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

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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(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
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=(xh)2 changes the graph of y=x2 for h>0
 Know how y=(xh)2 changes the graph of y=x2 for h<0
 Graph y=(xh)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(xh)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 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