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Tables of Contents for A Graphical Approach to College Algebra and Trigonometry
Chapter/Section Title
Page #
Page Count
Linear Functions, Equations, and Inequalities
1
89
Real Numbers and Coordinate Systems
2
9
Sets of Real Numbers
Coordinate Systems
Viewing Windows
Roots
Distance and Midpoint Formulas
Introduction to Relations and Functions
11
12
Set-Builder Notation and Interval Notation
Relations, Domain, and Range
Functions
Tables
Function Notation
Reviewing Basic Concepts (Sections 1.1 and 1.2)
23
1
Linear Functions
23
13
Basic Concepts about Linear Functions
Slope of a Line
Slope-Intercept Form of the Equation of a Line
Equations of Lines and Linear Models
36
15
Point-Slope Form of the Equation of a Line
Other Forms of the Equation of a Line
Parallel and Perpendicular Lines
Linear Models and Regression
Reviewing Basic Concepts (Sections 1.3 and 1.4)
51
1
Linear Equations and Inequalities
51
16
Solving Linear Equations
Graphical Approaches to Solving Linear Equations
Identities and Contradictions
Solving Linear Inequalities
Graphical Approaches to Solving Linear Inequalities
Three-Part Inequalities
Applications of Linear Functions
67
22
Problem-Solving Strategies
Applications of Linear Equations
Break-Even Analysis
Direct Variation
Formulas
Reviewing Basic Concepts (Sections 1.5 and 1.6)
77
1
Chapter 1 Summary
78
4
Chapter 1 Review Exercises
82
4
Chapter 1 Test
86
2
Chapter 1 Project Predicting Heights and Weights of Athletes
88
1
Analysis of Graphs of Functions
89
83
Graphs of Basic Functions and Relations; Symmetry
90
14
Continuity
Increasing and Decreasing Functions
The Identity Function
The Squaring Function and Symmetry with Respect to the y-Axis
The Cubing Function and Symmetry with Respect to the Origin
The Square Root and Cube Root Functions
The Absolute Value Function
The Relation x = y2 and Symmetry with Respect to the x-Axis
Even and Odd Functions
Vertical and Horizontal Shifts of Graphs
104
10
Vertical Shifts
Horizontal Shifts
Combinations of Vertical and Horizontal Shifts
Effects of Shifts on Domain and Range
Horizontal Shifts Applied to Equations for Modeling
Stretching, Shrinking, and Reflecting Graphs
114
12
Vertical Stretching
Vertical Shrinking
Reflecting Across an Axis
Combining Transformations of Graphs
Reviewing Basic Concepts (Sections 2.1 -- 2.3)
124
2
Absolute Value Functions: Graphs, Equations, Inequalities, and Applications
126
11
The Graph of y = |f(x)|
Properties of Absolute Value
Equations and Inequalities Involving Absolute Value
An Application Involving Absolute Value
Piecewise-Defined Functions
137
11
Graphing Piecewise-Defined Functions
The Greatest Integer Function
Applications of Piecewise-Defined Functions
Operations and Composition
148
24
Operations on Functions
The Difference Quotient
Composition of Functions
Applications of Operations and Composition
Reviewing Basic Concepts (Sections 2.4--2.6)
161
1
Chapter 2 Summary
162
3
Chapter 2 Review Exercises
165
3
Chapter 2 Test
168
1
Chapter 2 Project Modeling the Movement of a Cold Front
169
3
Polynomial Functions
172
100
Complex Numbers
173
7
The Number i
Operations with Complex Numbers
Quadratic Functions and Graphs
180
13
Completing the Square
Graphs of Quadratic Functions
Vertex Formula
Extreme Values
Applications and Modeling
Quadratic Equations and Inequalities
193
14
Zero-Product Property
Solving x2 = k
Quadratic Formula and the Discriminant
Solving Quadratic Equations
Solving Quadratic Inequalities
Formulas Involving Quadratics
Another Quadratic Model
Reviewing Basic Concepts (Sections 3.1--3.3)
207
1
Further Applications of Quadratic Functions and Models
207
10
Applications of Quadratic Functions
Quadratic Models
Higher-Degree Polynomial Functions and Graphs
217
15
Cubic Functions
Quartic Functions
Extrema
End Behavior
x-Intercepts (Real Zeros)
Comprehensive Graphs
Curve Fitting and Polynomial Models
Reviewing Basic Concepts (Sections 3.4 and 3.5)
231
1
Topics in the Theory of Polynomial Functions (I)
232
10
Intermediate Value Theorem
Division of Polynomials and Synthetic Division
Remainder and Factor Theorems
Topics in the Theory of Polynomial Functions (II)
242
9
Complex Zeros and the Fundamental Theorem of Algebra
Number of Zeros
Rational Zeros Theorem
Polynomial Equations and Inequalities; Further Applications and Models
251
21
Polynomial Equations and Inequalities
Complex nth Roots
Applications and Polynomial Models
Reviewing Basic Concepts (Sections 3.6--3.8)
261
1
Chapter 3 Summary
262
3
Chapter 3 Review Exercises
265
4
Chapter 3 Test
269
1
Chapter 3 Project Creating a Social Security Polynomial
270
2
Rational, Power, and Root Functions
272
70
Rational Functions and Graphs
273
6
The Reciprocal Function
The Rational Function Defined by f(x) = 1/x2
Mode and Window Choices for Calculator Graphs
More on Graphs of Rational Functions
279
14
Vertical and Horizontal Asymptotes
Graphing Techniques
Oblique Asymptotes
Graphs with Points of Discontinuity
Rational Equations, Inequalities, Applications, and Models
293
16
Solving Rational Equations and Inequalities
Applications and Models of Rational Functions
Inverse Variation
Combined and Joint Variation
Reviewing Basic Concepts (Sections 4.1--4.3)
308
1
Functions Defined by Powers and Roots
309
11
Power and Root Functions
Modeling Using Power Functions
Graphs of
Graphing Circles and Horizontal Parabolas Using Root Functions
Equations, Inequalities, and Applications Involving Root Functions
320
22
Equations and Inequalities
Applications
Reviewing Basic Concepts (Sections 4.4 and 4.5)
330
1
Chapter 4 Summary
331
2
Chapter 4 Review Exercises
333
4
Chapter 4 Test
337
2
Chapter 4 Project How Rugged Is Your Coastline?
339
3
Inverse, Exponential, and Logarithmic Functions
342
77
Inverse Functions
343
10
Inverse Operations
One-to-One Functions
Inverse Functions and Their Graphs
An Application of Inverse Functions
Exponential Functions
353
12
Real-Number Exponents
Graphs of Exponential Functions
Exponential Equations (Type I)
The Number e
Compound Interest
Logarithms and Their Properties
365
12
Definition of Logarithm
Common Logarithms
Natural Logarithms
Properties of Logarithms
Change-of-Base Rule
Reviewing Basic Concepts (Sections 5.1--5.3)
376
1
Logarithmic Functions
377
9
Graphs of Logarithmic Functions
Applying Earlier Work to Logarithmic Functions
A Logarithmic Model
Exponential and Logarithmic Equations and Inequalities
386
10
Exponential Equations and Inequalities (Type 2)
Logarithmic Equations and Inequalities
Equations and Inequalities Involving Both Exponentials and Logarithms
Formulas Involving Exponentials and Logarithms
Reviewing Basic Concepts (Sections 5.4 and 5.5)
395
1
Further Applications and Modeling with Exponential and Logarithmic Functions
396
23
Physical Science Applications
Financial Applications
Biological and Medical Applications
Modeling Data with Exponential and Logarithmic Functions
Chapter 5 Summary
410
3
Chapter 5 Review Exercises
413
3
Chapter 5 Test
416
1
Chapter 5 Project Modeling Motor Vehicle Sales in the United States (with a lesson about careless use of mathematical models)
417
2
Analytic Geometry
419
47
Circles and Parabolas
420
13
Conic Sections
Equations and Graphs of Circles
An Application of Circles
Equations and Graphs of Parabolas
An Application of Parabolas
Ellipses and Hyperbolas
433
13
Equations and Graphs of Ellipses
Applications of Ellipses
Equations and Graphs of Hyperbolas
Reviewing Basic Concepts (Sections 6.1 and 6.2)
446
1
Summary of the Conic Sections
446
8
Characteristics
Identifying Conic Sections
Eccentricity
Parametric Equations
454
12
Graphs of Parametric Equations and Their Rectangular Equivalents
Alternative Forms of Parametric Equations
An Application
Reviewing Basic Concepts (Sections 6.3 and 6.4)
459
1
Chapter 6 Summary
459
2
Chapter 6 Review Exercises
461
2
Chapter 6 Test
463
1
Chapter 6 Project Modeling the Path of a Bouncing Ball
464
2
Matrices and Systems of Equations and Inequalities
466
100
Systems of Equations
467
13
Linear Systems
Substitution Method
Elimination Method
Special Systems
Nonlinear Systems
Applications of Systems
Solution of Linear Systems by the Echelon Method
480
8
Geometric Considerations
Analytic Solution of Systems in Three Variables
Applications of Systems
Curve Fitting Using a System
Solution of Linear Systems by Row Transformations
488
13
Matrices and Technology
Matrix Row Transformations
Row Echelon Method
Reduced Row Echelon Method
Special Cases
An Application
Reviewing Basic Concepts (Sections 7.1--7.3)
501
1
Matrix Properties and Operations
501
15
Terminology of Matrices
Operations on Matrices
Applying Matrix Algebra
Determinants and Cramer's Rule
516
11
Determinants of 2 X 2 Matrices
Determinants of Larger Matrices
Derivation of Cramer's Rule
Using Cramer's Rule to Solve Systems
Solution of Linear Systems by Matrix Inverses
527
11
Identity Matrices
Multiplicative Inverses of Square Matrices
Solving Linear Systems Using Inverse Matrices
Curve Fitting Using a System
Reviewing Basic Concepts (Sections 7.4--7.6)
537
1
Systems of Inequalities and Linear Programming
538
10
Solving Linear Inequalities
Solving Systems of Inequalities
Linear Programming
Partial Fractions
548
18
Decomposition of Rational Expressions
Distinct Linear Factors
Repeated Linear Factors
Distinct Linear and Quadratic Factors
Repeated Quadratic Factors
Reviewing Basic Concepts (Sections 7.7 and 7.8)
555
1
Chapter 7 Summary
556
2
Chapter 7 Review Exercises
558
4
Chapter 7 Test
562
1
Chapter 7 Project Finding a Polynomial Whose Graph Passes Through Any Number of Given Points
563
3
Trigonometric Functions and Applications
566
107
Angles and Their Measures
567
16
Basic Terminology
Degree Measure
Standard Position and Coterminal Angles
Radian Measure
Arc Lengths and Sectors
Angular and Linear Speed
Trigonometric Functions and Fundamental Identities
583
12
Trigonometric Functions
Quadrantal Angles
Reciprocal Identities
Signs and Ranges of Function Values
Pythagorean Identities
Quotient Identities
Reviewing Basic Concepts (Sections 8.1 and 8.2)
594
1
Evaluating Trigonometric Functions
595
12
Definitions of the Trigonometric Functions
Trigonometric Function Values of Special Angles
Cofunction Identities
Reference Angles
Special Angles as Reference Angles
Finding Function Values with a Calculator
Finding Angle Measures
Applications of Right Triangles
607
13
Significant Digits
Solving Triangles
Angles of Elevation or Depression
Bearing
Further Applications
Reviewing Basic Concepts (Sections 8.3 and 8.4)
619
1
The Circular Functions
620
8
Circular Functions
Applications of Circular Functions
Graphs of the Sine and Cosine Functions
628
18
Periodic Functions
Graph of the Sine Function
Graph of the Cosine Function
Graphing Techniques, Amplitude, and Period
Translations
Determining a Trigonometric Model Using Curve Fitting
Reviewing Basic Concepts (Sections 8.5 and 8.6)
645
1
Graphs of the Other Circular Functions
646
10
Graphs of the Cosecant and Secant Functions
Graphs of the Tangent and Cotangent Functions
Addition of Ordinates
Harmonic Motion
656
17
Simple Harmonic Motion
Damped Oscillatory Motion
Reviewing Basic Concepts (Sections 8.7 and 8.8)
659
1
Chapter 8 Summary
660
4
Chapter 8 Review Exercises
664
5
Chapter 8 Test
669
2
Chapter 8 Project Modeling Sunset Times
671
2
Trigonometric Identities and Equations
673
66
Trigonometric Identities
674
10
Fundamental Identities
Using the Fundamental Identities
Verifying Identities
Sum and Difference Identities
684
10
Cosine Sum and Difference Identities
Sine and Tangent Sum and Difference Identities
Reviewing Basic Concepts (Sections 9.1 and 9.2)
693
1
Further Identities
694
12
Double-Number Identities
Product-to-Sum and Sum-to-Product Identities
Half-Number Identities
The Inverse Circular Functions
706
12
Inverse Sine Function
Inverse Cosine Function
Inverse Tangent Function
Remaining Inverse Trigonometric Functions
Inverse Function Values
Reviewing Basic Concepts (Sections 9.3 and 9.4)
717
1
Trigonometric Equations and Inequalities (I)
718
6
Equations Solvable by Linear Methods
Equations Solvable by Factoring
Equations Solvable by the Quadratic Formula
Using Trigonometric Identities to Solve Equations
Trigonometric Equations and Inequalities (II)
724
15
Equations and Inequalities Involving Multiple-Number Identities
Equations and Inequalities Involving Half-Number Identities
An Application
Reviewing Basic Concepts (Sections 9.5 and 9.6)
730
1
Chapter 9 Summary
731
2
Chapter 9 Review Exercises
733
3
Chapter 9 Test
736
1
Chapter 9 Project Modeling a Damped Pendulum
737
2
Applications of Trigonometry; Vectors
739
77
The Law of Sines
740
13
Congruency and Oblique Triangles
Derivation of the Law of Sines
Applications
Ambiguous Case
The Law of Cosines and Area Formulas
753
10
Derivation of the Law of Cosines
Applications
Area Formulas
Vectors and Their Applications
763
13
Basic Terminology
Algebraic Interpretation of Vectors
Operations with Vectors
Dot Product and the Angle between Vectors
Applications of Vectors
Reviewing Basic Concepts (Sections 10.1--10.3)
775
1
Trigonometric (Polar) Form of Complex Numbers
776
8
The Complex Plane and Vector Representation
Trigonometric (Polar) Form
Products of Complex Numbers in Trigonometric Form
Quotients of Complex Numbers in Trigonometric Form
Powers and Roots of Complex Numbers
784
7
Powers of Complex Numbers (De Moivre's Theorem)
Roots of Complex Numbers
Reviewing Basic Concepts (Sections 10.4 and 10.5)
790
1
Polar Equations and Graphs
791
10
Polar Coordinate System
Graphs of Polar Equations
Classifying Polar Equations
Converting Equations
More Parametric Equations
801
15
Parametric Equations with Trigonometric Functions
The Cycloid
Applications of Parametric Equations
Reviewing Basic Concepts (Sections 10.6 and 10.7)
808
1
Chapter 10 Summary
808
3
Chapter 10 Review Exercises
811
3
Chapter 10 Test
814
1
Chapter 10 Project When Is a Circle Really a Polygon?
814
2
Further Topics in Algebra
816
74
Sequences and Series
817
10
Sequences
Series and Summation Notation
Summation Properties
Arithmetic Sequences and Series
827
8
Arithmetic Sequences
Arithmetic Series
Geometric Sequences and Series
835
12
Geometric Sequences
Geometric Series
Infinite Geometric Series
Annuities
Reviewing Basic Concepts (Sections 11.1--11.3)
846
1
The Binomial Theorem
847
7
A Binomial Expansion Pattern
Pascal's Triangle
n-Factorial
Binomial Coefficients
The Binomial Theorem
rth Term of a Binomial Expansion
Mathematical Induction
854
7
Proof by Mathematical Induction
Proving Statements
Generalized Principle of Mathematical Induction
Proof of the Binomial Theorem
Reviewing Basic Concepts (Sections 11.4 and 11.5)
861
1
Counting Theory
861
9
Fundamental Principle of Counting
Permutations
Combinations
Distinguishing between Permutations and Combinations
Probability
870
20
Basic Concepts
Complements and Venn Diagrams
Odds
Union of Two Events
Binomial Probability
Reviewing Basic Concepts (Sections 11.6 and 11.7)
881
1
Chapter 11 Summary
881
4
Chapter 11 Review Exercises
885
2
Chapter 11 Test
887
1
Chapter 11 Project Using Experimental Probabilities to Simulate Family Makeup
888
2
Reference: Basic Algebraic Concepts and Geometry Formulas
890
36
Review of Exponents and Polynomials
891
6
Rules for Exponents
Terminology for Polynomials
Adding and Subtracting Polynomials
Multiplying Polynomials
Review of Factoring
897
7
Factoring Out the Greatest Common Factor
Factoring by Grouping
Factoring Trinomials
Factoring Special Products
Factoring by Substitution
Review of Rational Expressions
904
7
Domain of a Rational Expression
Lowest Terms of a Rational Expression
Multiplying and Dividing Rational Expressions
Adding and Subtracting Rational Expressions
Complex Fractions
Review of Negative and Rational Exponents
911
6
Negative Exponents and the Quotient Rule
Rational Exponents
Review of Radicals
917
7
Radical Notation
Rules for Radicals
Simplifying Radicals
Operations with Radicals
Rationalizing Denominators
Geometry Formulas
924
2
Appendix A: Vectors in Space
926
6
Appendix B: Polar Form of Conic Sections
932
4
Appendix C: Rotation of Axes
936
Answers to Selected Exercises
1
1
Index of Applications
1
6
Index
7