Algebra and Trigonometry

2.1 Linear Equations in One Variable

• Examples
• Example 3: Solving Absolute Value Equations
• Example 4: Solving Linear Equations for One Variable
• Example 5: Calculating Average Speed
• Example 6: Calculating Average Interest Rate

2.2 Linear Inequalities in One Variable

• Examples
• Example 1: Multiplying Inequalities by Negative Numbers
• Example 2: Solving Linear Inequalities
• Example 3: Graphing Intervals of Real Numbers
• Example 4: Calculating Final Grades
• Example 5: Solving Double Linear Inequalities
• Example 6: Solving Linear Absolute Value Inequalities
• Example 7: Translating Inequality Phrases
• Example 8: Applications of Inequalities

2.3 Quadratic Equations in One Variable

• Examples
• Example 2: Perfect Square Quadratic Equations
• Example 3: Completing the Square
• Example 4: Using the Quadratic Formula
• Example 5: The Discriminant
• Example 6: Methods of Solving Quadratic Equations
• Example 7: Gravity Problems

2.4 Polynomial and Polynomial-Like Equations in One Variable

• Examples
• Example 1: Quadratic-Like Equations
• Example 2: Solving Equations by Factoring
• Example 3: Solving Equations by Factoring

2.5 Rational Equations in One Variable

• Examples
• Example 1: Solving Rational Equations
• Example 3: Filling a Pool
• Example 4: Filling a Pool Poorly

2.6 Radical Equations in One Variable

• Examples
• Example 1: Causing Extraneous Solutions
• Example 2: Radical Equations
• Example 3: Rational Exponents
• Example 4: Escape Speed

3.1 The Cartesian Coordinate System

• Examples
• Example 1: Plotting Points in the Cartesian Coordinate System
• Example 2: Graphing Equations in Two Variables
• Example 3: Using the Distance Formula
• Example 4: Using the Midpoint Formula

3.2 Circles

• Examples
• Example 1: Standard Form of a Circle
• Example 2: Standard Form of a Circle
• Example 3: Standard Form of a Circle
• Example 4: Graphing Circles
• Example 5: Graphing Circles and Completing the Square

3.3 Linear Equations in Two Variables

• Examples
• Example 1: Linear Equations in Two Variables
• Example 2: Finding Intercepts and Graphing Linear Equations
• Example 3: Graphing Horizontal and Vertical Lines

3.4 Slope and Forms of Linear Equations

• Examples
• Example 1: Calculating the Slope of a Line
• Example 2: Calculating the Slope of a Line
• Example 3: Slope-Intercept Form of a Line
• Example 4: Slope-Intercept Form of a Line
• Example 5: Using the Point-Slope Form of a Line
• Example 6: Using the Point-Slope Form of a Line

3.5 Parallel and Perpendicular Lines

• Examples
• Example 1: Finding Equations of Parallel Lines
• Example 2: Finding Equations of Parallel Lines
• Example 3: Identifying a Quadrilateral
• Example 4: Finding Equations of Perpendicular Lines
• Example 5: Finding Equations of Perpendicular Lines
• Example 6: Identifying Parallel and Perpendicular Lines

3.6 Linear Inequalities in Two Variables

• Examples
• Example 1: Solving Linear Inequalities
• Example 2: Solving Linear Inequalities
• Example 3: Solving Absolute Value Linear Inequalities

4.1 Relations and Functions

• Examples
• Example 1: Relations, Domains, and Ranges
• Example 2: Is the Relation a Function?
• Example 3: Functions and the Vertical Line Test
• Example 4: Function Notation
• Example 6: Evaluating Functions
• Example 7: Domain, Codomain, and Range
• Example 8: Implied Domain of a Function

4.2 Linear Functions

• Examples
• Example 1: Graphing Linear Functions

4.3 Quadratic Functions

• Examples
• Example 1: Graphing Quadratic Functions
• Example 2: Using the Vertex Formula
• Example 6: Fencing a Garden

4.4 Other Common Functions

• Examples
• Example 1: Power Functions of the Form axⁿ
• Example 3: The Absolute Value Function
• Example 4: Piecewise-Defined Function

4.5 Variation and Multivariable Functions

• Examples
• Example 1: Direct Variation
• Example 2: Inverse Variation
• Example 3: Joint Variation
• Example 4: Finding the Force of Gravity

4.6 Mathematical Models

5.1 Transformations of Functions

• Examples
• Example 1: Horizontal Shifting/Translation
• Example 3: Horizontal and Vertical Shifting
• Example 5: Vertical Stretching and Compressing
• Example 8: Order of Transformations

5.2 Properties of Functions

• Examples
• Example 1: Using Symmetry to Graph Relations
• Example 2: Determining Intervals of Monotonicity
• Example 3: Modeling the Water Level of a River

5.3 Combining Functions

• Examples
• Example 1: Combining Functions Arithmetically
• Example 2: Combining Functions Arithmetically
• Example 3: Combining Functions Arithmetically
• Example 4: Composing Functions
• Example 5: Domains of Compositions of Functions
• Example 6: Domains of Compositions of Functions
• Example 7: Decomposing Functions

5.4 Inverses of Functions

• Examples
• Example 1: Finding the Inverse of a Relation
• Example 2: Inverse Functions

6.1 Polynomial Functions and Polynomial Inequalities

• Examples
• Example 1: Solutions of Polynomial Equations
• Example 2: Graphing Polynomial Functions
• Example 4: Solving Polynomial Inequalities Using a Graph

6.2 Polynomial Division and the Division Algorithm

• Examples
• Example 1: Polynomial Long Division
• Example 2: Polynomial Long Division
• Example 3: Polynomial Long Division with Complex Numbers
• Example 5: Synthetic Division (with Complex Numbers)
• Example 6: Constructing Polynomials

6.3 Locating Real Zeros of Polynomial Functions

• Examples
• Example 1: The Rational Zero Theorem
• Example 2: Descartes' Rule of Signs
• Example 3: Finding Bounds of Real Zeros
• Example 4: Finding the Zeros of a Polynomial
• Example 5: Intermediate Value Theorem

6.4 The Fundamental Theorem of Algebra

• Examples
• Example 1: Graphing Polynomial Functions
• Example 2: Factoring Polynomials
• Example 3: Constructing Polynomials

6.5 Rational Functions and Rational Inequalities

• Examples
• Example 1: Vertical Asymptotes
• Example 2: Horizontal and Oblique Asymptotes
• Example 3: Graphing Rational Functions
• Example 4: Solving Rational Inequalities
• Example 5: Solving Rational Inequalities

7.1 Exponential Functions and Their Graphs

• Examples
• Example 1: Graphing Exponential Functions
• Example 3: Solving Elementary Exponential Equations

7.2 Exponential Models

• Examples
• Example 1: Population Growth
• Example 3: Compound Interest Formula
• Example 4: Compound Interest Formula
• Example 5: Continuous Compounding Formula

7.3 Logarithmic Functions and Their Graphs

• Examples
• Example 1: Exponential and Logarithmic Equations
• Example 2: Graphing Logarithmic Functions
• Example 5: Solving Logarithmic Equations
• Example 6: Evaluating Logarithmic Expressions

7.4 Logarithmic Properties and Models

• Examples
• Example 1: Continuously Compounded Interest
• Example 2: Solving Exponential Equations
• Example 3: Expanding Logarithmic Expressions
• Example 4: Condensing Logarithmic Expressions
• Example 5: Change of Base Formula
• Example 6: The pH Scale
• Example 8: The Decibel Scale

7.5 Exponential and Logarithmic Equations

• Examples
• Example 1: Solving Exponential Equations
• Example 2: Solving Exponential Equations
• Example 3: Solving Logarithmic Equations
• Example 4: Solving Logarithmic Equations
• Example 5: Compounding Interest
• Example 6: Radiocarbon Dating

8.1 Radian and Degree Measure

• Examples
• Example 1: Converting between Degrees and Radians
• Example 2: Determining the Measure of an Angle
• Example 3: Applying the Arc Length Formula
• Example 4: Finding the Angular and Linear Speeds
• Example 5: Finding the Area of a Sector

8.2 Trigonometric Functions and Right Triangles

• Examples
• Example 1: Finding the Values of the Six Trigonometric Functions
• Example 2: Evaluating Trigonometric Functions
• Example 4: Using Trigonometric Functions
• Example 5: Using Trigonometric Functions
• Example 6: Using Trigonometric Functions

8.3 Trigonometric Functions and the Unit Circle

• Examples
• Example 4: Finding the Reference Angle
• Example 6: Using Cofunction Identities
• Example 7: Using the Relationships between Trigonometric Functions
• Example 8: Using the Relationships between Trigonometric Functions

8.4 Graphs of Sine and Cosine Functions

• Examples
• Example 7: Graphing Transformed Trigonometric Functions
• Example 8: Using Simple Harmonic Motion
• Example 9: Modeling Damped Harmonic Motion

8.5 Graphs of Other Trigonometric Functions

8.6 Inverse Trigonometric Functions

• Examples
• Example 2: Evaluating Compositions of Trigonometric Functions
• Example 3: Evaluating Compositions of Trigonometric Functions
• Example 4: Using Inverse Trigonometric Functions
• Example 5: Using Inverse Trigonometric Functions

9.1 Fundamental Trigonometric Identities

• Examples
• Example 1: Simplifying Trigonometric Expressions
• Example 2: Simplifying Trigonometric Expressions
• Example 3: Verifying Trigonometric Identities
• Example 4: Verifying Trigonometric Identities
• Example 5: Verifying Trigonometric Identities
• Example 6: Using Trigonometric Substitutions

9.2 Sum and Difference Identities

• Examples
• Example 1: Using the Sum and Difference Identities for Exact Evaluation
• Example 3: Using the Sum and Difference Identities for Exact Evaluation
• Example 4: Using the Sum and Difference Identities for Exact Evaluation
• Example 5: Using the Sum and Difference Identities
• Example 7: Using the Sum and Difference Identities
• Example 8: Using the Sum and Difference Identities
• Example 9: Using the Sum of Sines and Cosines Identity

9.3 Product-Sum Identities

• Examples
• Example 1: Using the Double-Angle Identities
• Example 2: Using Trigonometric Identities
• Example 3: Using the Power-Reducing Identities
• Example 4: Using the Half-Angle Identities
• Example 5: Using the Half-Angle Identities for Exact Evaluation
• Example 6: Using the Product-to-Sum Identities
• Example 7: Using the Sum-to-Product Identities

9.4 Trigonometric Equations

• Examples
• Example 1: Solving Equations by Isolating the Trigonometric Function
• Example 2: Solving Equations by Isolating the Trigonometric Function
• Example 3: Solving Trigonometric Equations by Factoring
• Example 4: Solving Equations Using Trigonometric Identities
• Example 5: Solving Trigonometric Equations by Graphing
• Example 6: Solving Equations Using Trigonometric Identities
• Example 7: Solving Equations by Isolating the Trigonometric Function
• Example 8: Solving Equations Using Inverse Trigonometric Functions
• Example 9: Solving Equations Using Inverse Trigonometric Functions

10.1 The Law of Sines

• Examples
• Example 1: Using the Law of Sines in an AAS Situation
• Example 2: Using the Law of Sines in an ASA Situation
• Example 3: Using the Law of Sines in an SSA Situation with Two Solutions
• Example 4: Using the Law of Sines in an SSA Situation with No Solution
• Example 5: Using the Sine Formula to Find the Area of a Triangle

10.2 The Law of Cosines

• Examples
• Example 1: Using the Law of Cosines in an SSS Situation
• Example 2: Using the Law of Cosines in an SAS Situation
• Example 3: Using Heron's Formula to Find the Area of a Triangle

10.3 Polar Coordinates and Polar Equations

• Examples
• Example 1: Plotting in Polar Coordinates
• Example 2: Converting from Polar to Cartesian Coordinates
• Example 4: Rewriting an Equation in Polar Form
• Example 5: Rewriting an Equation in Rectangular Form
• Example 6: Graphing Polar Equations
• Example 7: Graphing Polar Equations
• Example 8: Graphing Common Polar Equations Using Symmetry
• Example 9: Graphing Common Polar Equations Using Symmetry

10.4 Parametric Equations

• Examples
• Example 1: Determining Parametric Equations and Their Graphs
• Example 2: Using Parametric Equations
• Example 3: Eliminating the Parameter
• Example 4: Eliminating the Parameter
• Example 5: Constructing Parametric Equations

10.5 Trigonometric Form of Complex Numbers

• Examples
• Example 1: Finding the Magnitude
• Example 2: Graphing Regions in the Complex Plane
• Example 3: Writing Complex Numbers in Trigonometric Form
• Example 4: Writing Complex Numbers in Standard Form
• Example 5: Multiplying Complex Numbers
• Example 6: Dividing Complex Numbers
• Example 7: Graphing Complex Numbers and Their Product
• Example 8: Using De Moivre's Theorem
• Example 9: Finding Roots of Complex Numbers
• Example 10: Finding Roots of Complex Numbers

10.6 Vectors in the Cartesian Plane

• Examples
• Example 1: Graphing the Sum of Two Vectors
• Example 2: Graphing Vectors
• Example 3: Using Vector Operations
• Example 4: Finding the Unit Vector and Linear Combination of a Vector
• Example 5: Finding the Vector Form of the Velocity
• Example 6: Applying Vector Operations
• Example 7: Applying Vector Operations

10.7 The Dot Product

• Examples
• Example 1: Calculating the Dot Product
• Example 2: Using the Properties of the Dot Product
• Example 3: Applying the Dot Product Theorem
• Example 4: Finding an Orthogonal Vector
• Example 5: Finding the Projection of u onto v
• Example 6: Applying the Dot Product
• Example 7: Applying the Dot Product

10.8 Hyperbolic Functions

11.1 Ellipses

• Examples
• Example 1: Graphing an Ellipse Centered at the Origin
• Example 2: Graphing an Ellipse with Equation in Standard Form
• Example 3: Graphing an Ellipse with Equation Not in Standard Form
• Example 4: Constructing the Equation for an Ellipse

11.2 Parabolas

• Examples
• Example 1: Graphing a Parabola with Equation in Standard Form
• Example 2: Graphing a Parabola with Equation Not in Standard Form
• Example 3: Standard Form Equation of a Parabola
• Example 4: Parabolic Mirrors

11.3 Hyperbolas

• Examples
• Example 1: Graphing a Hyperbola with Equation in Standard Form
• Example 2: Graphing a Hyperbola with Equation Not in Standard Form
• Example 3: Standard Form of the Equation of a Hyperbola
• Example 4: LORAN

11.4 Rotation of Conic Sections

• Examples
• Example 1: Finding $x^{\mathrm{\prime }}{y}^{\mathrm{\prime }}$-Coordinates
• Example 2: Using the Rotation Relations
• Example 4: Classifying and Graphing Conic Sections

11.5 Polar Equations of Conic Sections

• Examples
• Example 1: Identifying the Type and Directrix of a Conic
• Example 2: Constructing a Polar Equation
• Example 3: Graphing Conic Sections in Polar Form
• Example 4: Graphing Conic Sections in Polar Form
• Example 5: Graphing Conic Sections Using Rotation in Polar Coordinates

12.1 Solving Systems of Linear Equations by Substitution and Elimination

• Examples
• Example 1: Solving an Independent System by Substitution
• Example 2: Solving a Dependent System by Substitution
• Example 3: Solving an Independent System by Elimination
• Example 4: Solving an Inconsistent System by Elimination
• Example 5: Solving an Independent System by Elimination
• Example 6: Solving a Dependent System by Elimination
• Example 7: Mixing Alloys
• Example 8: Determining Ages

12.2 Matrix Notation and Gauss-Jordan Elimination

• Examples
• Example 1: Matrices and Matrix Notation
• Example 2: Augmented Matrices
• Example 3: Row Echelon Form
• Example 4: Elementary Row Operations
• Example 5: Gaussian Elimination
• Example 6: Reduced Row Echelon Form

12.3 Determinants and Cramer's Rule

• Examples
• Example 1: Determinant of a 2×2 Matrix
• Example 2: Minors and Cofactors
• Example 3: Determinant of an n×n Matrix
• Example 4: Properties of Determinants
• Example 5: Cramer's Rule
• Example 6: Cramer's Rule

12.4 Basic Matrix Operations

• Examples
• Example 1: Matrix Addition
• Example 2: Matrix Equations
• Example 3: Scalar Multiplication
• Example 4: Matrix Subtraction
• Example 5: Matrix Multiplication
• Example 6: Matrix Multiplication

12.5 Inverses of Matrices

• Examples
• Example 1: Matrix Equation
• Example 2: Finding the Inverse of a Matrix
• Example 3: Finding the Inverse of a Matrix
• Example 4: Inverse Matrix Method

12.6 Partial Fraction Decomposition

• Examples
• Example 1: Finding the Partial Fraction Decomposition of a Function
• Example 2: Finding the Partial Fraction Decomposition of a Function
• Example 3: Finding the Partial Fraction Decomposition of a Function
• Example 4: Finding the Partial Fraction Decomposition of a Function
• Example 5: Finding the Partial Fraction Decomposition of a Function

12.7 Systems of Linear Inequalities and Linear Programming

• Examples
• Example 2: Feasible Regions
• Example 3: Linear Programming
• Example 4: Linear Programming

12.8 Systems of Nonlinear Equations and Inequalities

• Examples
• Example 1: Solving Systems of Nonlinear Equations by Graphing
• Example 2: Solving Systems of Nonlinear Equations by Graphing
• Example 3: Solving Systems of Nonlinear Equations Algebraically
• Example 4: Solving Systems of Nonlinear Equations Algebraically
• Example 5: Solving Systems of Nonlinear Equations Algebraically
• Example 6: Solving Systems of Nonlinear Inequalities by Graphing

13.1 Sequences and Series

• Examples
• Example 1: Sequences
• Example 2: Recursively Defined Sequences
• Example 3: Finding the Formula for a Sequence
• Example 4: Evaluating Sums
• Example 5: Evaluating Sums
• Example 6: Evaluating Partial Sums and Series

13.2 Arithmetic Sequences and Series

• Examples
• Example 1: General Term of an Arithmetic Sequence
• Example 3: Partial Sums of Arithmetic Sequences
• Example 4: Partial Sums of Arithmetic Sequences

13.3 Geometric Sequences and Series

• Examples
• Example 1: General Term of a Geometric Sequence
• Example 2: General Term of a Geometric Sequence
• Example 3: Comparing Salary Offers
• Example 4: Partial Sum of a Geometric Sequence
• Example 5: Infinite Geometric Series

13.4 Mathematical Induction

• Examples
• Example 1: Proof by Mathematical Induction
• Example 2: Proof by Mathematical Induction
• Example 3: Proof by Mathematical Induction
• Example 4: Proof by Mathematical Induction

13.5 Combinatorics

• Examples
• Example 2: Using the Multiplication Principle of Counting
• Example 3: Calculating Permutations
• Example 4: Calculating Permutations
• Example 5: Using the Permutation Formula
• Example 6: Calculating Permutations and Combinations
• Example 7: The Combination Formula and Forming Committees
• Example 8: Counting Rules and Forming "Words"
• Example 11: Using the Multinomial Theorem

13.6 Probability

• Examples
• Example 1: Probabilities When Outcomes Are Equally Likely
• Example 2: Computing Probabilities
• Example 3: Computing Probabilities
• Example 4: Probability of a Union of Two Events
• Example 5: Unions of Mutually Exclusive Events
• Example 6: Independent Events