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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 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
  • 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
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

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