 ### Intermediate Algebra

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

• 82 Video
Lessons
• 82 Embedded Video
Questions
• 451 Practice
Assignments
##### 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
• 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
• 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
##### 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
• Domain
##### Quadratic Equations Application: Even Numbers
• Factoring
• Multiplying Polynomials
• Factoring Trinomials
• 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
• 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
• 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
• 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
##### 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
• 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
• Rationalize a monomial or binomial denominator
• Graph radical functions using transformations
• Solve square roots by squaring both sides
• Check for extraneous solutions
##### 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 