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Tables of Contents for Algebra and Trigonometry

Chapter/Section Title

Page #

Page Count

PREFACE

XV

CHAPTER 1 MODELING AND PROBLEM SOLVING

1

28

1-1 A Case for Algebra

1

6

Algebra Is an Analytical Tool

4

1

Algebra Is Rich in Concepts and Contexts

4

1

Algebra Is Part of Our Intellectual Heritage

5

1

You May Actually Use Algebra!

5

1

Some Suggestions for Feeling at Home with This Book

5

2

1-2 Models of Quantitative Relationships

7

12

Three Types of Model

7

1

Information Provided by the Three Types of Model

8

1

Creating One Model from Another

9

10

1-3 Problem-Solving Strategies

19

8

First Step: Understand the Problem

20

1

Second Step: Devise a Plan

21

1

Third Step: Carry Out the Plan

22

1

Fourth Step: Look Back

23

4

Chapter Review

27

2

CHAPTER 2 FUNCTIONS

29

44

2-1 Three Views of Functions

29

11

The Definition of Function

29

2

A Numerical View of Functions

31

1

An Analytical View of Functions

32

1

A Graphical View of Functions

33

3

Reasons to Study Functions and Their Models

36

3

Supplementary Topic: Nonnumerical Functions

39

1

2-2 The Concept of Function As Process

40

11

Functional Notation

41

1

Combining Functions

42

9

2-3 Domain and Range

51

14

Domain and Range of Abstract Functions

52

3

Domain and Range of Functions in a Physical Context

55

2

Sequences

57

2

Dynamic Behavior: Increasing and Decreasing Functions

59

6

Supplementary Topic: Domain and Range of Combinations of Functions

65

1

2-4 Solving Equations and Inequalities Graphically

65

6

Solving Equations Graphically

66

2

Solving Inequalities Graphically

68

3

Chapter Review

71

2

CHAPTER 3 LINEAR FUNCTIONS

73

42

3-1 Three Views of Linear Functions

73

10

A Numerical View of Linear Functions

73

4

A Graphical View of Linear Functions

77

1

An Analytical View of Linear Functions

78

2

Summary

80

3

3-2 Modeling and Problem Solving with Linear Functions

83

8

Modeling of Linear Relationships

83

2

Linear Variation

85

1

Arithmetic Sequences

86

5

3-3 Linear Modeling of Nonlinear Relationships

91

10

Linearization

91

2

Average Rate of Change

93

7

Supplementary Topic: Linearizing a Situation

100

1

Supplementary Topic: Local Linearization: A Propinquity Principle

101

1

3-4 Piecewise Linear Functions

101

12

Three Views of Piecewise Linear Functions

102

2

Three Views of Linear Absolute Value Functions

104

5

Solving Linear Absolute Value Inequalities

109

4

Chapter Review

113

2

CHAPTER 4 LINEAR SYSTEMS

115

34

4-1 Systems of Linear Equations

115

10

Methods of Solving 2 x 2 Systems

115

2

Methods of Solving 3 x 3 Systems

117

2

Number of Solutions to 2 x 2 and 3 x 3 Systems

119

5

Supplementary Topic: Number of Solutions to m x n Linear Systems

124

1

4-2 Matrix Solutions of Systems of Linear Equations

125

14

Augmented Matrix of a System

125

1

Matrix Row Operations

126

2

Gauss-Jordan Elimination

128

2

Reduced Row-Echelon Matrices

130

1

Matrices and Numbers of Solutions

131

5

Summary: Matrix Versus Nonmatrix Methods

136

3

4-3 Systems of Linear Inequalities and Linear Programming

139

8

Graphs of Linear Inequalities in Two Variables

139

2

Graphs of Systems of Linear Inequalities in Two Variables

141

1

Linear Programming

142

5

Chapter Review

147

2

CHAPTER 5 QUADRATIC FUNCTIONS

149

28

5-1 Three Views of Quadratic Functions

149

17

A Graphical View of Quadratic Functions

150

4

An Analytical View of Quadratic Functions

154

6

A Numerical View of Quadratic Functions

160

5

Supplementary Topic: Effect of the Viewing Window on the Apparent Steepness of Graphs

165

1

Supplementary Topic: Complex Factors

165

1

5-2 Modeling and Problem Solving with Quadratic Functions

166

9

Optimization Problems

166

2

Solving Quadratic Inequalities

168

2

Fitting a Quadratic Function to a Table

170

4

Supplementary Topic: Validity of a Quadratic Model

174

1

Chapter Review

175

2

CHAPTER 6 QUADRATIC RELATIONS

177

74

6-1 Relations

177

10

Three Views of Relations

179

5

Implicit Functions

184

3

6-2 A Graphical View of Conic Sections

187

15

A Graphical View of Parabolas

189

1

A Graphical View of Ellipses

190

5

A Graphical View of Hyperbolas

195

5

Exceptional Graphs

200

2

6-3 Graphical Transformations

202

15

Stretches and Compressions

203

2

Shifts

205

1

Reflections

206

2

Applying a Sequence of Transformations

208

2

Summary

210

4

Supplementary Topic: Graphical Transformation in the Context of Functions

214

3

6-4 An Analytical View of Conic Sections

217

15

An Analytical View of Parabolas

217

4

An Analytical View of Ellipses

221

4

An Analytical View of Hyperbolas

225

7

6-5 Square Root Functions

232

6

6-6 Systems of Quadratic Equations and Inequalities

238

11

Solving Systems of Quadratic Equations Analytically

239

2

Solving Systems of Quadratic Equations Graphically

241

2

Solving Systems of Quadratic Inequalities

243

6

Chapter Review

249

2

CHAPTER 7 POLYNOMIAL FUNCTIONS

251

44

7-1 Power Functions

251

8

Basic Power Functions and Their Graphs

251

2

Graphical Transformations

253

1

Polynomial Variation

254

5

7-2 An Analytical View of Polynomial Functions

259

8

Factors and Zeros

260

2

The Fundamental Theorem of Algebra

262

4

Supplementary Topic: The Rational Root Theorem

266

1

7-3 A Graphical View of Polynomial Functions

267

14

Points of Interest

268

5

End Behavior

273

3

Summary

276

5

7-4 A Numerical View of Polynomial Functions

281

6

nth-Order Differences

281

1

Fitting a Polynomial Function to a Table

282

1

A Numerical Method for Finding Zeros

283

2

A Comparison of Methods for Finding Zeros

285

2

7-5 Solving Polynomial Inequalities

287

6

Solving Polynomial Inequalities Graphically

287

2

Solving Polynomial Inequalities Analytically

289

3

Supplementary Topic: Using Scan Method to Graph Polynomial Functions

292

1

Chapter Review

293

2

CHAPTER 8 RATIONAL FUNCTIONS

295

28

8-1 Reciprocal Power Functions

295

9

Three Views of Reciprocal Power Functions

296

2

Inverse Variation

298

2

Graphical Transformations

300

4

8-2 Discontinuities and End Behavior of Rational Functions

304

10

Discontinuities and Vertical Asymptotes

305

3

End Behavior and Horizontal Asymptotes

308

6

Supplementary Topic: Slant Asymptotes

314

1

8-3 Solving Rational Inequalities

314

6

A Graphical Method

316

1

The Test-Value Method

316

1

The Scan Method

317

3

Chapter Review

320

3

CHAPTER 9 EXPONENTIAL AND LOGARITHMIC FUNCTIONS

323

68

9-1 Exponential Functions

323

15

A Numerical View of Exponential Functions

324

2

An Analytical View of Exponential Functions

326

3

A Graphical View of Exponential Functions

329

3

Geometric Sequences and Series

332

6

9-2 The Special Number e

338

7

The Definition of e

338

3

Using the Base to Express Exponential Functions

341

4

9-3 Inverse Functions

345

13

One-to-One Functions

346

2

Finding Inverses of One-to-One Functions

348

10

9-4 Logarithmic Functions

358

16

A Numerical View of Logarithmic Functions

360

3

A Graphical View of Logarithmic Functions

363

2

An Analytical View of Logarithmic Functions

365

9

Supplementary Topic: Slide Rules

374

1

9-5 Curve Fitting

374

14

Fitting Linear Functions to Data

375

3

Fitting Logarithmic Functions to Data

378

2

Fitting Exponential Functions to Data

380

1

Fitting Power Functions to Data

381

2

Curve Fitting on Your Calculator

383

5

Chapter Review

388

3

CHAPTER 10 TRIGONOMETRIC FUNCTIONS

391

60

10-1 Introduction to the Sine and Cosine Functions

391

13

Two Ways of Defining the Sine and Cosine Functions

392

4

Compatibility of the Two Definitions

396

3

Summary

399

4

Supplementary Topic: Angular Velocity

403

1

10-2 Trigonometric Functions in Right Triangles

404

9

The Right Triangle Definitions of the Trigonometric Functions

405

3

Calculations Using Trigonometric Functions in Right Triangles

408

5

10-3 Three Views of Sine and Cosine Functions

413

13

A Numerical View

414

2

An Analytical View

416

1

A Graphical View

417

7

Supplementary Topic: Equivalent Expressions for Sine and Cosine Functions

424

2

10-4 Trigonometric Functions in the Unit Circle

426

10

The Unit Circle Definitions of the Trigonometric Functions

426

2

An Analytical View of the Trigonometric Functions

428

2

A Numerical View of the Trigonometric Functions

430

1

A Graphical View of the Trigonometric Functions

430

3

Summary

433

3

10-5 The Inverse Trigonometric Functions

436

12

The Inverse Sine Function

437

3

The Inverse Cosine Function

440

2

The Inverse Tangent Function

442

2

The Inverse Cotangent, Secant, and Cosecant Functions

444

1

Compositions of Trigonometric and Inverse Trigonometric Functions

445

3

Chapter Review

448

3

CHAPTER 11 TRIGONOMETRIC FUNCTIONS AS ANALYTICAL TOOLS

451

44

11-1 The Law of Cosines and the Law of Sines

451

9

The Law of Cosines

451

4

The Law of Sines

455

5

11-2 Trigonometric Identities

460

14

Fundamental Identities

461

1

Reflection and Rotation Identities

461

3

Sum and Difference Identities for Sine and Cosine

464

1

Double-Angle and Half-Angle Identities

465

3

Product-to-Sum Identities

468

1

Methods for Distinguishing Identities from Conditional Equations

469

5

11-3 Vectors

474

12

Geometric Representation of Vectors

475

4

Analytical Representation of Vectors

479

7

11-4 Trigonometric Equations

486

6

Three Special Types of Equation

488

1

Other Trigonometric Equations

489

3

Chapter Review

492

3

CHAPTER 12 TRIGONOMETRIC FUNCTIONS AS GRAPHING TOOLS

495

12-1 Graphs of General Conic Sections

495

9

Coordinate Systems Related by Rotation of Axes

495

5

Choosing an Appropriate Angle of Rotation

500

2

Summary

502

2

Supplementary Topic: The Invariance of B(2) - 4AC

504

1

12-2 The Polar Coordinate System

504

8

Polar Coordinates of Points

506

3

Relationships Between Rectangular and Polar Coordinates

509

3

12-3 Graphing in Polar Coordinates

512

11

Rectangular and Polar Equations for the Same Curve

512

2

Graphs of Polar Equations

514

9

12-4 The Geometry of Complex Numbers

523

10

The Complex Plane

524

1

The Polar Form of a Complex Number

525

1

Adding and Subtracting Complex Numbers Geometrically

526

1

Multiplying and Dividing Complex Numbers Geometrically

527

1

De Moivre's Theorem

528

5

Chapter Review

533

APPENDIX A BASIC ALGEBRA REFERENCES

A1

A-1 Accuracy and Precision

A1

3

A-2 Linear Equations

A4

1

A-3 The Coordinate Plane

A5

2

A-4 The Pythagorean Theorem and the Distance Formula

A7

1

A-5 Basic Graphing Techniques

A8

3

A-6 Graphing Linear Equations

A11

4

A-7 Intervals

A15

2

A-8 Linear Inequalities

A17

2

A-9 Absolute Value Equations and Inequalities

A19

3

A-10 Systems of Linear Equations

A22

4

A-11 The Laws of Exponents

A26

2

A-12 Factoring

A28

4

A-13 Quadratic Equations

A32

4

A-14 Operations with Complex Numbers

A36

2

A-15 Division and Synthetic Division of Polynomials

A38

3

A-16 Algebraic Fractions

A41

7

A-17 Equations with Algebraic Fractions

A48

1

A-18 Radicals and Rational Exponents

A49

5

A-19 Equations and Radicals

A54

APPENDIX B TIPS FOR GRAPHING FUNCTIONS WITH A CALCULATOR

B1

B-1 The Viewing Window

B1

5

B-2 Graphing Linear Functions

B6

1

B-3 Graphing Quadratic Functions

B7

2

B-4 Graphing Quadratic Relations

B9

2

B-5 Graphing Polynomial Functions

B11

3

B-6 Graphing Rational Functions

B14

ANSWERS TO ODD-NUMBERED EXERCISES

ANS-1

INDEX

I-1