Gary Rockswold focuses on teaching algebra in context, answering the question, “Why am I learning this?” and ultimately motivating the students to succeed in this class. In addition, the author's understanding of what instructors need from a text (great 'real' examples and lots of exercises) makes this book fun and easy to teach from. Integrating this textbook into your course will be a worthwhile endeavor.
Table of Contents:
Chapter 1: INTRODUCTION TO FUNCTIONS AND GRAPHS 1.1 Numbers, Data, and Problem Solving
Sets of Numbers
Scientific Notation
Problem Solving
1.2 Visualization of Data
One-Variable Data
Two Variable Data
The Distance Formula
The Midpoint Formula
Graphing with a Calculator (Optional)
Checking Basic Concepts for Sections 1.1 and 1.2
1.3 Functions and Their Representations
Basic Concepts
Representations of Functions
Formal Definition of a Function
Graphing Calculators and Functions (Optional)
Identifying Functions
1.4 Types of Functions and Their Rates of Change
Constant Functions
Linear Functions
Slope as a Rate of Change
Nonlinear Functions
Average Rate of Change
The Difference Quotient
Checking Basic Concepts for Sections 1.3 and 1.4
Chapter 1 Summary
Chapter 1 Review Exercises
Chapter 1 Extended and Discovery Exercises
Chapter 2: LINEAR FUNCTIONS AND EQUATIONS 2.1 Linear Functions and Models
Exact and Approximate Models
Representations of Linear Functions
Modeling with Linear Functions
Linear Regression (Optional)
2.2 Equations of Lines
Forms for Equations of Lines
Determining Intercepts
Horizontal, Vertical, Parallel, and Perpendicular Lines
Modeling Data (Optional)
Interpolation and Extrapolation
Direct Variation
Checking Basic Concepts for Sections 2.1 and 2.2 2.3 Linear Equations
Equations
Symbolic Solutions
Graphical and Numerical Solutions
Problem-Solving Strategies
2.4 Linear Inequalities
Inequalities
Interval Notion
Techniques for Solving Inequalities
Compound Inequalities
Checking Basic Concepts for Sections 2.3 and 2.4 2.5 Piecewise-Defined Functions
Evaluating and Graphing Piecewise-Defined Functions
The Greatest Integer Function
The Absolute Value Function
Equations and Inequalities Involving Absolute Values
Checking Basic Concepts for Section 2.5 Chapter 2 Summary
Chapter 2 Review Exercises
Chapter 2 Extended and Discovery Exercises
Chapter 1-2 Cumulative Review Exercises
Chapter 3: QUADRATIC FUNCTIONS AND EQUATIONS
3.1 Quadratic Functions and Models
Basic Concepts
Completing the Square and the Vertex Formula
Applications and Models
Quadratic Regression (Optional)
3.2 Quadratic Equations and Problem Solving
Basic Concepts
Solving Quadratic Equations
Problem Solving
Checking Basic Concepts for Sections 3.1 and 3.2
3.3 Quadratic Inequalities
Graphical Solutions
Symbolic Solutions
3.4 Transformations of Graphs
Vertical and Horizontal Translations
Stretching and Shrinking
Reflection of Graphs
Combining Transformations
Modeling with Transformations (Optional)
Checking Basic Concepts for Sections 3.3 and 3.4
Chapter 3 Summary
Chapter 3 Review Exercises
Chapter 3 Extended and Discovery Exercises
Chapter 4: NONLINEAR FUNCTIONS AND EQUATIONS 4.1 Nonlinear Functions and Their Graphs
Polynomial Functions
Increasing and Decreasing Functions
Extrema of Nonlinear Functions
Symmetry
4.2 Polynomial Functions and Models
Graphs of Polynomial Functions
Piecewise-Defined Polynomial Functions
Polynomial Regression (Optional)
Checking Basic Concepts for Sections 4.1 and 4.2 4.3 Real Zeros of Polynomial Functions
Division of Polynomials
Factoring Polynomials
Graphs and Multiple Zeros
Rational Zeros
Polynomial Equations
4.4 The Fundamental Theorem of Algebra
Complex Numbers
Quadratic Equations with Complex Solutions
Fundamental Theorem of Algebra
Polynomial Equations with Complex Solutions
Checking Basic Concepts for Sections 4.3 and 4.4 4.5 Rational Functions and Models
Rational Functions
Vertical Asymptotes
Horizontal Asymptotes
Identifying Asymptotes
Rational Equations
Variation
4.6 Polynomial and Rational Inequalities
Polynomial Inequalities
Rational Inequalities
Checking Basic Concepts for Sections 4.5 and 4.6 4.7 Power Functions and Radical Equations
Rational Exponents and Radical Notation
Power Functions and Models
Equations Involving Rational Exponents
Equations Involving Radicals
Power Regression (Optional)
Checking Basic Concepts for Section 4.7 Chapter 4 Summary
Chapter 4 Review Exercises
Chapter 4 Extended and Discovery Exercises
Chapters 1-4 Cumulative Review Exercises
Chapter 5: EXPONENTIAL AND LOGARITHMIC FUNCTIONS 5.1 Combining Functions
Arithmetic Operations on Functions
Composition of Functions
5.2 Inverse Functions and Their Representations
Inverse Operations
One-to-One Functions
Symbolic Representations of Inverse Functions
Other Representations of Inverse Functions
Checking Basic Concepts for Sections 5.1 and 5.2
5.3 Exponential Functions and Models
Linear and Exponential Growth
Exponential Models
Compound Interest
The Natural Exponential Function
5.4 Logarithmic Functions and Models
The Common Logarithmic Function
Basic Equations
Logarithms with Other Bases
General Logarithmic Equations
Checking Basic Concepts for Sections 5.3 and 5.4
5.5 Properties of Logarithms
Basic Properties of Logarithms
Change of Base Formula
5.6 Exponential and Logarithmic Equations
Exponential Equations
Logarithmic Equations
Checking Basic Concepts for 5.5 and 5.6
5.7 Constructing Nonlinear Models
Exponential Model
Logarithmic Model
Logistic Model
Checking Basic Concepts for Section 5.7 Chapter 5 Summary
Chapter 5 Review Exercises
Chapter 5 Extended and Discovery Exercises
Chapter 6: TRIGONOMETRIC FUNCTIONS
6.1 Angles and Their Measure
Angles
Degree Measure
Radian Measure
Arc Length
Area of a Sector
6.2 Right Triangle Trigonometry
Basic Concepts of Trigonometric Functions
Applications of Right Triangle Trigonometry
Complementary Angles and Cofunctions
Checking Basic Concepts for 6.1 and 6.2 6.3 The Sine and Cosine Functions and Their Graphs
Definitions
The Unit Circle
Representations of the Sine and the Cosine Functions
Applications of the Sine and Cosine Functions
Modeling with the Sine Function (Optional)
6.4 Other Trigonometric Functions and Their Graphs
Definitions and Basic Identities
Representations of Other Trigonometric Functions
Applications of Trigonometric Functions
Checking Basic Concepts for Sections 6.3 and 6.4 6.5 Graphing Trigonometric Functions
Transformations of Trigonometric Graphs
Graphing Trigonometric Functions by Hand
Simple Harmonic Motion
Models Involving Trigonometric Functions (Optional)
6.6 Inverse Trigonometric Functions
Review of Inverses
The Inverse Sine Function
The Inverse Cosine Function
The Inverse Tangent Function
Solving Triangles and Equations
Checking Basic Concepts for Sections 6.5 and 6.6
Chapter 6 Summary
Chapter 6 Review Exercises
Chapter 6 Extended and Discovery Exercises
Chapters 1-6 Cumulative Review Exercises
Chapter 7: TRIGONOMETRIC IDENTITIES AND EQUATIONS 7.1 Fundamental Identities
Reciprocal and Quotient Identities
Pythagorean Identities
Negative-Angle Identities
7.2 Verifying Identities
Simplifying Trigonometric Expressions
Verification of Identities
Checking Basic Concepts for Section 7.1 and 7.2 7.3 Trigonometric Equations
Reference Angles
Solving Trigonometric Equations
Solving Inverse Trigonometric Equations
7.4 Sum and Difference Identities
Sum and Difference Identities for Cosine
Other Sum and Difference Identities
Checking Basic Concepts for Section 7.3 and 7.4 7.5 Multiple-Angle Identities
Double-Angle Identities
Half-Angle Formulas
Solving Equations
Product-to-Sum and Sum-to-Product Identities
Checking Basic Concepts for Section 7.5
Chapter 7 Summary
Chapter 7 Review Exercises
Chapter 7 Extended and Discovery Exercises
Chapter 8: FURTHER TOPICS IN TRIGONOMETRY
8.1 Law of Sines
Oblique Triangles
Solving Triangles
The Ambiguous Case
8.2 Law of Cosines
Derivation of the Law of Cosines
Solving Triangles
Area Formulas
Checking Basic Concepts for Sections 8.1 and 8.2 8.3 Vectors
Basic Concepts
Operations on Vectors
The Dot Product
Work
8.4 Parametric Equations
Basic Concepts
Applications of Parametric Equations
Checking Basic Concepts for Sections 8.3 and 8.4 8.5 Polar Equations
The Polar Coordinate System
Graphs of Polar Equations
Graphing Calculators and Polar Equations (Optional)
Solving Polar Equations
8.6 Trigonometric Form and Roots of Complex Numbers
Trigonometric Form
Products and Quotients of Complex Numbers
De Moivre’s Theorem
Roots of Complex Numbers
Checking Basic Concepts for Sections 8.5 and 8.6 Chapter 8 Summary
Chapter 8 Review Exercises
Chapter 8 Extended and Discovery Exercises
Chapters 1-8 Cumulative Review Exercises
Chapter 9: SYSTEMS OF EQUATIONS AND INEQUALITIES 9.1 Functions and Equations in Two Variables
Functions of Two Variables
Systems of Equations
The Method of Substitution
Graphical and Numerical Methods
Joint Variation
9.2 Systems of Equations and Inequalities in Two Variables
Types of Linear Systems in Two Variables
The Elimination Method
Systems of Linear and Nonlinear Inequalities
Linear Programming
Checking Basic Concepts for 9.1 and 9.2 9.3 Systems of Linear Equations in Three Variables
Basic Concepts
Solving with Elimination and Substitution
Systems with No Solutions
Systems with Infinitely Many Solutions
9.4 Solutions to Linear Systems Using Matrices
Representing Systems of Linear Equations with Matrices
Row-Echelon Form
Gaussian Elimination
Solving Systems of Linear Equations with Technology (Optional)
Checking Basic Concepts for Sections 9.3 and 9.4 9.5 Properties and Applications of Matrices
Matrix Notation
Sums, Differences, and Scalar Multiples of Matrices
Matrix Products
Technology and Matrices (Optional)
9.6 Inverses of Matrices
Matrix Inverses
Finding Inverses Symbolically
Representing Linear Systems with Matrix Equations
Solving Linear Systems with Inverses
Checking Basic Concepts for Sections 9.5 and 9.6 9.7 Determinants
Definition and Calculation of Determinants
Area of Regions
Cramer’s Rule
Limitations on the Method of Cofactors and Cramer’s Rule
Checking Basic Concepts for Section 9.7
Chapter 9 Summary
Chapter 9 Review Exercises
Chapter 9 Extended and Discovery Exercise
Chapter 10: CONIC SECTIONS 10.1 Parabolas
Equations and Graphs of Parabolas
Reflective Property of Parabolas
Translations of Parabolas
10.2 Ellipses
Equations and Graphs of Ellipses
Reflective Property of Ellipses
Translations of Ellipses
Circles
Solving Systems of Equations and Inequalities
Checking Basic Concepts for Section 10.1 and 10.2 10.3 Hyperbolas
Equations and Graphs of Hyperbolas
Reflective Property of Hyperbolas
Translations of Hyperbolas
Solving Systems of Nonlinear Equations
Checking Basic Concepts for Section 10.3
Chapter 10 Summary
Chapter 10 Review Exercises
Chapter 10 Extended and Discovery Exercises
Chapter 11: FURTHER TOPICS IN ALGEBRA 11.1 Sequences
Basic Concepts
Representations of Sequences
Arithmetic Sequences
Geometric Sequences
11.2 Series
Basic Concepts
Arithmetic Series
Geometric Series
Summation Notation
Checking Basic Concepts for Sections 11.1 and 11.2 11.3 Counting
Fundamental Counting Principle
Permutations
Combinations
11.4 The Binomial Theorem
Derivation of the Binomial Theorem
Pascal’s Triangle
Checking Basic Concepts for Sections 11.3 and 11.4 11.5 Mathematical Induction
Mathematical Induction
Proving Statements
Generalized Principle of Mathematical Induction
11.6 Probability
Definition of Probability
Compound Events
Independent and Dependent Events
Checking Basic Concepts for Sections 11.5 and 11.6 Chapter 11 Summary
Chapter 11 Review Exercises
Chapter 11 Extended and Discovery Exercises
Chapters 1-11 Cumulative Review Exercises
Chapter R: REFERENCE- BASIC CONCEPTS FROM ALGEBRA AND GEOMETRY
R.1 Formulas from Geometry
Geometric Shapes in a Plane
The Pythagorean Theorem
Three-Dimensional Objects
Similar Triangles
A Summary of Geometric Formulas
R.2 Circles
Equations and Graphs of Circles
Finding the Center and Radius of a Circle
R.3 Integer Exponents
Bases and Positive Exponents
Zero and Negative Exponents
Product, Quotient, and Power Rules
R.4 Polynomial Expressions
Addition and Subtraction of Monomials
Addition and Subtraction of Polynomials
Distributive Properties
Multiplying Polynomials
Some Special Products
R.5 Factoring Polynomials
Common Factors
Factoring by Grouping
Factoring x2 + bx + c
Factoring Trinomials by Grouping
Factoring Trinomials with FOIL
Difference of Two Squares
Perfect Square Trinomials
Sum and Difference of Two Cubes
R.6 Rational Expressions
Simplifying Rational Expressions
Multiplication and Division of Rational Expressions
Least Common Multiples
Common Denominators
Addition and Subtraction of Rational Expressions
Clearing Fractions
Complex Fractions
R.7 Radical Notation and Rational Exponents
Radical Notation
Rational Exponents
Properties of Rational Exponents
R.8 Radical Expressions
Product Rule for Radical Expressions
Quotient Rule for Radical Expressions
Addition and Subtraction
Multiplication
Rationalizing the Denominator
APPENDIX A: A Library of Functions
APPENDIX B: Using the Graphing Calculator
APPENDIX C: Partial Fractions APPENDIX D: Rotation of Axes
Bibliography
Answers to Selected Exercises
Index of Applications
Index
