conic Sections 577
7.1 The Parabola 578
Parabolas / Finding Standard Form by Completing the Square / Applications
7.2 The Circle and the Ellipse 586
Circles / Ellipses / Applications
MID-CHAPTER MIXED REVIEW 595
7.3 The Hyperbola 596
Standard Equations of Hyperbolas / Applications
7.4 Nonlinear Systems of Equations and Inequalities 606
Nonlinear Systems of Equations / Modeling and Problem Solving /
Nonlinear Systems of Inequalities
VISUALIZING THE GRAPH 613
Study Guide 618 Review Exercises 622 Chapter Test 625
8
Sequences, Series, and Combinatorics 627
8.1 Sequences and Series 628
Sequences / Finding the General Term / Sums and Series / Sigma
Notation / Recursive Definitions
8.2 Arithmetic Sequences and Series 636
Arithmetic Sequences / Sum of the First n Terms of an Arithmetic
Sequence / Applications
8.3 Geometric Sequences and Series 645
Geometric Sequences / Sum of the First n Terms of a Geometric
Sequence / Infinite Geometric Series / Applications
VISUALIZING THE GRAPH 653