Thinkwell.College.Algebra.Course

Thinkwell.College.Algebra.Course




- Prerequisites

* 1.1 Introduction
o 1.1.1 An Introduction to Algebra
o 1.1.2 The Top Ten List of Mistakes

* 1.2 Inequalities
o 1.2.1 Concepts of Inequality
o 1.2.2 Inequalities and Interval Notation

* 1.3 Absolute Value
o 1.3.1 Properties of Absolute Value
o 1.3.2 Evaluating Absolute Value Expressions

* 1.4 Exponents
o 1.4.1 An Introduction to Exponents
o 1.4.2 Evaluating Exponential Expressions
o 1.4.3 Applying the Rules of Exponents
o 1.4.4 Evaluating Expressions with Negative Exponents

* 1.5 Converting between Notations
o 1.5.1 Converting between Decimal and Scientific Notation
o 1.5.2 Converting Rational Exponents and Radicals

* 1.6 Radical Expressions
o 1.6.1 Simplifying Radical Expressions
o 1.6.2 Simplifying Radical Expressions with Variables
o 1.6.3 Rationalizing Denominators

* 1.7 Polynomial Expressions
o 1.7.1 Determining Components and Degree
o 1.7.2 Adding, Subtracting, and Multiplying Polynomials
o 1.7.3 Multiplying Big Products
o 1.7.4 Using Special Products

* 1.8 Factoring
o 1.8.1 Factoring Using the Greatest Common Factor
o 1.8.2 Factoring by Grouping
o 1.8.3 Factoring Trinomials Completely

* 1.9 Factoring Patterns
o 1.9.1 Factoring Perfect Square Trinomials
o 1.9.2 Factoring the Difference of Two Squares
o 1.9.3 Factoring Sums and Differences of Cubes
o 1.9.4 Factoring by Any Method

* 1.10 Rational Expressions
o 1.10.1 Rational Expressions and Domain
o 1.10.2 Working with Fractions
o 1.10.3 Writing Rational Expressions in Lowest Terms

* 1.11 Working with Rationals
o 1.11.1 Multiplying and Dividing Rational Expressions
o 1.11.2 Adding and Subtracting Rational Expressions
o 1.11.3 Rewriting Complex Fractions

* 1.12 Complex Numbers
o 1.12.1 Introducing and Writing Complex Numbers
o 1.12.2 Rewriting Powers of i
o 1.12.3 Adding and Subtracting Complex Numbers
o 1.12.4 Multiplying Complex Numbers
o 1.12.5 Dividing Complex Numbers

- Equations and Inequalities

* 2.1 Linear Equations
o 2.1.1 An Introduction to Solving Equations
o 2.1.2 Solving a Linear Equation
o 2.1.3 Solving a Linear Equation with Rationals
o 2.1.4 Solving a Linear Equation That Has Restrictions

* 2.2 Word Problems with Linear Equations: Math Topics
o 2.2.1 An Introduction to Solving Word Problems
o 2.2.2 Solving for Perimeter
o 2.2.3 Solving a Linear Geometry Problem
o 2.2.4 Solving for Consecutive Numbers
o 2.2.5 Solving to Find the Average

* 2.3 Word Problems with Linear Equations: Applications
o 2.3.1 Solving for Constant Velocity
o 2.3.2 Solving a Problem about Work
o 2.3.3 Solving a Mixture Problem
o 2.3.4 Solving an Investment Problem
o 2.3.5 Solving Business Problems

* 2.4 Quadratic Equations: Some Solution Techniques
o 2.4.1 Solving Quadratics by Factoring
o 2.4.2 Solving Quadratics by Completing the Square
o 2.4.3 Completing the Square: Another Example

* 2.5 Quadratic Equations and the Quadratic Formula
o 2.5.1 Proving the Quadratic Formula
o 2.5.2 Using the Quadratic Formula
o 2.5.3 Predicting the Type of Solutions Using the Discriminant

* 2.6 Quadratic Equations: Special Topics
o 2.6.1 Solving for a Squared Variable
o 2.6.2 Finding Real Number Restrictions
o 2.6.3 Solving Fancy Quadratics

* 2.7 Word Problems with Quadratics: Math Topics
o 2.7.1 An Introduction to Word Problems with Quadratics
o 2.7.2 Solving a Quadratic Geometry Problem
o 2.7.3 Solving with the Pythagorean Theorem

* 2.8 Word Problems with Quadratics: Applications
o 2.8.1 Solving a Motion Problem
o 2.8.2 Solving a Projectile Problem
o 2.8.3 Solving Other Problems

* 2.9 Radical Equations
o 2.9.1 Determining Extraneous Roots
o 2.9.2 Solving an Equation Containing a Radical
o 2.9.3 Solving an Equation with Two Radicals
o 2.9.4 Solving an Equation with Rational Exponents

* 2.10 Variation
o 2.10.1 An Introduction to Variation
o 2.10.2 Direct Proportion
o 2.10.3 Inverse Proportion

* 2.11 Solving Inequalities
o 2.11.1 An Introduction to Solving Inequalities
o 2.11.2 Solving Compound Inequalities
o 2.11.3 More on Compound Inequalities
o 2.11.4 Solving Word Problems Involving Inequalities

* 2.12 Inequalities: Quadratics
o 2.12.1 Solving Quadratic Inequalities
o 2.12.2 Solving Quadratic Inequalities: Another Example

* 2.13 Inequalities: Rationals and Radicals
o 2.13.1 Solving Rational Inequalities
o 2.13.2 Solving Rational Inequalities: Another Example
o 2.13.3 Determining the Domains of Expressions with Radicals

* 2.14 Absolute Value
o 2.14.1 Matching Number Lines with Absolute Values
o 2.14.2 Solving Absolute Value Equations
o 2.14.3 Solving Equations with Two Absolute Value Expressions
o 2.14.4 Solving Absolute Value Inequalities
o 2.14.5 Solving Absolute Value Inequalities: More Examples

- Relations and Functions

* 3.1 Graphing Basics
o 3.1.1 Using the Cartesian System
o 3.1.2 Thinking Visually

* 3.2 Relationships between Two Points
o 3.2.1 Finding the Distance between Two Points
o 3.2.2 Finding the Second Endpoint of a Segment

* 3.3 Relationships among Three Points
o 3.3.1 Collinearity and Distance
o 3.3.2 Triangles

* 3.4 Circles
o 3.4.1 Finding the Center-Radius Form of the Equation of a Circle
o 3.4.2 Finding the Center and Radius of a Circle
o 3.4.3 Decoding the Circle Formula
o 3.4.4 Solving Word Problems Involving Circles

* 3.5 Graphing Equations
o 3.5.1 Graphing Equations by Locating Points
o 3.5.2 Finding the x- and y-Intercepts of an Equation

* 3.6 Function Basics
o 3.6.1 Functions and the Vertical Line Test
o 3.6.2 Identifying Functions
o 3.6.3 Function Notation and Finding Function Values

* 3.7 Working with Functions
o 3.7.1 Determining Intervals Over Which a Function Is Increasing
o 3.7.2 Evaluating Piecewise-Defined Functions for Given Values
o 3.7.3 Solving Word Problems Involving Functions

* 3.8 Function Domain and Range
o 3.8.1 Finding the Domain and Range of a Function
o 3.8.2 Domain and Range: One Explicit Example
o 3.8.3 Satisfying the Domain of a Function

* 3.9 Linear Functions: Slope
o 3.9.1 An Introduction to Slope
o 3.9.2 Finding the Slope of a Line Given Two Points
o 3.9.3 Interpreting Slope from a Graph
o 3.9.4 Graphing a Line Using Point and Slope

* 3.10 Equations of a Line
o 3.10.1 Writing an Equation in Slope-Intercept Form
o 3.10.2 Writing an Equation Given Two Points
o 3.10.3 Writing an Equation in Point-Slope Form
o 3.10.4 Matching a Slope-Intercept Equation with Its Graph
o 3.10.5 Slope with Parallel and Perpendicular Lines

* 3.11 Linear Functions: Applications
o 3.11.1 Constructing Linear Function Models of a Set of Data
o 3.11.2 Linear Cost and Revenue Functions

* 3.12 Graphing Functions
o 3.12.1 Graphing Some Important Functions
o 3.12.2 Graphing Piecewise-Defined Functions
o 3.12.3 Matching Equations with Their Graphs

* 3.13 The Greatest Integer Function
o 3.13.1 The Greatest Integer Function
o 3.13.2 Graphing the Greatest Integer Function

* 3.14 Composite Functions
o 3.14.1 Using Operations on Functions
o 3.14.2 Composite Functions
o 3.14.3 Components of Composite Functions
o 3.14.4 Finding Functions That Form a Given Composite
o 3.14.5 Finding the Difference Quotient of a Function

* 3.15 Quadratic Functions: Basics
o 3.15.1 Deconstructing the Graph of a Quadratic Function
o 3.15.2 Nice-Looking Parabolas
o 3.15.3 Using Discriminants to Graph Parabolas
o 3.15.4 Maximum Height in the Real World

* 3.16 Quadratic Functions: The Vertex
o 3.16.1 Finding the Vertex by Completing the Square
o 3.16.2 Using the Vertex to Write the Quadratic Equation
o 3.16.3 Finding the Maximum or Minimum of a Quadratic
o 3.16.4 Graphing Parabolas

* 3.17 Manipulating Graphs: Shifts and Stretches
o 3.17.1 Shifting Curves along Axes
o 3.17.2 Shifting or Translating Curves along Axes
o 3.17.3 Stretching a Graph
o 3.17.4 Graphing Quadratics Using Patterns

* 3.18 Manipulating Graphs: Symmetry and Reflections
o 3.18.1 Determining Symmetry
o 3.18.2 Reflections
o 3.18.3 Reflecting Specific Functions

- Polynomial and Rational Functions

* 4.1 Polynomials: Long Division
o 4.1.1 Using Long Division with Polynomials
o 4.1.2 Long Division: Another Example

* 4.2 Polynomials: Synthetic Division
o 4.2.1 Using Synthetic Division with Polynomials
o 4.2.2 More Synthetic Division

* 4.3 The Remainder Theorem
o 4.3.1 The Remainder Theorem
o 4.3.2 More on the Remainder Theorem

* 4.4 The Factor Theorem
o 4.4.1 The Factor Theorem and Its Uses
o 4.4.2 Factoring a Polynomial Given a Zero

* 4.5 The Rational Root Theorem
o 4.5.1 Presenting the Rational Zero Theorem
o 4.5.2 Considering Possible Solutions

* 4.6 Zeros of Polynomials
o 4.6.1 Finding Polynomials Given Zeros, Degree, and One Point
o 4.6.2 Finding all Zeros and Multiplicities of a Polynomial
o 4.6.3 Finding the Real Zeros for a Polynomial
o 4.6.4 Using Descartes' Rule of Signs
o 4.6.5 Finding the Zeros of a Polynomial from Start to Finish

* 4.7 Graphing Polynomials
o 4.7.1 Matching Graphs to Polynomial Functions
o 4.7.2 Sketching the Graphs of Basic Polynomial Functions

* 4.8 Rational Functions
o 4.8.1 Understanding Rational Functions
o 4.8.2 Basic Rational Functions

* 4.9 Graphing Rational Functions
o 4.9.1 Vertical Asymptotes
o 4.9.2 Horizontal Asymptotes
o 4.9.3 Graphing Rational Functions
o 4.9.4 Graphing Rational Functions: More Examples

- Exponential and Logarithmic Functions

* 5.1 Function Inverses
o 5.1.1 Understanding Inverse Functions
o 5.1.2 The Horizontal Line Test
o 5.1.3 Are Two Functions Inverses of Each Other?
o 5.1.4 Graphing the Inverse

* 5.2 Finding Function Inverses
o 5.2.1 Finding the Inverse of a Function
o 5.2.2 Finding the Inverse of a Function with Higher Powers

* 5.3 Exponential Functions
o 5.3.1 An Introduction to Exponential Functions
o 5.3.2 Graphing Exponential Functions: Useful Patterns
o 5.3.3 Graphing Exponential Functions: More Examples

* 5.4 Applying Exponential Functions
o 5.4.1 Using Properties of Exponents to Solve Exponential Equations
o 5.4.2 Finding Present Value and Future Value
o 5.4.3 Finding an Interest Rate to Match Given Goals

* 5.5 The Number e
o 5.5.1 e
o 5.5.2 Applying Exponential Functions

* 5.6 Logarithmic Functions
o 5.6.1 An Introduction to Logarithmic Functions
o 5.6.2 Converting between Exponential and Logarithmic Functions

* 5.7 Solving Logarithmic Functions
o 5.7.1 Finding the Value of a Logarithmic Function
o 5.7.2 Solving for x in Logarithmic Equations
o 5.7.3 Graphing Logarithmic Functions
o 5.7.4 Matching Logarithmic Functions with Their Graphs

* 5.8 Properties of Logarithms
o 5.8.1 Properties of Logarithms
o 5.8.2 Expanding a Logarithmic Expression Using Properties
o 5.8.3 Combining Logarithmic Expressions

* 5.9 Evaluating Logarithmic Functions
o 5.9.1 Evaluating Logarithmic Functions Using a Calculator
o 5.9.2 Using the Change of Base Formula

* 5.10 Applying Logarithmic Functions
o 5.10.1 The Richter Scale
o 5.10.2 The Distance Modulus Formula

* 5.11 Solving Exponential and Logarithmic Equations
o 5.11.1 Solving Exponential Equations
o 5.11.2 Solving Logarithmic Equations
o 5.11.3 Solving Equations with Logarithmic Exponents

* 5.12 Applying Exponents and Logarithms
o 5.12.1 Compound Interest
o 5.12.2 Predicting Change

* 5.13 Word Problems Involving Exponential Growth and Decay
o 5.13.1 An Introduction to Exponential Growth and Decay
o 5.13.2 Half-Life
o 5.13.3 Newton's Law of Cooling
o 5.13.4 Continuously Compounded Interest

- Systems of Equations

* 6.1 Linear Systems of Equations
o 6.1.1 An Introduction to Linear Systems
o 6.1.2 Solving Systems with Substitution
o 6.1.3 Solving Systems by Elimination

* 6.2 Linear Systems in Three Variables
o 6.2.1 An Introduction to Linear Systems in Three Variables
o 6.2.2 Solving Linear Systems in Three Variables
o 6.2.3 Solving Inconsistent Systems
o 6.2.4 Solving Dependent Systems
o 6.2.5 Solving Systems with Two Equations

* 6.3 Applying Linear Systems
o 6.3.1 Investments
o 6.3.2 Solving with Partial Fractions

* 6.4 Nonlinear Systems of Equations
o 6.4.1 Solving Nonlinear Systems Using Elimination
o 6.4.2 Solving Nonlinear Systems with Substitution

* 6.5 Matrices
o 6.5.1 An Introduction to Matrices
o 6.5.2 The Arithmetic of Matrices
o 6.5.3 Multiplying Matrices by a Scalar
o 6.5.4 Multiplying Matrices
o 6.5.5 Multiplying Matrices: Can They Multiply?

* 6.6 The Gauss-Jordan Method of Solving Matrices
o 6.6.1 Using the Gauss-Jordan Method
o 6.6.2 Using Gauss-Jordan: Another Example

* 6.7 Evaluating Determinants
o 6.7.1 Evaluating 2x2 Determinants
o 6.7.2 Evaluating nxn Determinants
o 6.7.3 Applying Determinants

* 6.8 Cramer's Rule
o 6.8.1 Using Cramer's Rule
o 6.8.2 Using Cramer's Rule in a 3x3 Matrix

* 6.9 Inverses and Matrices
o 6.9.1 An Introduction to Inverses
o 6.9.2 Inverses: 2x2 Matrices
o 6.9.3 Another Look at 2x2 Inverses
o 6.9.4 Inverses: 3x3 Matrices
o 6.9.5 Solving a System of Equations with Inverses

* 6.10 Working with Inequalities
o 6.10.1 An Introduction to Graphing Linear Inequalities
o 6.10.2 Graphing Linear and Nonlinear Inequalities
o 6.10.3 Graphing the Solution Set of a System of Inequalities

* 6.11 Linear Programming
o 6.11.1 Solving for Maxima-Minima
o 6.11.2 Applying Linear Programming

- Conic Sections

* 7.1 Parabolas
o 7.1.1 An Introduction to Conic Sections
o 7.1.2 An Introduction to Parabolas
o 7.1.3 Determining Information about a Parabola from Its Equation
o 7.1.4 Writing an Equation for a Parabola

* 7.2 Ellipses
o 7.2.1 An Introduction to Ellipses
o 7.2.2 Finding the Equation for an Ellipse
o 7.2.3 Applying Ellipses: Satellites

* 7.3 Hyperbolas
o 7.3.1 An Introduction to Hyperbolas
o 7.3.2 Finding the Equation for a Hyperbola
o 7.3.3 Applying Hyperbolas: Navigation

* 7.4 Conic Sections
o 7.4.1 Identifying a Conic
o 7.4.2 Name That Conic

- Further Topics in Algebra

* 8.1 The Binomial Theorem
o 8.1.1 Using the Binomial Theorem
o 8.1.2 Binomial Coefficients

* 8.2 Sequences
o 8.2.1 Understanding Sequence Problems
o 8.2.2 Solving Problems Involving Arithmetic Sequences
o 8.2.3 Solving Problems Involving Geometric Sequences

* 8.3 Induction
o 8.3.1 Proving Formulas Using Mathematical Induction
o 8.3.2 Examples of Induction

* 8.4 Combinations and Probability
o 8.4.1 Solving Problems Involving Permutations
o 8.4.2 Solving Problems Involving Combinations
o 8.4.3 Independent Events
o 8.4.4 Inclusive and Exclusive Events

- Conclusion

* 9.1 Conclusion
o 9.1.1 Final Close