Precalculus Crash Course | Author Jonathan David

Before Starting Precalculus

Based on the Sullivan Precalculus Textbook Table of Contents

1 Graphs

1.1 The Distance and Midpoint Formulas

Use the Distance Formula
Use the Midpoint Formula

1.2 Graphs of Equations in Two Variables; Intercepts; Symmetry

Graph Equations by Plotting Points
Find Intercepts from a Graph
Find Intercepts from an Equation
Test an Equation for Symmetry with Respect to the x-Axis, the y-Axis, and the Origin
Know How to Graph Key Equations

1.3 Lines

Calculate and Interpret the Slope of a Line
Graph Lines Given a Point and the Slope
Find the Equation of a Vertical Line
Use the Point-Slope Form of a Line
Identify Horizontal Lines
Use the Slope-Intercept Form of a Line
Find the Equation of a Line Given Two Points
Graph Lines Written in General Form Using Intercepts
Find Equations of Parallel Lines
Find Equations of Perpendicular Lines

1.4 Circles

Write the Standard Form of the Equation of a Circle
Graph a Circle
Work with the General Form of the Equation of a Circle

2 Functions and Their Graphs

2.1 Functions

Describe a Relation
Determine Whether a Relation Represents a Function
Use Function Notation; Find the Value of a Function
Find the Difference Quotient of a Function
Find the Domain of a Function Defined by an Equation
Form the Sum, Difference, Product, and Quotient of Two Functions

2.2 The Graph of a Function

Identify the Graph of a Function
Obtain Information from or about the Graph of a Function

2.3 Properties of Functions

Identify Even and Odd Functions from a Graph
Identify Even and Odd Functions from an Equation
Use a Graph to Determine Where a Function is Increasing, Decreasing, or Constant
Use a Graph to Locate Local Maxima and Local Minima
Use a Graph to Locate the Absolute Maximum and the Absolute Minimum
Use a Graphing Utility to Approximate Local Maxima and Local Minima and to Determine Where a Function Is Increasing or Decreasing
Find the Average Rate of Change of a Function

2.4 Library of Functions; Piecewise-defined Functions

Graph the Functions Listed in the Library of Functions
Analyze a Piecewise-defined Function

2.5 Graphing Techniques: Transformations

Graph Functions Using Vertical and Horizontal Shifts
Graph Functions Using Compressions and Stretches
Graph Functions Using Reflections about the x-Axis and the y-Axis

2.6 Mathematical Models: Building Functions

Build and Analyze Functions

3 Linear and Quadratic Functions

3.1 Properties of Linear Functions and Linear Models

Graph Linear Functions
Use Average Rate of Change to Identify Linear Functions
Determine Whether a Linear Function Is Increasing, Decreasing, or Constant
Build Linear Models from Verbal Descriptions

3.2 Building Linear Models from Data

Draw and Interpret Scatter Plots
Distinguish between Linear and Nonlinear Relations
Use a Graphing Utility to Find the Line of Best Fit

3.3 Quadratic Functions and Their Properties

Graph a Quadratic Function Using Transformations
Identify the Vertex and Axis of Symmetry of a Parabola
Graph a Quadratic Function Using Its Vertex, Axis, and Intercepts
Find a Quadratic Function Given Its Vertex and One Other Point
Find the Maximum or Minimum Value of a Quadratic Function

3.4 Building Quadratic Models from Verbal Descriptions and from Data

Build Quadratic Models from Verbal Descriptions
Build Quadratic Models from Data

3.5 Inequalities Involving Quadratic Functions

Solve Inequalities Involving a Quadratic Function

4 Polynomial and Rational Functions

4.1 Polynomial Functions

Identify Polynomial Functions and Their Degree
Graph Polynomial Functions Using Transformations
Identify the Real Zeros of a Polynomial Function and Their Multiplicity

4.2 Graphing Polynomial Functions; Models

Graph a Polynomial Function
Graph a Polynomial Function Using a Graphing Utility
Build Cubic Models from Data

4.3 Properties of Rational Functions

Find the Domain of a Rational Function
Find the Vertical Asymptotes of a Rational Function
Find a Horizontal or an Oblique Asymptote of a Rational Function

4.4 The Graph of a Rational Function

Graph a Rational Function
Solve Applied Problems Involving Rational Functions

4.5 Polynomial and Rational Inequalities

Solve Polynomial Inequalities
Solve Rational Inequalities

4.6 The Real Zeros of a Polynomial Function

Use the Remainder and Factor Theorems
Use Descartes’ Rule of Signs to Determine the Number of Positive and the Number of Negative Real Zeros of a Polynomial Function
Use the Rational Zeros Theorem to List the Potential Rational Zeros of a Polynomial Function
Find the Real Zeros of a Polynomial Function
Solve Polynomial Equations
Use the Theorem for Bounds on Zeros
Use the Intermediate Value Theorem

4.7 Complex Zeros; Fundamental Theorem of Algebra

Use the Conjugate Pairs Theorem
Find a Polynomial Function with Specified Zeros
Find the Complex Zeros of a Polynomial Function

5 Exponential and Logarithmic Functions

5.1 Composite Functions

Form a Composite Function
Find the Domain of a Composite Function

5.2 One-to-One Functions; Inverse Functions

Determine Whether a Function Is One-to-One
Obtain the Graph of the Inverse Function from the Graph of a One-to-One Function
Verify an Inverse Function
Find the Inverse of a Function Defined by an Equation

5.3 Exponential Functions

Evaluate Exponential Functions
Graph Exponential Functions
Define the Number e
Solve Exponential Equations

5.4 Logarithmic Functions

Change Exponential Statements to Logarithmic Statements and Logarithmic Statements to Exponential Statements
Evaluate Logarithmic Expressions
Determine the Domain of a Logarithmic Function
Graph Logarithmic Functions
Solve Logarithmic Equations

5.5 Properties of Logarithms

Work with the Properties of Logarithms
Write a Logarithmic Expression as a Sum or Difference of Logarithms
Write a Logarithmic Expression as a Single Logarithm
Evaluate Logarithms Whose Base Is Neither 10 Nor e

5.6 Logarithmic and Exponential Equations

Solve Logarithmic Equations
Solve Exponential Equations
Solve Logarithmic and Exponential Equations Using a Graphing Utility

5.7 Financial Models

Determine the Future Value of a Lump Sum of Money
Calculate Effective Rates of Return
Determine the Present Value of a Lump Sum of Money
Determine the Rate of Interest or the Time Required to Double a Lump Sum of Money

5.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models

Model Populations That Obey the Law of Uninhibited Growth
Model Populations That Obey the Law of Uninhibited Decay
Use Newton’s Law of Cooling
Use Logistic Models

5.9 Building Exponential, Logarithmic, and Logistic Models from Data

Build an Exponential Model from Data
Build a Logarithmic Model from Data
Build a Logistic Model from Data

6 Trigonometric Functions

6.1 Angles, Arc Length, and Circular Motion

Angles and Degree Measure
Convert between Decimal and Degree, Minute, Second Measures for Angles
Find the Length of an Arc of a Circle
Convert from Degrees to Radians and from Radians to Degrees
Find the Area of a Sector of a Circle
Find the Linear Speed of an Object Traveling in Circular Motion

6.2 Trigonometric Functions: Unit Circle Approach

Find the Exact Values of the Trigonometric Functions Using a Point on the Unit Circle
Find the Exact Values of the Trigonometric Functions of Quadrantal Angles
Find the Exact Values of the Trigonometric Functions of π/4 = 45°
Find the Exact Values of the Trigonometric Functions of π/6 = 30° and π/3 = 60°
Find the Exact Values of the Trigonometric Functions for Integer Multiples of π/6 = 30°, π/4 = 45°, and π/3 = 60°
Use a Calculator to Approximate the Value of a Trigonometric Function
Use a Circle of Radius r to Evaluate the Trigonometric Functions

6.3 Properties of the Trigonometric Functions

Determine the Domain and the Range of the Trigonometric Functions
Determine the Period of the Trigonometric Functions
Determine the Signs of the Trigonometric Functions in a Given Quadrant
Find the Values of the Trigonometric Functions Using Fundamental Identities
Find the Exact Values of the Trigonometric Functions of an Angle Given One of the Functions and the Quadrant of the Angle
Use Even-Odd Properties to Find the Exact Values of the Trigonometric Functions

6.4 Graphs of the Sine and Cosine Functions

Graph the Sine Function y = sin x and Functions of the Form y = A sin (ωx)
Graph the Cosine Function y = cos x and Functions of the Form y = A cos (ωx)
Determine the Amplitude and Period of Sinusoidal Functions
Graph Sinusoidal Functions Using Key Points
Find an Equation for a Sinusoidal Graph

6.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions

Graph the Tangent Function y = tan x and the Cotangent Function y = cot x
Graph Functions of the Form y = A tan (ωx) + B and y = A cot (ωx) + B
Graph the Cosecant Function y = csc x and the Secant Function y = sec x
Graph Functions of the Form y = A csc (ωx) + B and y = A sec (ωx) + B

6.6 Phase Shift; Sinusoidal Curve Fitting

Graph Sinusoidal Functions of the Form y = A sin (ωx – ϕ) + B
Build Sinusoidal Models from Data

7 Analytic Trigonometry

7.1 The Inverse Sine, Cosine, and Tangent Functions

Define the Inverse Sine Function
Find the Value of an Inverse Sine Function
Define the Inverse Cosine Function
Find the Value of an Inverse Cosine Function
Define the Inverse Tangent Function
Find the Value of an Inverse Tangent Function
Use Properties of Inverse Functions to Find Exact Values of Certain Composite Functions
Find the Inverse Function of a Trigonometric Function
Solve Equations Involving Inverse Trigonometric Functions

7.2 The Inverse Trigonometric Functions (Continued)

Define the Inverse Secant, Cosecant, and Cotangent Functions
Find the Value of Inverse Secant, Cosecant, and Cotangent Functions
Find the Exact Value of Composite Functions Involving the Inverse Trigonometric Functions
Write a Trigonometric Expression as an Algebraic Expression

7.3 Trigonometric Equations

Solve Equations Involving a Single Trigonometric Function
Solve Trigonometric Equations Using a Calculator
Solve Trigonometric Equations Quadratic in Form
Solve Trigonometric Equations Using Fundamental Identities
Solve Trigonometric Equations Using a Graphing Utility

7.4 Trigonometric Identities

Use Algebra to Simplify Trigonometric Expressions
Establish Identities

7.5 Sum and Difference Formulas

Use Sum and Difference Formulas to Find Exact Values
Use Sum and Difference Formulas to Establish Identities
Use Sum and Difference Formulas Involving Inverse Trigonometric Functions
Solve Trigonometric Equations Linear in Sine and Cosine

7.6 Double-angle and Half-angle Formulas

Use Double-angle Formulas to Find Exact Values
Use Double-angle Formulas to Establish Identities
Use Half-angle Formulas to Find Exact Values

7.7 Product-to-Sum and Sum-to-Product Formulas

Rewrite Products as Sums
Express Sums as Products

8 Applications of Trigonometric Functions

8.1 Right Triangle Trigonometry; Applications

Find the Value of Trigonometric Functions of Acute Angles Using Right Triangles
Use the Complementary Angle Theorem
Solve Right Triangles
Solve Applied Problems

8.2 The Law of Sines

Solve SAA or ASA Triangles
Solve SSA Triangles
Solve Applied Problems

8.3 The Law of Cosines

Solve SAS Triangles
Solve SSS Triangles
Solve Applied Problems

8.4 Area of a Triangle

Find the Area of SAS Triangles
Find the Area of SSS Triangles

8.5 Simple Harmonic Motion; Damped Motion; Combining Waves

Build a Model for an Object in Simple Harmonic Motion
Analyze Simple Harmonic Motion
Analyze an Object in Damped Motion
Graph the Sum of Two Functions

9 Polar Coordinates; Vectors

9.1 Polar Coordinates

Plot Points Using Polar Coordinates
Convert from Polar Coordinates to Rectangular Coordinates
Convert from Rectangular Coordinates to Polar Coordinates
Transform Equations between Polar and Rectangular Forms

9.2 Polar Equations and Graphs

Identify and Graph Polar Equations by Converting to Rectangular Equations
Test Polar Equations for Symmetry
Graph Polar Equations by Plotting Points

9.3 The Complex Plane; De Moivre’s Theorem

Plot Points in the Complex Plane
Convert a Complex Number between Rectangular Form and Polar Form or Exponential Form
Find Products and Quotients of Complex Numbers
Use De Moivre’s Theorem
Find Complex Roots

9.4 Vectors

Graph Vectors
Find a Position Vector
Add and Subtract Vectors Algebraically
Find a Scalar Multiple and the Magnitude of a Vector
Find a Unit Vector
Find a Vector from Its Direction and Magnitude
Model with Vectors

9.5 The Dot Product

Find the Dot Product of Two Vectors
Find the Angle between Two Vectors
Determine Whether Two Vectors Are Parallel
Determine Whether Two Vectors Are Orthogonal
Decompose a Vector into Two Orthogonal Vectors
Compute Work

9.6 Vectors in Space

Find the Distance between Two Points in Space
Find Position Vectors in Space
Perform Operations on Vectors
Find the Dot Product
Find the Angle between Two Vectors
Find the Direction Angles of a Vector

9.7 The Cross Product

Find the Cross Product of Two Vectors
Know Algebraic Properties of the Cross Product
Know Geometric Properties of the Cross Product
Find a Vector Orthogonal to Two Given Vectors
Find the Area of a Parallelogram

10 Analytic Geometry

10.1 Conics

Know the Names of the Conics

10.2 The Parabola

Analyze Parabolas with Vertex at the Origin
Analyze Parabolas with Vertex at (h, k)
Solve Applied Problems Involving Parabolas

10.3 The Ellipse

Analyze Ellipses with Center at the Origin
Analyze Ellipses with Center at (h, k)
Solve Applied Problems Involving Ellipses

10.4 The Hyperbola

Analyze Hyperbolas with Center at the Origin
Find the Asymptotes of a Hyperbola
Analyze Hyperbolas with Center at (h, k)
Solve Applied Problems Involving Hyperbolas

10.5 Rotation of Axes; General Form of a Conic

Identify a Conic
Use a Rotation of Axes to Transform Equations
Analyze an Equation Using a Rotation of Axes
Identify Conics without Rotating the Axes

10.6 Polar Equations of Conics

Analyze and Graph Polar Equations of Conics
Convert the Polar Equation of a Conic to a Rectangular Equation

10.7 Plane Curves and Parametric Equations

Graph Parametric Equations
Find a Rectangular Equation for a Plane Curve Defined Parametrically
Use Time as a Parameter in Parametric Equations
Find Parametric Equations for Plane Curves Defined by Rectangular Equations

11 Systems of Equations and Inequalities

11.1 Systems of Linear Equations: Substitution and Elimination

Solve Systems of Equations by Substitution
Solve Systems of Equations by Elimination
Identify Inconsistent Systems of Equations Containing Two Variables
Express the Solution of a System of Dependent Equations Containing Two Variables
Solve Systems of Three Equations Containing Three Variables
Identify Inconsistent Systems of Equations Containing Three Variables
Express the Solution of a System of Dependent Equations Containing Three Variables

11.2 Systems of Linear Equations: Matrices

Write the Augmented Matrix of a System of Linear Equations
Write the System of Equations from the Augmented Matrix
Perform Row Operations on a Matrix
Solve a System of Linear Equations Using Matrices

11.3 Systems of Linear Equations: Determinants

Evaluate 2 by 2 Determinants
Use Cramer’s Rule to Solve a System of Two Equations Containing Two Variables
Evaluate 3 by 3 Determinants
Use Cramer’s Rule to Solve a System of Three Equations Containing Three Variables
Know Properties of Determinants

11.4 Matrix Algebra

Find the Sum and Difference of Two Matrices
Find Scalar Multiples of a Matrix
Find the Product of Two Matrices
Find the Inverse of a Matrix
Solve a System of Linear Equations Using an Inverse Matrix

11.5 Partial Fraction Decomposition

Decompose P/Q Where Q Has Only Nonrepeated Linear Factors
Decompose P/Q Where Q Has Repeated Linear Factors
Decompose P/Q Where Q Has a Nonrepeated Irreducible Quadratic Factor
Decompose P/Q Where Q Has a Repeated Irreducible Quadratic Factor

11.6 Systems of Nonlinear Equations

Solve a System of Nonlinear Equations Using Substitution
Solve a System of Nonlinear Equations Using Elimination

11.7 Systems of Inequalities

Graph an Inequality
Graph a System of Inequalities

11.8 Linear Programming

Set Up a Linear Programming Problem
Solve a Linear Programming Problem

12 Sequences; Induction; the Binomial Theorem

12.1 Sequences

List the First Several Terms of a Sequence
List the Terms of a Sequence Defined by a Recursive Formula
Use Summation Notation
Find the Sum of a Sequence

12.2 Arithmetic Sequences

Determine Whether a Sequence Is Arithmetic
Find a Formula for an Arithmetic Sequence
Find the Sum of an Arithmetic Sequence

12.3 Geometric Sequences; Geometric Series

Determine Whether a Sequence Is Geometric
Find a Formula for a Geometric Sequence
Find the Sum of a Geometric Sequence
Determine Whether a Geometric Series Converges or Diverges
Solve Annuity Problems

12.4 Mathematical Induction

Prove Statements Using Mathematical Induction

12.5 The Binomial Theorem

Evaluate (n choose r)
Use the Binomial Theorem

13 Counting and Probability

13.1 Counting

Find All the Subsets of a Set
Count the Number of Elements in a Set
Solve Counting Problems Using the Multiplication Principle

13.2 Permutations and Combinations

Solve Counting Problems Using Permutations Involving n Distinct Objects
Solve Counting Problems Using Combinations
Solve Counting Problems Using Permutations Involving n Nondistinct Objects

13.3 Probability

Construct Probability Models
Compute Probabilities of Equally Likely Outcomes
Find Probabilities of the Union of Two Events
Use the Complement Rule to Find Probabilities

14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function

14.1 Investigating Limits Using Tables and Graphs

Investigate a Limit Using a Table
Investigate a Limit Using a Graph

14.2 Algebraic Techniques for Finding Limits

Find the Limit of a Sum, a Difference, and a Product
Find the Limit of a Polynomial
Find the Limit of a Power or a Root
Find the Limit of a Quotient
Find the Limit of an Average Rate of Change

14.3 One-sided Limits; Continuity

Find the One-sided Limits of a Function
Determine Whether a Function Is Continuous at a Number

14.4 The Tangent Problem; The Derivative

Find an Equation of the Tangent Line to the Graph of a Function
Find the Derivative of a Function
Find Instantaneous Rates of Change
Find the Instantaneous Velocity of an Object

14.5 The Area Problem; The Integral

Approximate the Area under the Graph of a Function
Approximate Integrals Using a Graphing Utility