College Algebra
Functions, transformations, and matrices.

111
Video
Lessons 
124
Embedded Video
Questions 
753
Practice
Problems
Table of Contents
The Sets of Numbers
 Know the definitions of the sets of numbers
 Determine which set or sets a number belongs to
Graphing Real Numbers
 Graph a fraction or mixed number on a number line
 Graph a decimal number on a number line
 Graph an irrational number on a number line
Order of Operations
 Know the order in which mathematical operations are followed
 Use the order of operations in the context of simplifying expressions
Properties of Real Numbers
 Know the closure properties of the real numbers
 Know the commutative, associative, and distributive properties
 Know what additive and multiplicative identity elements and inverses are
 Know the multiplication and division properties of zero
Using Formulas to Find Area
 Find the area of common shapes using formulas
Using Formulas to Find Volume
 Find the volume of geometric solids using formulas
Arithmetic Properties of Exponents
 Know the rules for exponents
 Manipulate and simplify expressions with positive exponents
Negative and Fractional Exponents
 Know the rules dictating how to deal with fractional and negative exponents
 Manipulate expressions with fractional and negative exponents
Scientific Notation
 Express numbers in scientific notation
 Convert numbers from scientific notation to decimal form
 Perform calculations with numbers expressed in scientific notation
Roots
 Identify the parts of a square root
 Find nth roots of whole numbers
Simplifying Radical Expressions
 Simplify radical expressions that contain fractions
 Simplify radical expressions with variables
 Simplify nth root expressions with variables
Adding and Subtracting Radicals
 Add and subtract like radical expressions
 Simplify radicals to be like radicals to add or subtract
Multiplying Radical Expressions
 Multiply radical expressions
Dividing Radical Expressions
 Divide radical expressions
 Rationalize a monomial or binomial denominator
Polynomial Vocabulary
 Determine whether an expression is a polynomial
 Classify a polynomial as a monomial, binomial, or trinomial
 Find the degree of a polynomial
Adding and Subtracting Polynomials
 Know the definition of "like terms"
 Identify like terms in an expression
 Combine like terms correctly
 Be able to add and subtract polynomial expressions by identifying like terms
Multiplying Polynomials
 Know the distributive property of multiplication
 Use the distributive property to multiply polynomials with polynomials
 Be able to multiply polynomials of any length
Factoring Trinomials with Lead Coefficient 1
 Factor trinomials with a leading coefficient of 1 into two binomials
Factoring Trinomials with Lead Coefficient a
 Factor trinomials with a leading coefficient of a into two binomials
Factoring Perfect Square Trinomials
 Identify a trinomial as being a perfect square
 Factor a perfect square into two binomials
Factoring Differences of Squares
 Identify a difference of two squares
 Factor a difference of two squares into two binomials
Factoring Sums and Differences of Cubes
 Identify a difference or a sum of two cubes
 Factor a difference or a sum of two cubes using the formulas
Factoring with Negative and Fractional Exponents
 Factor expressions involving fractional and negative exponents
Simplifying Rational Expressions
 Use factoring to simplify rational expressions
Multiplication and Division of Rational Expressions
 Simplify rational expressions with products and quotients
Addition and Subtraction of Rational Expressions
 Simplify rational expressions with sums and differences
Composition of Rational Expressions
 Simplify more complicated rational expressions
The Cartesian Plane
 Plot a point on the Cartesian plane if given its coordinates
 Find the coordinates of a point on the Cartesian plane
Tables of Values for Linear Equations
 Construct a table of values given a formula
 Graph a linear equation by constructing a table of values
Characteristics of Graphs
 Identify xintercepts and yintercepts of graphs
 Solve to find x and yintercepts
 Find zeros of a function
The Distance Formula
 Find the distance between two points
 Find the distance between two locations
The Midpoint Formula
 Find the midpoint between two points
Solving Equations
 Know when to use the properties of equality
 Know how to use the distributive property
 Employ the properties of equality
 Employ the distributive property
 Solve equations involving properties together
Solving Equations with Rational Expressions
 Use the techniques from the previous lectures to solve equations involving rational expressions
The Slope of a Line
 Know what the slope of a line is
 Find the slope of a line if given two points
 Find the slope of a line from a graph
 Graph a line that goes through a point and has a given slope
SlopeIntercept Form of a Line
 Know the slopeintercept form of a line
 Find the equation of a line in slope intercept form if given the graph
 Find the equation of a line in slopeintercept form if given two points
 Graph a line if its equation is in slopeintercept form
PointSlope Form of a Line
 Know the pointslope form of a line
 Find the equation of a line in pointslope form if given a graph
 Find the equation of a line in pointslope form if given two points
 Graph a line if its equation is in pointslope form
Standard Form of a Line
 Know the standard form of a line
 Graph a line if its equation is in standard form
Parallel and Perpendicular Lines
 Know the definition of parallel lines
 Know the definition of perpendicular lines
 Determine if two lines are parallel from their equations
 Determine if two lines are perpendicular from their equations
Percent Increase and Decrease
 Set up and solve percent increase problems
 Set up and solve percent decrease problems
Markup and Discount
 Formulate markup and discount problems as percent increase and decrease problems
 Solve markup and discount problems
Simple Interest
 Use the simple interest formula to find simple interest due
 Find the maturity value of a simple interest loan
Linear Equations Application: Ladder Problem
 Solving Equations
 Adding and Subtracting Polynomials
Using factoring to Solve Equations
 Know the zero product property
 Understand the zero product property
 Use the zero product property in association with factoring to find solutions to equations
Completing the Square
 Know the completing the square technique
 Apply completing the square to find all solutions to a quadratic equation
The Quadratic Formula
 Know the quadratic formula
 Use the quadratic formula to find solutions to quadratic equations
The Discriminant of a Quadratic
 Use the discriminant to determine the number and kind of solutions to a quadratic equation
The Number i
 Define i
 Evaluate powers of i
Add, Subtract, and Multiply Complex Numbers
 Add complex numbers
 Subtract complex numbers
 Multiply complex numbers
Dividing Complex Numbers
 Identify complex conjugates
 Divide complex numbers by multiplying by the conjugate
The Complex Plane
 Plot complex numbers in the complex plane
 Interpret points in the complex plane as complex numbers
Solving Equations with Absolute Value
 Know the definition of absolute value
 Find the absolute value of any integer
 Solve equations involving absolute value
Solving Radical Equations 1
 Solve square roots by squaring both sides
 Check for extraneous solutions
Solving Radical Equations 2
 Solve radical equations which result in quadratic equations
Solving Equations that are Quadratic in Form 1
 Identify equations that are quadratic in form
 Solve equations that are quadratic in form using substitution
Solving Equations that are Quadratic in Form 2
 Solve equations that are quadratic in form without using substitution
Introduction to Inequalities
 Know the meaning of a solution set
 Represent the solution set of an inequality on a number line
Interval Notation
 Know the definition of interval notation
 Use interval notation to rewrite an inequality
Manipulating Inequalities
 Manipulate inequalities to isolate a variable and represent the solution set on a number line
Intersection and Union
 Know the definition of intersection and union
 Find the intersection of two sets
 Find the union of two sets
AND Inequalities
 Know the word "and" should be viewed as intersection
 Simplify and graph compound inequalities with "and"
OR Inequalities
 Know the word "or" should be viewed as union
 Simplify and graph compound inequalities with "or"
Inequalities and Absolute Value
 Translate an absolute value inequality into a compound inequality in order to find the solution set
What is a Function?
 Know the definition of a function
 Describe a function using a formula
 Describe a function using a table
 Recognize if a given relation is a function or not
Evaluating Functions
 Evaluate a function on numerical input
 Evaluate a function on variable input
Domain of a Function
 Know the definition of the domain of a function
 Find the domain of a function using a graph
 Find the domain of a function if given a formula
Range of a Function
 Know the definition of the range of a function
 Find the range of a function using a graph
 Find the range of a function if given a formula
OnetoOne Functions
 Know the definition of a onetoone function
 Know how to use the Horizontal Line Test for onetoone functions
 Determine if a function is onetoone using a graph
Piecewise Functions
 Know the definition of a piecewise function
 Know an example of a piecewise function
 Evaluate a piecewise function on a numerical value
 Graph a piecewise function
Arithmetic on Functions
 Add, subtract, divide, and multiply functions and find their domains
 Find the graph of f+g using the graphs of f and g
Composition of Functions
 Know the definition of the composition of two functions
 Find the composition of two functions
 Find the domain of the composition of two functions
 Evaluate the composition of two functions if given their graphs
Inverse Functions
 Know the definition of an inverse function
 Determine if a function has an inverse using a table
 Determine if a function has an inverse using its graph
Finding the Inverse of a Function
 Find a formula for the inverse of a function
 Find the inverse of a function using its graph
Vertical Shifts
 Know how f(x)+k changes the graph for k>0
 Know how f(x)+k changes the graph for k<0
 Graph a function if given the equation f(x)+k
 Find the equation of a graph shifted vertically by k units
Horizontal Shifts
 Know how f(xh) changes the graph for h>0
 Know how f(xh) changes the graph for h<0
 Graph a function if given the equation f(xh)
 Find the equation of a graph shifted horizontally by h units
Vertical Scaling
 Know how a ⋅ f(x) changes the graph of f(x) for 0<a<1
 Know how a ⋅ f(x) changes the graph of f(x) for a>1
 Graph the function a ⋅ f(x) if you know how to graph f(x)
 Find the formula for a function if given the graph scaled vertically by a units
Horizontal Scaling
 Know how f(ax) changes the graph of f(x) for 0<a<1
 Know how f(ax) changes the graph of f(x) for a>1
 Know how f(ax) changes the graph of f(x) for a<1
 Graph the function f(ax) if you know how to graph f(x)
 Find the formula for a function if given the graph scaled horizontally by a units
Average Rate of Change
 Find the average rate of change from a graph
 Find the average rate of change from a formula
 Find the average rate of change from a situation
Increasing and Decreasing Functions
 Use a graph to determine when a function is increasing, decreasing, or constant
Extreme Values
 Use a graph to find the local and absolute extrema of a function
Absolute Value Functions
 Find the domain and range of absolute value functions
 Graph absolute value functions
Multiplicity of Roots
 Know the definition of multiplicity of a root
 Determine the multiplicity of a root
 Understand how multiplicity of roots affects the shape of the graph of a polynomial
Graphing Polynomials
 Know how to graph power functions and monomials
 Know how to graph linear, quadratic, cubic, and quartic functions
 Use arrow notation
 Understand how the degree of a polynomial affects the shape of the graph
The Intermediate Value Theorem
 Use the intermediate value theorem to find intervals that contain roots
Polynomial Long Division
 Divide polynomials using long division
Synthetic Division
 Perform synthetic division
 Interpret the results of synthetic division
The Remainder and Factor Theorems
 Understand the remainder theorem and the factor theorem
 Use the remainder theorem and synthetic division to evaluate a polynomial
 Use the factor theorem to solve a polynomial equation
The Rational Root Theorem 1
 Understand when to use the rational root theorem
 Use the rational root theorem to find rational roots of polynomials
The Rational Root Theorem 2
 Use the rational root theorem to find rational roots of polynomials
Descartes' Rule of Signs 1
 Use Descartes' rule of signs to count the positive and negative roots of a polynomial
Descartes' Rule of Signs 2
 Use Descartes' rule of signs and the rational root theorem together to find rational roots of polynomials
Complex Roots and The Fundamental Theorem of Algebra 1
 Know how the complex roots of a polynomial with real coefficients are related
 Know the fundamental theorem of algebra
Complex Roots and The Fundamental Theorem of Algebra 2
 Use the fundamental theorem of algebra and other theorems to find all the roots of a polynomial
Vertical Asymptotes of Rational Functions
 Find the vertical asymptotes of a rational function
 Find the removable discontinuities of a rational function
Horizontal and Slant Asymptotes of Rational Functions
 Find horizontal asymptotes of rational functions
 Find slant asymptotes of rational functions
 Determine the end behavior of a rational function
Introduction to Exponential Functions
 Know the definition of an exponential function
 Know properties of graphs of exponential functions
 Use the onetoone property to solve exponential equations
Introduction to Logarithms
 Know the definition of a logarithm
 Know how to sketch a graph of a logarithm
 Compute basic logarithms
 Find the domain and range of a logarithmic function
Inverse Properties of Logarithms
 Know the inverse properties of logarithms
 Solve various equations using the inverse properties of logarithms
Properties of Logarithms
 Know the properties of logarithms
 Know the change of base formula
 Condense and expand expressions using the properties of logarithms
 Use the change of base formula to relate logarithms with different bases
More Solving of Logarithmic and Exponential Equations
 Know the onetoone property
 Use the strategies introduced for solving equations with exponents and logs
Exponential Modeling
 Know the formula for interest compounded n times per year
 Know the formula for continuously compounded interest
 Know the exponential growth and decay formula
 Compute compound interest
 Use the exponential growth and decay models in the context of real world problems
Systems of Equations: Substitution Method
 Know how to isolate a variable
 Know how to use the isolated variable to make a substitution
 Know how to solve for a variable after substitution
 Use the substitution method to solve a system of equations
Systems of Equations: Linear Combination Method
 Know the method of elimination
 Know the idea behind using linear combinations of equations to eliminate a variable
 Use the linear combination method to solve a system of equations
 Be able to recognize a solution of "all real numbers" or "no solution"
Dependent and Inconsistent Systems
 Identify inconsistent systems of equations
 Identify dependent and independent systems of equations
 Express the solution of a dependent system of equations
Linear Systems with Three Variables
 Solve systems of equations involving three variables
 Solve dependent systems of equations involving three variables
Systems of Nonlinear Equations 1
 Solve a nonlinear system of equations by substitution
Systems of Nonlinear Equations 2
 Solve a nonlinear system of equations by elimination
Matrix Basics
 Know what a matrix is
 Know what the elements of a matrix are
 Identify the rows and columns of a matrix
Operations with Matrices
 Find the sum of two matrices
 Find the difference of two matrices
Matrix Multiplication 1
 Identify when two matrices can be multiplied
 Multiply matrices
Matrix Multiplication 2
 Identify when two matrices can be multiplied
 Multiply matrices