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College Algebra and
Trigonometry:
Building Concepts and Connections
Revathi Narasimhan
Kean University
HOUGHTON MIFFLIN HARCOURT PUBLISHING COMPANY Boston New York
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Copyright © 2009 by Houghton Mifflin Harcourt Publishing Company.All rights reserved.
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Instructor’s Annotated Edition:
ISBN-10:0-618-41290-5
ISBN-13:978-0-618-41290-7
For orders,use student text ISBNs:
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1 2 3 4 5 6 7 89-CRK-12 11 10 09 08
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Contents
About the Author ix
Preface xi
1
Algebra and Geometry Review
Chapter P
2
P.1 The Real Number System
11
P.2 Integer Exponents and Scientific Notation
20
P.3 Roots, Radicals, and Rational Exponents
27
P.4 Polynomials
33
P.5 Factoring
41
P.6 Rational Expressions
48
P.7 Geometry Review
53
P.8 Solving Basic Equations
Chapter P Summary 57
Review P Exercises 63
Chapter P Test 66
67
Functions, Graphs, and Applications
Chapter 1
68
1.1 Functions
80
1.2 Graphs of Functions
90
1.3 Linear Functions
106
1.4 Modeling with Linear Functions; Variation
120
1.5 Intersections of Lines and Linear Inequalities
Chapter 1 Summary 133
Review 1 Exercises 136
Chapter 1 Test 139
141
More About Functions and Equations
Chapter 2
142
2.1 Coordinate Geometry:Distance, Midpoints, and Circles
151
2.2 The Algebra of Functions
163
2.3 Transformations of the Graph of a Function
178
2.4 Symmetry and Other Properties of Functions
188
2.5 Equations and Inequalities Involving Absolute Value
195
2.6 Piecewise-Defined Functions
Chapter 2 Summary 203
Review 2 Exercises 207
Chapter 2 Test 211
v
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vi ■ Contents
213
Quadratic Functions
Chapter 3
214
3.1 Graphs of Quadratic Functions
231
3.2 Quadratic Equations
246
3.3 Complex Numbers and Quadratic Equations
256
3.4 Quadratic Inequalities
3.5 Equations That Are Reducible to Quadratic Form; Rational
266
and Radical Equations
Chapter 3 Summary 276
Review 3 Exercises 280
Chapter 3 Test 282
283
Polynomial and Rational Functions
Chapter 4
284
4.1 Graphs of Polynomial Functions
299
4.2 More on Graphs of Polynomial Functions and Models
308
4.3 Division of Polynomials; the Remainder and Factor Theorems
316
4.4 Real Zeros of Polynomials; Solutions of Equations
325
4.5 The Fundamental Theorem of Algebra; Complex Zeros
331
4.6 Rational Functions
348
4.7 Polynomial and Rational Inequalities
Chapter 4 Summary 354
Review 4 Exercises 359
Chapter 4 Test 362
363
Exponential and Logarithmic Functions
Chapter 5
364
5.1 Inverse Functions
376
5.2 Exponential Functions
391
5.3 Logarithmic Functions
407
5.4 Properties of Logarithms
415
5.5 Exponential and Logarithmic Equations
425
5.6 Exponential, Logistic, and Logarithmic Models
Chapter 5 Summary 436
Review 5 Exercises 441
Chapter 5 Test 444
445
Trigonometric Functions
Chapter 6
446
6.1 Angles and Their Measures
460
6.2 Trigonometric Functions of Acute Angles
473
6.3 Trigonometric Functions of Any Angle Using Right Triangles
485
6.4 Trigonometric Functions of Any Angle Using the Unit Circle
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Contents ■ vii
502
6.5 Graphs of Sine and Cosine Functions
519
6.6 Graphs of Other Trigonometric Functions
529
6.7 Inverse Trigonometric Functions
Chapter 6 Summary 541
Review 6 Exercises 549
Chapter 6 Test 551
553
Trigonometric Identities and Equations
Chapter 7
554
7.1 Verifying Identities
561
7.2 Sum and Difference Identities
574
7.3 Multiple-Angle Identities; Sum and Product Identities
586
7.4 Trigonometric Equations
Chapter 7 Summary 597
Review 7 Exercises 600
Chapter 7 Test 602
603
Additional Topics in Trigonometry
Chapter 8
604
8.1 The Law of Sines
616
8.2 The Law of Cosines
625
8.3 Polar Coordinates
636
8.4 Graphs of Polar Equations
649
8.5 Vectors
661
8.6 Dot Product of Vectors
670
8.7 Trigonometric Form of a Complex Number
Chapter 8 Summary 677
Review 8 Exercises 683
Chapter 8 Test 686
687
Systems of Equations and Inequalities
Chapter 9
688
9.1 Systems of Linear Equations and Inequalities in Two Variables
706
9.2 Systems of Linear Equations in Three Variables
718
9.3 Solving Systems of Equations Using Matrices
734
9.4 Operations on Matrices
749
9.5 Matrices and Inverses
761
9.6 Determinants and Cramer’s Rule
771
9.7 Partial Fractions
778
9.8 Systems of Nonlinear Equations
Chapter 9 Summary 786
Review 9 Exercises 790
Chapter 9 Test 793
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795
Conic Sections
Chapter 10
796
10.1 The Parabola
809
10.2 The Ellipse
823
10.3 The Hyperbola
837
10.4 Rotation of Axes; General Form of Conic Sections
847
10.5 Polar Equations of Conic Sections
853
10.6 Parametric Equations
Chapter 10 Summary 861
Review 10 Exercises 865
Chapter 10 Test 867
869
More Topics in Algebra
Chapter 11
870
11.1 Sequences
882
11.2 Sums of Terms of Sequences
895
11.3 General Sequences and Series
903
11.4 Counting Methods
914
11.5 Probability
925
11.6 The Binomial Theorem
931
11.7 Mathematical Induction
Chapter 11 Summary 936
Review 11 Exercises 941
Chapter 11 Test 943
945
Keystroke Guide for the TI-83/84 Calculator Series
Appendix A
946
A.1 Keys on Your Calculator
946
A.2 Getting Started
948
A.3 Editing and Deleting
948
A.4 Entering and Evaluating Common Expressions
950
A.5 Entering and Evaluating Functions
951
A.6 Building a Table
953
A.7 Graphing Linear, Quadratic, and Piecewise-Defined Functions
955
A.8 Graphing Polynomials, Rational Functions, and Inequalities
957
A.9 Solving Equations
959
A.10 Finding the Maximum and Minimum of a Function
960
A.11 Complex Numbers
961
A.12 Fitting Curves to Data (Regression)
962
A.13 Matrices
966
A.14 Sequences and Series
967
A.15 Trigonometry
970
A.16 Parametric Equations
Answers to Check It Out Exercises S1
Answers to Odd-Numbered Exercises A1
Index I1
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ABOUT THE AUTHOR
R
evathi Narasimhan received her Ph.D.in Applied Mathematics from the Uni-
versity of Maryland at College Park. She grew up in Mesa,Arizona and re-
ceived her undergraduate degree at Arizona State University.She is currently
on the faculty of the Mathematics Department at Kean University in Union,
New Jersey.
Professionally trained to apply the principles of analysis and algebra,she is keen on
helping her students understand the “big picture” concepts in mathematics, whether
at the graduate or undergraduate level. In addition to this textbook, she has written
scholarly articles for academic journals and technology supplements for other text-
books.
She and her husband,a research microbiologist,have two sons.Reva likes to garden
and sew and is an avid reader.
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PREFACE
Our Mission
The goal of College Algebra and Trigonometry: Building Concepts and Connections is to
teach students to more clearly see how mathematical concepts connect and relate.We
set out to accomplish this goal in two fundamental ways.
Functions as a Unifying Theme
First,we considered the order in which functions should be presented relative to their
corresponding equations.Accordingly,rather than present a comprehensive review of
equations and equation solving in Chapter 1,we introduce functions in Chapter 1.We
then present related equations and techniques for solving those equations in the con-
text of their associated functions. When equations are presented in conjunction with
their “functional”counterparts in this way,students come away with a more coherent
picture of the mathematics.
Pedagogical Reinforcement
We also created a pedagogy that “recalls” previous topics and skills by way of linked
examples and Just in Time exercises and references.Through these devices, students
receive consistent prompts that enable them to better remember and apply what they
have learned.
Ultimately,our hope is that through College Algebra and Trigonometry: Building Concepts
and Connections, students will develop a better conceptual understanding of the sub-
ject and achieve greater preparedness for future math courses.
Which Textbook is Right for You?
We recognize that instructors’needs in this course area are diverse.By offering varia-
tion in the coverage of trigonometry—in particular, variation in the right triangle ap-
proach relative to the unit circle approach—this series strives to meet everyone’s needs.
College Algebra and Trigonometry: Building Concepts and Connections
Do you put as much emphasis on the right triangle ap-
proach as you do the unit circle approach to find the
values of trigonometric functions of non-acute angles?
If so,we recommend College Algebra and Trigonometry.
Precalculus: Building Concepts and Connections
Do you emphasize use of the unit circle to find the val-
ues of trigonometric functions of non-acute angles,
more so than the right triangle approach?
If so,we recommend Precalculus.
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