 ### College Algebra

Functions, transformations, and matrices.

• 111 Video
Lessons
• 124 Embedded Video
Questions
• 753 Practice
Problems
##### 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
• Simplify radical expressions that contain fractions
• Simplify radical expressions with variables
• Simplify nth root expressions with variables
• 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
• 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 x-intercepts and y-intercepts of graphs
• Solve to find x- and y-intercepts
• 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
##### Slope-Intercept Form of a Line
• Know the slope-intercept 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 slope-intercept form if given two points
• Graph a line if its equation is in slope-intercept form
##### Point-Slope Form of a Line
• Know the point-slope form of a line
• Find the equation of a line in point-slope form if given a graph
• Find the equation of a line in point-slope form if given two points
• Graph a line if its equation is in point-slope 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
##### 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 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
• 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
• Solve square roots by squaring both sides
• Check for extraneous solutions
##### 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
##### One-to-One Functions
• Know the definition of a one-to-one function
• Know how to use the Horizontal Line Test for one-to-one functions
• Determine if a function is one-to-one 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(x-h) changes the graph for h>0
• Know how f(x-h) changes the graph for h<0
• Graph a function if given the equation f(x-h)
• 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 one-to-one 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 one-to-one 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 