Pre-Calculus Scope and Sequence

Tutor: Kim Nelson

Text: PreCalculus, Larson/Hostetler, 6^th Edition

Chapter 1: Functions and Their Graphs

Sec. 1.1 Graphs of Equations

Graphing using tables

Intercepts

Symmetry

Circles

Application

Sec. 1.2 Linear Equations in 2 Variables

Using Slope

Finding the Slope

Writing Linear Equations

Parallel & Perpendicular lines

Application

Sec. 1.3 Functions

Introductions to Functions

Function Notation

Domain of a Function

Applications

Sec. 1.4 Analyzing Graphs of Functions

Graph of a Function

Zeros of a Function

Increasing and Decreasing Functions

Even and Odd Functions

Sec. 1.5 A Library of Functions

Linear and Squaring Functions

Cubic, Square Root, and Reciprocal Functions

Step and Piecewise-defined Functions

Common Functions

Sec. 1.6 Shifting, Reflecting, and Stretching Graphs

Shifting Graphs

Reflecting Graphs

Nonrigid Transformations

Sec. 1.7 Combinations of Functions

Artihmetic Combinations of Functions

Composition of Functions

Application

Sec. 1.8 Inverse Functions

Inverse Functions

Graph of an Inverse Function

One-to-One Functions

Finding Inverse Functions Algebraically

Sec. 1.9 Mathematical Modeling

Introduction

Direct Variation

Direct Variationas an Nth Power

Inverse Variation

Joint Variation

CHAPTER 1 REVIEW

Chapter 2: Polynomial and Rational Functions

Sec. 2.1 Quadratic Functions

Graph of a Quadratic Function

Standard Form of a Quadratic Function

Application

Sec. 2.2 Polynomial Functions of Higher Degree

Graphs of Polynomial Functions

Leading Coefficient Test

Zeros of Polynomial Functions

Intermediate Value Theorem

Section 2.3 Polynomial and Synthetic Division

Long Division of Polynomials

Synthetic Division

The Remainder and Factor Theorems

Section 2.4 Complex Numbers

The Imaginary Unit i

Operations with Complex Numbers

Complex Conjugates

Complex Solutions of Quadratic Equations

Sec. 2.5 Zeros of Polynomial Functions

The Fundamental Theorem of Algebra

The Rational Zero Test

Conjugate Pairs

Factoring a Polynomial

Other Tests of Zeros of Polynomials

Sec. 2.6 Rational Functions

Horizontal and Vertical Asymptotes

Analyzing Graphs of Rational Functions

Slant Asymptotes

Applications

CHAPTER 2 REVIEW

Chapter 3: Exponential and Logarithmic Functions

Sec. 3.1 Exponential Functionsand Their Graphs

Exponential Functions

Graphs of Exponential Functions

Natural Base e

Applications

Sec. 3.2 Logarithmic Functions and Their Graphs

Logarithmic Functions

Graphs of Logarithmic Functions

The Natural Logarithmic Function

Application

Sec. 3.3 Properties of Logarithms

Change of Base

Properties of Logarithms

Rewriting Logarithmic Expressions

Application

Sec. 3.4 Exponential and Logarithmic Equations

Introduction

Solving Exponential Equations

Solving Logarithmic Equations

Application

CHAPTER 3 REVIEW

Chapter 4: Trigonometry

Sec. 4.1 Radian and Degree Measure

Angles

Radian Measure

Degree Measure

Applications

Trigonometric Functions: The Unit Circle

The Unit Circle

Trigonometric Functions

Domain and Period of Sine and Cosine

Evaluating Trigonometric Functions With a Calculator

Sec. 4.3 Right Triangle Trigonometry

The Six Trigonometric Functions

Trigonometric Identities

Evaluating Trigonometric Functions with a Calculator

Applications Involving Right Triangles

Sec. 4.4 Trigonometric Functions of Any Angle

Introduction

Reference Angles

Trigonometric Functions of Real Numbers

Sec. 4.5 Graphs of Sine and Cosine Functions

Basic Sine and Cosine Curves

Amplitude and Period

Translation of Sine and Cosine Curves

Mathematical Modeling

Sec. 4.6 Graphs of other Trig Functions

Graph of the Tangent Function

Graph of the Cotangent Function

Graphs of Reciprocal Functions

Damped Trigonometric Graphs

Sec. 4.7 Inverse Trigonometric Functions

Inverse Sine Function

Other Inverse Trigonometric Functions

Composition of Functions

Sec. 4.8 Applications and Models

Applications Involving Right Triangles

Trigonometry and Bearings

Harmonic Motion

CHAPTER 4 REVIEW

Chapter 5: Analytic Trigonometry

Sec. 5.1 Using Fundamental Identities

Introduction

Using the Fundamental Identities

Sec. 5.2 Verifying Trig Identities

Introduction

Verifying Trigonometric Identities

Sec. 5.3 Solving Trig Equations

Introduction

Equations of Quadratic Type

Functions Involving Multiple Angles

Using Inverse Functions

Sec. 5.4 Sum and Difference Formulas

Using sum and Difference Formulas

Sec. 5.5 Multiple-Angle and Product-to-Sum Formulas

Multiple-Angle Formulas

Power-Reducing formulas

Half-Angle Formulas

CHAPTER 5 REVIEW

Chapter 6: Additional Topics in Trigonometry

Sec. 6.1 Law of Sines

Introduction

The Ambiguous Case (SSA)

Area of an Oblique Triangle

Application

Sec. 6.2 Law of Cosines

Introduction

Applications

Heron's Area Formula

Sec. 6.3 Vectors in the plane

Introduction

Component Form of a Vector

Vector Operations

Sec. 6.4 Vectors and Dot Products

The Dot Product of Two Vectors

The Angle Between Two Vectors

CHAPTER 6 REVIEW

Chapter 7: Systems of Equations and Inequalities

Sec. 7.1 Solving Systems of Equations

The Method of Substitution

Graphical Approach to Finding Solutions

Applications

Sec. 7.2 Two-Variable Linear Systems

The Method of Elimination

Graphical Interpretation of Solutions

Applications

Sec. 7.3 Multivariable Linear Systems

Row-Echelon Form and Back-Substitution

Gaussian Elimination

Nonsquare Systems

Applications

Sec. 7.4 Systems of Inequalities

The Graph of an Inequality

Systems of Inequalities

Applications

Sec. 7.5 Linear Programming

Linear Programming: A Graphical Approach

Applications

CHAPTER 7 REVIEW

Chapter 8: Matrices and Determinants

Sec. 8.1 Matrices and Systems of Equations

Matrices

Elementary Row Operations

Gaussian Elimination with Back-Substitution

Gauss-Jordon Elimination

Sec. 8.2 Operations with Matrices

Equality of Matrices

Matrix Addition and Scalar Multiplication

Matrix Multiplication

Applications

Sec. 8.3 The Inverse of a Square Matrix

The Inverse of a Matrix

Finding Inverse Matrices

The Inverse of a 2 x 2 Matrix

Systems of Linear Equations

Sec. 8.4 The Determinant of a Square Matrix

The Determinant of a 2 x 2 Matrix

Minors and Cofactors

The Determinant of a Square Matrix

Sec. 8.5 Applications of Matrices and Determinants

Cramer’s Rule

Area of a Triangle

Lines in a Plane

Cryptography

CHAPTER 8 REVIEW

Chapter 9: Sequences, Series, and Probability

Sec. 9.1 Sequences and Series

Sequences

Factorial Notation

Summation Notation

Application

Sec. 9.2 Arithmetic Sequences and Partial Sums

Arithmetic Sequences

The Sum of a Finite Arithmetic Sequence

Applications

Sec. 9.3 Geometric Sequences and Series

Geometric Sequences

The Sum of a Finite Geometric Sequence

Geometric Series

Application

Sec. 9.5 The Binomial Theorem

Binomial Coefficients

Pascal’s Triangle

Binomial Expansions

Sec. 9.6 Counting Principles

Simple Counting Problems

The Fundamental Counting Principle

Permutations

Combinations

Sec. 9.7 Probability

The Probability of an Event

Mutually Exclusive Events

Independent Events

The Complement of an Event

CHAPTER 9 REVIEW

Chapter 10: Topics in Analytic Geometry

Sec. 10.1 Lines

Inclination of a Line

The Angle Between Two Lines

The Distance Between a Point and a Line

Sec. 10.2 Introduction to Conics: Parabolas

Conics

Parabolas

Application

Sec. 10.3 Ellipses

Introduction

Application

Eccentricity

Sec. 10.4 Hyperbolas

Introduction

Asymptotes of a Hyperbola

Applications

General Equations of Conics

Sec. 10.6 Parametric Equations

Plane Curve

Sketching a Plane Curve

Eliminating a Parameter

Finding Parametric Equations for Graph

Sec. 10.7 Polar Coordinates

Introduction

Coordinate Conversion

Equation Conversion

Sec. 10.8 Graphs of Polar Equations

Introduction

Symmetry

Zeros and Maximum r-Values

Sketch Polar Graphs