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Intermediate Algebra

Working with functions, systems of equations and inequalities, and exponential and logarithmic functions.

  • 82 Video
    Lessons
  • 82 Embedded Video
    Questions
  • 451 Practice
    Problems
Table of Contents
What is a Function?
  • 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
Graph of a Function
  • Understand the definition of the graph of a function
  • Know how to determine if a given plot represents a function
  • Plot the graph of a function
  • Use a graph to evaluate a function
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
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
Absolute Value Functions
  • Find the domain and range of absolute value functions
  • Graph absolute value functions
Step Functions
  • Know what ⌊x⌋ and ⌈x⌉ are
  • Know the definition of a step function
  • Evaluate a step function on numerical input
  • Graph a step 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
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
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
Factoring a Common Term
  • Know the definition of "greatest common factor"
  • Find the greatest common factor among terms of an expression
  • Factor a common term out of a polynomial expression
Factoring by Grouping
  • Use the grouping technique to factor certain polynomials
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
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
The Number i
  • Define i
  • Evaluate powers of i
Adding, Subtracting, and Multiplying 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
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
Finding the Intersection of Parabolas
  • Determine the intersection point of two parabolas by using a graph
  • Determine the intersection point of two parabolas using algebraic methods
Quadratic Equations Application: Rectangle Problem
  • Domain
  • Quadratic Functions
Quadratic Equations Application: Even Numbers
  • Factoring
  • Multiplying Polynomials
  • Factoring Trinomials
Vertical Shifts of Quadratics
  • Know how y=x2+k changes the graph of y=x2 for k>0
  • Know how y=x2+k changes the graph of y=x2 for k<0
  • Graph y=x2+k
  • Find the equations of a parabola shifted vertically by k units
Horizontal Shifts of Quadratics
  • Know how y=(x-h)2 changes the graph of y=x2 for h>0
  • Know how y=(x-h)2 changes the graph of y=x2 for h<0
  • Graph y=(x-h)2
  • Find the equation of a parabola shifted horizontally by h units
Scaling of Quadratics
  • Know how y=ax2 changes the graph of y=x2 for 0<a<1
  • Know how y=ax2 changes the graph of y=x2 for a>1
  • Know how y=ax2 changes the graph of y=x2 for a<1
  • Graph y=ax2
  • Find the equation of a parabola scaled vertically by a units
Characteristics of Parabolas
  • Know what the x and y intercepts of parabolas are
  • Know what the vertex of a parabola is
  • Know what the axis of symmetry of a parabola is
  • Find the x and y intercepts of parabolas
  • Find the vertex of a parabola
  • Find the axis of symmetry of a parabola
Vertex Form of Quadratics
  • Understand the vertex form of a quadratic equation
  • Graph the equation y=a(x-h)2+k
  • Find the equation of a parabola given its graph
  • Put a quadratic equation into vertex form
Horizontal and Vertical Parabolas
  • Graph horizontal parabolas
  • Identify whether a parabola is horizontal or vertical based on its equation
Maximum and Minimum Problems
  • Identify if a quadratic has a maximum or minimum and find it
  • Solve maximization and minimization problems using parabolas
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
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
Simplifying Rational Expressions
  • Use factoring to simplify rational expressions
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
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
Solving Equations with Rational Expressions
  • Use the techniques from the previous lectures to solve equations involving rational expressions
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
Quadratic Inequalities
  • Solve quadratic inequalities graphically
  • Solve quadratic inequalities algebraically
Rational Inequalities
  • Solve rational inequalities
Roots
  • Identify the parts of a square root
  • Find nth roots of whole numbers
Negative and Fractional Exponents
  • Know the rules dictating how to deal with fractional and negative exponents
  • Manipulate expressions with fractional and negative exponents
Simplifying Radical Expressions
  • Simplify radical expressions that contain fractions
  • Simplify radical expressions with variables
  • Simplify nth root expressions with variables
Multiplying and Dividing Radical Expressions
  • Multiply and divide radical expressions
  • Rationalize a monomial denominator
  • Rationalize a binomial denominator
Dividing radical expressions
  • Divide radical expressions
  • Rationalize a monomial or binomial denominator
Graphing Radical Functions 1
  • Graph basic radical functions
Graphing Radical Functions 2
  • Graph radical functions using transformations
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
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

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