Precalculus

2.1 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

2.2 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

2.3 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: Point-Slope Form of a Line
• Example 6: Point-Slope Form of a Line

2.4 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

2.5 Linear Inequalities in Two Variables

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

2.6 Introduction to 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

3.1 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 5: Evaluating Functions
• Example 6: Domain, Codomain, and Range
• Example 7: Implied Domain of a Function

3.2a Linear and Quadratic Functions

• Examples
• Example 1: Graphing Linear Functions
• Example 2: Graphing Quadratic Functions
• Example 3: Using the Vertex Formula

3.2b Max/Min Applications of Quadratic Functions

• Examples
• Example 1: Fencing a Garden

3.3 Other Common Functions

• Examples
• Example 1: Functions of the Form axⁿ
• Example 2: Functions of the Form ${\displaystyle \frac{a}{x^n}}$
• Example 3: The Absolute Value Function
• Example 4: Piecewise-Defined Function

3.4 Variation and Multi-Variable Functions

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

3.5 Transformations of Functions

• Examples
• Example 1: Horizontal Shifting/Translation
• Example 3: Horizontal and Vertical Shifting
• Example 5: Vertical Stretching and Compressing
• Example 6: Order of Transformations
• Example 7: Symmetry of Equations
• Example 8
• Example 9

3.6 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

3.7 Inverses of Functions

• Examples
• Example 1: Finding the Inverse of a Relation
• Example 2: Inverse Functions
• Example 3: Finding Formulas of Inverse Functions

4.1 Introduction to Polynomial Equations and Graphs

• Examples
• Example 1: Solutions of Polynomial Equations
• Example 2: Graphing Polynomial Functions
• Example 3: Solving Polynomial Inequalities Using a Graph
• Example 4: Solving Polynomial Inequalities: Sign-Test Method

4.2 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 4: Synthetic Division
• Example 5: Synthetic Division (with Complex Numbers)
• Example 6: Constructing Polynomials

4.3 Locating Real Zeros of Polynomials

• 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

4.4 The Fundamental Theorem of Algebra

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

4.5a Rational Functions

• Examples
• Example 1: Vertical Asymptotes
• Example 2: Horizontal and Oblique Asymptotes
• Example 3: Graphing Rational Functions

4.5b Rational Inequalities

• Examples
• Example 1: Solving Rational Inequalities
• Example 2: Rational Inequalities

5.1 Exponential Functions and their Graphs

• Examples
• Example 1: Graphing Exponential Functions
• Example 2: Graphing Exponential Functions
• Example 3: Solving Elementary Exponential Equations

5.2 Applications of Exponential Functions

• Examples
• Example 1: Population Growth
• Example 2: Radioactive Decay
• Example 3: Compound Interest Formula
• Example 4: Compound Interest Formula
• Example 5: Continuous Compounding Formula

5.3 Logarithmic Functions and their Graphs

• Examples
• Example 1: Exponential and Logarithmic Equations
• Example 2: Graphing Logarithmic Functions
• Example 3: Graphing Logarithmic Functions
• Example 4: Logarithmic Expressions
• Example 5: Solving Logarithmic Equations
• Example 6: Evaluating Logarithmic Expressions

5.4 Properties and Applications of Logarithms

• 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 7: The Richter Scale
• Example 8: The Decibel Scale

5.5 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

6.1 Radian and Degree Measure of Angles

• 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

6.2 Trigonometric Functions of Acute Angles

• 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

6.3 Trigonometric Functions of Any Angle

• Examples
• Example 1: Finding the Values of the Six Trigonometric Functions
• Example 2: Finding the Values of the Six Trigonometric Functions
• Example 3: Finding the Reference Angle
• Example 4: Evaluating Trigonometric Functions
• Example 5: Using Cofunction Identities
• Example 6: Using the Relationships Between Trigonometric Functions
• Example 7: Using the Relationships Between Trigonometric Functions

6.4 Graphs of Trigonometric Functions

• Examples
• Example 1: Graphing the Cosecant Function
• Example 2: Finding the Period of Trigonometric Functions
• Example 3: Graphing the Tangent Function
• Example 4: Using Simple Harmonic Motion
• Example 5: Graphing Transformed Trigonometric Functions
• Example 6: Graphing Transformed Trigonometric Functions
• Example 7: Graphing Transformed Trigonometric Functions
• Example 8: Modeling Damped Harmonic Motion

6.5 Inverse Trigonometric Functions

• Examples
• Example 1: Defining and Plotting the Arccosine Function
• Example 2: Evaluating Inverse Trigonometric Functions
• Example 3: Evaluating Compositions of Trigonometric Functions
• Example 4: Evaluating Compositions of Trigonometric Functions
• Example 5: Using Inverse Trigonometric Functions
• Example 6: Using Inverse Trigonometric Functions

7.1 Fundamental Identities and their Uses

• 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

7.2 Sum and Difference Identities

• Examples
• Example 1: Using the Sum and Difference Identities for Exact Evaluation
• Example 2: 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 6: Using the Sum and Difference Identities
• Example 7: Using the Sum and Difference Identities
• Example 8: Using the Sum of Sines and Cosines Identity

7.3 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

7.4 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

8.1a The Law of Sines and the Law of Cosines

• 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: Determining If a Triangle Exists
• Example 4: Determining If a Triangle Exists
• Example 5: Using the Law of Cosines in a SSS Situation
• Example 6: Using the Law of Cosines in an SAS Situation

8.1b Area of Triangles

• Examples
• Example 1: Using the Sine Formula to Find the Area of a Triangle
• Example 2: Using Heron's Formula to Find the Area of a Triangle

8.2 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 Coordinates
• Example 5: Rewriting an Equation in Rectangular Coordinates
• 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

8.3a Parametric Equations - Graphing and Applications

• Examples
• Example 1: Determining Parametric Equations and Their Graphs
• Example 2: Using Parametric Equations

8.3b Parametric Equations - Eliminating the Parameter

• Examples
• Example 3: Eliminating the Parameter
• Example 4: Eliminating the Parameter

8.3c Parametric Equations - Constructing Equations

• Examples
• Example 5: Constructing Parametric Equations

8.4a 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

8.4b Operations with Complex Numbers

• Examples
• Example 1: Multiplying Complex Numbers
• Example 2: Dividing Complex Numbers
• Example 3: Graphing Complex Numbers and Their Product
• Example 4: Using De Moivre's Theorem
• Example 5: Finding Roots of Complex Numbers
• Example 6: Finding Roots of Complex Numbers

8.5 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

8.6 The Dot Product and Its Uses

• 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

9.1 The Ellipse

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

9.2 The Parabola

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

9.3 The Hyperbola

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

9.4 Rotation of Conics

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

9.5 Polar Form 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

10.1 Solving Systems by Substitution and Elimination

• Examples
• Example 1: Solving a System of Equations by Substitution
• Example 2: Solving a System of Equations by Substitution
• Example 3: Solving a System of Equations by Elimination
• Example 4: Solving a System of Equations by Elimination
• Example 5: Solving a System of Equations by Elimination
• Example 6: Solving a System of Equations by Elimination
• Example 7: Mixing Alloys
• Example 8: Determining Ages

10.2 Matrix Notation and Gaussian 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

10.3 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

10.4 The Algebra of Matrices

• 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

10.5 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

10.6 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

10.7 Linear Programming

• Examples
• Example 1: Feasible Regions
• Example 2: Linear Programming
• Example 3: Linear Programming
• Example 4: Linear Programming

10.8 Nonlinear Systems of Equations

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

11.1 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

11.2 Arithmetic Sequences and Series

• Examples
• Example 1: General Term of an Arithmetic Sequence
• Example 2: Modeling Population Growth
• Example 3: Partial Sums of Arithmetic Sequences
• Example 4: Partial Sums of Arithmetic Sequences

11.3 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

11.4 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

11.5a An Introduction to Combinatorics - Counting, Permutations, and Combinations

• Examples
• Example 1: Counting Phone Numbers
• Example 2: The Multiplication Principle of Counting
• Example 3: Permutations
• Example 4: Permutations
• Example 5: Permutation Formula
• Example 6: Permutations and Combinations
• Example 7: Forming Committees
• Example 8: Forming "Words"

11.5b An Introduction to Combinatorics - The Binomial and Multinomial Theorems

• Examples
• Example 1: Binomial Expansion
• Example 2: The Binomial Theorem
• Example 3: The Multinomial Theorem

11.6 An Introduction to 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

A.1 Mathematical Models

A.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

A.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

A.4 Limits in the Plane

• Examples
• Example 3: Finding One-Sided Limits near Vertical Asymptotes
• Example 4: Finding One-Sided Limits near Vertical Asymptotes
• Example 6: Using Limits at Infinity to Identify Horizontal Asymptotes