College Algebra Crash Course

Before You Start College Algebra Video Lesson(s)

Course Outline is Based off of the Michael Sullivan “College Algebra” Textbooks


1. Equations and Inequalities

1.1 Linear Equations

1.2 Quadratic Equations

1.3 Complex Numbers; Quadratic Equations in the Complex Number System

1.4 Radical Equations; Equations Quadratic in Form; Factorable Equations

  • Solve Radical Equations
  • Solve Equations Quadratic in Form
  • Solve Equations by Factoring

1.5 Solving Inequalities

  • Use Interval Notation
  • Use Properties of Inequalities
  • Solve Inequalities
  • Solve Combined Inequalities

1.6 Equations and Inequalities Involving Absolute Value

  • Solve Equations Involving Absolute Value
  • Solve Inequalities Involving Absolute Value

1.7 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications

  • Translate Verbal Descriptions into Mathematical Expressions
  • Solve Interest Problems
  • Solve Mixture Problems
  • Solve Uniform Motion Problems
  • Solve Constant Rate Job Problems

2. Graphs

2.1 The Distance and Midpoint Formulas

  • Use the Distance Formula
  • Use the Midpoint Formula

2.2 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, y-Axis, and the Origin
  • Know How to Graph Key Equations

2.3 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 an 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

2.4 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

2.5 Variation

  • Construct a Model Using Direct Variation
  • Construct a Model Using Inverse Variation
  • Construct a Model Using Joint Variation

3. Functions and Their Graphs

3.1 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

3.2 The Graph of a Function

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

3.3 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
  • Determine Where a Function Is Increasing or Decreasing
  • Find the Average Rate of Change of a Function

3.4 Library of Functions; Piecewise-defined Functions

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

3.5 Graphing Techniques: Transformations (Page 254)

  • 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

3.6 Mathematical Models: Building Functions

  • Build and Analyze Functions

4. Linear and Quadratic Functions

4.1 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

4.2 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

4.3 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

4.4 Building Quadratic Models from Verbal Descriptions and from Data

  • Build Quadratic Models from Verbal Descriptions
  • Build Quadratic Models from Data

4.5 Inequalities Involving Quadratic Functions

  • Solve Inequalities Involving a Quadratic Function

5. Polynomial and Rational Functions

5.1 Polynomial Functions

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

5.2 Graphing Polynomial Functions; Models

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

5.3 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

5.4 The Graph of a Rational Function

  • Graph a Rational Function
  • Solve Applied Problems Involving Rational Functions

5.5 Polynomial and Rational Inequalities

  • Solve Polynomial Inequalities
  • Solve Rational Inequalities

5.6 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 Negative Real Zeros
  • Use the Rational Zeros Theorem to List the Potential Rational Zeros
  • Find the Real Zeros of a Polynomial Function
  • Solve Polynomial Equations
  • Use the Theorem for Bounds on Zeros
  • Use the Intermediate Value Theorem

5.7 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

6. Exponential and Logarithmic Functions

6.1 Composite Functions

  • Form a Composite Function
  • Find the Domain of a Composite Function

6.2 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

6.3 Exponential Functions

  • Evaluate Exponential Functions
  • Graph Exponential Functions
  • Define the Number ee
  • Solve Exponential Equations

6.4 Logarithmic Functions

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

6.5 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 ee

6.6 Logarithmic and Exponential Equations

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

6.7 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

6.8 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

6.9 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

7. Analytic Geometry

7.1 Conics

  • Know the Names of the Conics

7.2 The Parabola

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

7.3 The Ellipse

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

7.4 The Hyperbola

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

8. Systems of Equations and Inequalities

8.1 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

8.2 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

8.3 Systems of Linear Equations: Determinants

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

8.4 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

8.5 Partial Fraction Decomposition

  • Decompose PQ\frac{P}{Q} Where QQ Has Only Nonrepeated Linear Factors
  • Decompose PQ\frac{P}{Q} Where QQ Has Repeated Linear Factors
  • Decompose PQ\frac{P}{Q} Where QQ Has a Nonrepeated Irreducible Quadratic Factor
  • Decompose PQ\frac{P}{Q} Where QQ Has a Repeated Irreducible Quadratic Factor

8.6 Systems of Nonlinear Equations

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

8.7 Systems of Inequalities

  • Graph an Inequality
  • Graph a System of Inequalities

8.8 Linear Programming

  • Set Up a Linear Programming Problem
  • Solve a Linear Programming Problem

9. Sequences; Induction; the Binomial Theorem

9.1 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

9.2 Arithmetic Sequences

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

9.3 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

9.4 Mathematical Induction

  • Prove Statements Using Mathematical Induction

9.5 The Binomial Theorem

  • Evaluate (nj)\binom{n}{j}
  • Use the Binomial Theorem

10. Counting and Probability

10.1 Counting

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

10.2 Permutations and Combinations

  • Solve Counting Problems Using Permutations Involving nn Distinct Objects
  • Solve Counting Problems Using Combinations
  • Solve Counting Problems Using Permutations Involving nn Nondistinct Objects

10.3 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

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