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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