**CHAPTER 0: Prerequisites***

**0.1 Real Numbers**

Representing Real Numbers • Order and Interval Notation • Basic Properties of Algebra • Integer Exponents • Scientific Notation

**0.2 Cartesian Coordinate System**

Cartesian Plane • Absolute Value of a Real Number • Distance Formulas • Midpoint Formulas • Equations of Circles

**0.3 Linear Equations and Inequalities**

Equations • Solving Equations • Linear Equations in One Variable • Linear Inequalities in One Variable

**0.4 Lines in the Plane**

Slope of a Line • Point-Slope Equation of a Line • Slope-Intercept Equation of a Line • Graphing Linear Equations in Two Variables • Parallel and Perpendicular Lines

**0.5 Solving Equations**

Solving Equations Graphically • Solving Quadratic Equations Algebraically • Approximating Solutions of Equations • Solving Equations by Finding Intersections

**0.6 Complex Numbers**

Complex Numbers • Operations with Complex Numbers • Complex Conjugates and Division • Complex Solutions of Quadratic Equations

**0.7 Solving Systems of Equations**

Method of Substitution • Solving Systems Graphically • Method of Elimination • Systems in Three Variables

**0.8 Solving Inequalities**

Solving Absolute Value Inequalities • Solving Quadratic Inequalities • Solving Other Inequalities

**CHAPTER 1: Functions and Their Properties**

**1.1 Modeling and Equation Solving**

Numerical Models • Algebraic Models • Graphical Models • The Zero Factor Property • Problem Solving and Modeling • Grapher Failure and Hidden Behavior • A Word About Proof

**1.2 Function Behavior**

Function Definition and Notation • Change in Tandem • Domain and Range • Increasing and Decreasing on Intervals • Concavity and Points of Inflection • Boundedness • Local and Absolute Extrema • Symmetry • Continuity, Asymptotes, and Holes • End Behavior

**1.3 Twelve Basic Functions**

What Graphs Can Reveal • Twelve Basic Functions • Analyzing Functions Graphically

**1.4 Building Functions from Functions**

Combining Functions Algebraically • Composition of Functions • Relations and Implicitly Defined Functions • Piecewise-Defined Functions

**1.5 Parametric Functions and Inverse Functions**

Parametric Functions • Relations Defined Parametrically • Inverse Relations • Inverse Functions

**1.6 Graphical Transformations**

Transformations • Vertical and Horizontal Translations • Reflections Across Axes • Vertical and Horizontal Dilations (Stretches and Shrinks) • Combining Transformations

**1.7 Modeling with Functions**

Modeling Change in Tandem • Functions from Formulas • Functions from Graphs • Functions from Verbal Descriptions • Functions from Data • Piecewise-Defined Models • Assumption Articulation

**CHAPTER 2: Polynomial and Rational Functions**

**2.1 Function Families, Linear Functions, and Linear Models**

Power Functions and Variation • Monomial Functions • Polynomial Functions • Linear Functions and Their Graphs • Graphical Transformations of the Identity Function • Average Rate of Change • Association, Correlation, and Linear Models

**2.2 Quadratic Functions and Modeling**

Quadratic Functions • Transformations of the Squaring Function • Local Linearity and Rates of Change • Rates of Change of Rates of Change • Quadratic Modeling • Regression and Residual Analysis

**2.3 Polynomial Functions and Modeling**

Graphs of Polynomial Functions • Concavity and Points of Inflection • End Behavior of Polynomial Functions • Zeros of Polynomial Functions • Intermediate Value Theorem • Modeling with Polynomial Functions

**2.4 Real Zeros of Polynomial Functions**

Long Division and the Division Algorithm • Remainder and Factor Theorems • Analyzing Graphs of Polynomial Functions in Factored Form • The Binomial Theorem

**2.5 Complex Zeros and the Fundamental Theorem of Algebra**

Fundamental Theorem and Linear Factorization Theorem • Complex Conjugate Zeros • Factoring Polynomials • The Complex Plane

**2.6 Rational Functions and Their Graphs**

Rational Functions • Transformations of the Reciprocal Function • Limits, Asymptotes, and Holes • Analyzing Graphs of Rational Functions • Direct and Inverse Variation Revisited • Modeling with Rational Functions

**2.7 Rational Equations, Inequalities, and Modeling**

Solving Polynomial Equations and Inequalities • Solving Rational Equations • Extraneous Solutions • Rational Inequalities and Graphical Analysis • Applications and Justifications

**CHAPTER 3: Exponential and Logarithmic Functions**

**3.1 Arithmetic and Geometric Sequences**

Arithmetic and Geometric Sequences • Graphs of Sequences • Linear and Exponential Growth and Decay • Infinite Series, Summation Notation, and Convergence*

**3.2 Exponential Functions**

Exponential Functions and Their Graphs • The Natural Base e • Logistic Functions and Their Graphs* • Population Models

**3.3 Exponential Modeling**

Constant Percentage Change and Exponential Functions • Exponential Growth and Decay Models • Choosing and Validating a Growth Model for Data

**3.4 Logarithmic Functions**

Inverses of Exponential Functions • Common (Base 10) Logarithms • Natural (Base e) Logarithms • Graphs of Logarithmic Functions • Logarithmic Scaling

**3.5 Properties of Logarithmic Functions**

Properties of Logarithms • Change of Base • Graphs of Logarithmic Functions with Base *b*

**3.6 Logarithmic Scaling and Semi-Log Plots**

Using Logarithmic Scaling to Linearize Exponential Data • Semi-Log Plots • Log-Log Plots*

**3.7 Equation Solving and Modeling**

Solving Exponential Equations • Solving Logarithmic Equations • Orders of Magnitude and Logarithmic Models • Change in Tandem Revisited

**CHAPTER 4: Trigonometric Functions**

**4.1 Angles and Their Measures**

The Problem of Angular Measure • Radians and Degrees • Circular Arc Length • Angular and Linear Motion

**4.2 Trigonometry of Acute Angles**

Right Triangle Trigonometry • Two Famous Triangles • Evaluating Trigonometric Functions • Applications of Right Triangle Trigonometry*

**4.3 The Trigonometric Functions Extended**

Trigonometric Functions of Any Angle • Trigonometric Functions of Real Numbers • Periodic Functions • The 16-Point Unit Circle

**4.4 Graphs of Sine and Cosine: Sinusoids**

The Basic Waves Revisited • Sinusoids and Transformations • Modeling Periodic Phenomena • Simple Harmonic Motion

**4.5 Tangent, Cotangent, Secant, and Cosecant**

The Tangent Function • The Cotangent Function • The Secant Function • The Cosecant Function

**4.6 Graphs of Composite Trigonometric Functions**

Combining Trigonometric and Algebraic Functions • Sums of Sinusoids

**4.7 Inverse Trigonometric Functions**

Inverse Sine Function • Inverse Cosine and Tangent Functions • Composing Trigonometric and Inverse Trigonometric Functions • Applications of Inverse Trigonometric Functions

**CHAPTER 5: Equivalent Trigonometric Representations and Polar Functions**

**5.1 Fundamental Identities**

Identities • Basic Trigonometric Identities • Pythagorean Identities • Cofunction Identities • Odd-Even Identities • Simplifying Trigonometric Expressions • Solving Trigonometric Equations

**5.2 Proving Trigonometric Identities**

A Proof Strategy • Proving Identities • Disproving Non-Identities • Identities Useful in Calculus

**5.3 Sum and Difference Identities**

Cosine of a Sum or Difference • Sine of a Sum or Difference • Tangent of a Sum or Difference • Double-Angle Identities • Solving Trigonometric Equations

**5.4 Laws of Sines and Cosines**

Deriving the Law of Sines • Solving Triangles (AAS, ASA) • The Ambiguous Case (SSA) • Deriving the Law of Cosines • Solving Triangles (SAS, SSS) • Triangle Area • Applications

**5.5 Polar Coordinates and the Complex Plane**

Polar Coordinate System • Coordinate Conversion • Equation Conversion • Finding Distance Using Polar Coordinates • Polar Form of Complex Numbers • Multiplication and Division of Complex Numbers • Powers and Roots of Complex Numbers

**5.6 Polar Functions and Their Graphs**

Graphing Polar and Parametric Functions • Symmetry • Analyzing Graphs of Polar Functions • Change in Tandem for Polar Functions • Rates of Change for Polar Functions

**CHAPTER 6: Vectors, Parametric Functions, and Conic Sections**

**6.1 Vectors in the Plane**

Vectors in the xy-Plane • Vector Operations • Unit Vectors • Direction Angles • The Dot Product and Its Applications

**6.2 Parametric and Implicit Functions Revisited**

Parametric and Vectored-Valued Functions • Redefining a Line • Modeling Planar Motion • How Equations in Two Variables Define Functions Implicitly • Rates of Change and Slopes of Curves

**6.3 Conic Sections and a New Look at Parabolas**

Conic Sections • Geometry of a Parabola • Horizontal Parabolas • Defining a Parabola Parametrically • Reflective Property of a Parabola*

**6.4 Circles and Ellipses**

Defining a Circle Parametrically • Equations of Circles in Standard Form • Equations of Ellipses in Standard Form • Defining an Ellipse Parametrically • Reflective Property of an Ellipse*

**6.5 Hyperbolas**

Equations of Hyperbolas in Standard Form • Asymptotes, Vertices, and Foci of Hyperbolas • Defining a Hyperbola Parametrically • Reflective Property of a Hyperbola*

**CHAPTER 7: Linear Transformations and Matrix-Vector Functions**

**7.1 Matrix Algebra**

Matrices • Matrix Addition and Subtraction • Matrix Multiplication • Identity and Inverse Matrices • Determinant of a Square Matrix • Applications*

**7.2 Linear Transformations and Matrices**

What are Linear Transformations? • Matrices and Linear Transformations • Compositions of Linear Transformations • Inverses of Linear Transformations

**7.3 Linear Transformations of the ***xy*-Plane

How Matrix Multiplication Transforms Vectors • Rotations in the *xy*-Plane • Dilations in the *xy*-Plane • Reflections over a Line • How Linear Transformations Transform the *xy*-Plane

**7.4 Transition Matrices and Modeling**

State Vectors and Transition Matrices • Predicting Future States • Calculating Past States • Calculating a Steady State

**CHAPTER 8: Limits, Derivatives, and Integrals***

**8.1 Limits and Motion: The Tangent Problem**

Average Velocity • Instantaneous Velocity • Limits Revisited • The Connection to Tangent Lines • The Derivative

**8.2 Limits and Motion: The Area Problem**

Distance Given a Constant Velocity • Distance Given a Changing Velocity • Limits at Infinity • Connection to Areas • The Definite Integral

**8.3 More on Limits**

A Little History • Defining a Limit Informally • Properties of Limits • Limits of Continuous Functions • One-Sided and Two-Sided Limits • Limits Involving Infinity

**8.4 Numerical Derivatives and Integrals**

Derivatives on a Calculator • Definite Integrals on a Calculator • Computing a Derivative from Data • Computing a Definite Integral from Data

**APPENDIX: Algebra Review***

**A.1 Radicals and Rational Exponents**

Radicals • Simplifying Radical Expressions • Rationalizing the Denominator • Rational Exponents

**A.2 Polynomials and Factoring**

Adding, Subtracting, and Multiplying Polynomials • Expanding Special Products • Basic Concepts of Factoring Polynomials • Factoring Polynomials Using Special Products • Factoring Trinomials • Factoring by Grouping

**A.3 Fractional Expressions**

Algebraic Expressions and Their Domain • Reducing Rational Expressions • Operations with Rational Expressions • Compound Rational Expressions

**Key Formulas** (End Papers/Endpapers)

**Formulas from Algebra**

**Formulas from Geometry**

**Formulas from Trigonometry**

**Formulas from Analytic Geometry**

**Gallery of Basic Functions**