Intermediate Algebra

Intermediate Algebra
Intermediate Algebra
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INT E RME DIATE A LG EB R A
Authors
AUTHOR
John Redden
COLLEGE OF THE SEQUOIAS
REVIEWERS
Katherine Adams
EASTERN MICHIGAN UNIVERSITY
Sheri Berger
LOS ANGELES VALLEY COLLEGE
Seung Choi
NORTHERN VIRGINIA COMMUNITY COLLEGE
Stephen DeLong
COLORADO MOUNTAIN COLLEGE
Keith Eddy
COLLEGE OF THE SEQUOIAS
Solomon Emeghara
WILLIAM PATERSON UNIVERSITY
Audrey Gillant
SUNY–MARITIME
Barbara Goldner
NORTH SEATTLE COMMUNITY COLLEGE
Joseph Grich
WILLIAM PATERSON UNIVERSITY
Caroll Hobbs
PENSACOLA STATE COLLEGE
Clark Ingham
MOTT COMMUNITY COLLEGE
Valerie LaVoice
NHTI, CONCORD COMMUNITY COLLEGE
Sandra Martin
BREVARD SCHOOLS
Bethany Mueller
PENSACOLA STATE COLLEGE
Tracy Redden
COLLEGE OF THE SEQUOIAS
James Riley
NORTHERN ARIZONA UNIVERSITY
Bamdad Samii
CALIFORNIA STATE UNIVERSITY–NORTHRIDGE
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INT E RME DIATE A LG EB R A
Table of Contents
Chapter 1: Graphing Functions and
Inequalities
GRAPHING THE BASIC FUNCTIONS
Piecewise Functions and Graphs of Basic Functions
• Define and graph seven basic functions
REL ATIONS AND FUNCTIONS
• Graph piecewise-defined functions
The Definition of a Function
• Evaluate piecewise-defined functions
• Representing relations on a Cartesian coordinate
plane
USING TRANSFORMATIONS TO GRAPH FUNCTIONS
• Determine the domain and range from a graph
Transformations of Functions
• Determine if a relation is a function given ordered
pairs
• Graph functions using vertical and horizontal
translations
• Determine if a relation is a function given a graph
• Graph functions using reflections about the xand y- axes
Function Notation
• Graph functions using dilations
• Use function notation with a numerical argument
• Use function notation with an algebraic argument
SOLVING ABSOLUTE VALUE EQUATIONS AND
• Evaluate a function from a graph
INEQUALITIES
• Find the input from the output in function
notation
Absolute Value Equations
• Understand the concept of absolute value
• Solve absolute value equations with one absolute
value expression
LINEAR FUNCTIONS AND THEIR GRAPHS
Graphs of Linear Functions
• Solve absolute value equations with two absolute
value expressions
• Identify and graph linear functions
• Determine a linear function given a graph
Absolute Value Inequalities
Horizontal and Vertical Lines and Graphical
Interpretations of Equations and Inequalities
• Solve absolute value inequalities involving less
than
• Understand properties of horizontal and vertical
lines
• Solve absolute value inequalities involving greater
than
• Represent linear equations and inequalities
graphically
SOLVING INEQUALITIES WITH TWO VARIABLES
Solve Linear Inequalities
MODELING LINEAR FUNCTIONS
• Determine if an ordered pair is in a solution set of
a linear inequality with two variables
Equations of Lines
• Determine a linear function using point-slope
form
• Graph the solution set of a linear inequality with
two variables
• Find a function that passes through a point and is
parallel to another function
Solve Nonlinear Inequalities
• Determine if an ordered pair is in a solution set of
a non-linear inequality with two variables
• Find a function that passes through a point and is
perpendicular to another function
• Graph the solution set of a non-linear inequality
with two variables
Modeling Linear Applications
• Write a linear mathematical model
• Use a linear model for interpolation and
extrapolation
• Use functions to represent revenue, cost and
profit
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INT E RME DIATE A LG EB R A : TA B L E OF CON T E N T S
Chapter 2: Solving Linear Systems
Determinants and Cramer’s Rule
• Calculate the determinant of a 2×2 matrix
LINEAR SYSTEMS WITH TWO VARIABLES
• Calculate the determinant of a 3×3 matrix
Graphing Linear Systems of Equations with Two
Variables
• Use Cramer’s rule to solve systems of linear
equations with two variables
• Determine if an ordered pair is a solution to a
system of linear equations
• Use Cramer’s rule to solve systems of linear
equations with three variables
• Solve systems of linear equations in two variables
by graphing
SYSTEMS OF INEQUALITIES WITH TWO VARIABLES
• Identify inconsistent and dependent systems
of linear equations graphically, and express the
solution of dependent equations
Solving Linear Systems of Inequalities
• Determine if an ordered pair is a solution to a
system of linear inequalities with two variables
Solving Linear Systems with Two Variables
• Graph the solution sets of systems of linear
inequalities
• Solve systems of linear equations in two variables
by substitution
Solving Nonlinear Systems of Inequalities
• Solve systems of linear equations in two variables
by elimination
• Determine if an ordered pair is a solution to
a system of non-linear inequalities with two
variables
• Choose a method to solve a system of linear
equations
• Graph the solution sets of systems of non-linear
inequalities
Applications of Linear Systems with Two Variables
• Write and solve a system of linear equations in
two variables in word problems
Chapter 3: Polynomial and Rational
Functions
• Write and solve mixture problems
POLYNOMIAL FUNCTIONS
• Write and solve distance problems
Evaluate Polynomial Functions
Solving Linear Systems with Three Variables
• Identify and evaluate polynomial functions
• Check solutions to linear systems with three
variables
• Evaluate polynomial functions for algebraic
expressions
• Solve linear systems with three variables by
elimination
• Use polynomial functions in applications
• Identify dependent and inconsistent linear
systems with three variables
Operations with Polynomial Functions and Adding
Functions Graphically
• Add and subtract polynomial functions
• Solve applications involving three unknowns
• Multiply and divide polynomial functions
SOLVING LINEAR SYSTEMS WITH MATRICES
• Add functions graphically
Matrices and Gaussian Elimination
• Use back substitution to solve linear systems in
upper triangular form
Factoring with a Greatest Common Factor
• Determine the greatest common factor of
monomials
• Convert linear systems to equivalent augmented
matrices
• Factor polynomials using the greatest common
factor
• Solve a linear system of equations with two
variables using Gaussian elimination
• Use grouping to factor polynomials
• Solve a linear system of equations with three
variables using Gaussian elimination
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INT E RME DIATE A LG EB R A : TA B L E OF CON T E N T S
Factoring Polynomials
Add and Subtract Rational Functions
• Factor the difference of squares
• Add and subtract rational functions
• Factor the sum and difference of cubes
• Add and subtract terms with negative exponents
• Factor special polynomials with multiple steps
Simplify Complex Fractions
• Simplify complex rational expressions
Factoring Quadratic Trinomials
• Factor a quadratic trinomial with a leading
coefficient of 1
• Simplify complex rational expressions with
negative exponents
• Factor a quadratic trinomial with a leading
coefficient of 1 and multiple variables
Solve Rational Equations
• Solve rational equations with monomials in the
denominator
Factoring Trinomials Quadratic in Form
• Factor a trinomial with a higher degree with a
leading coefficient of 1
• Solve rational equations with binomials in the
denominator
• Factor a quadratic trinomial with a leading
coefficient greater than 1
• Solve rational equations with factoring to find the
LCD
• Factor a quadratic trinomial with a leading
coefficient greater than 1
• Solve rational equations using cross multiplication
Applications of Rational Equations
• Use multiple strategies to factor polynomials
• Solve literal equations
SOLVE POLYNOMIAL EQUATIONS
• Solve for an unknown number using reciprocals
Solve a Quadratic or Polynomial Equation by
Factoring
• Solve uniform motion problems
• Solve work-rate problems
• Solve polynomial equations given in factored form
using the zero-product property
Variation
• Solve quadratic trinomial equations by factoring
• Solve problems involving direct variation
• Solve polynomial equations by factoring
• Solve problems involving inverse variation
• Solve problems involving joint variation
Find the Root of a Quadratic or Polynomial Function
Chapter 4: Radical Functions and
Equations
• Find the root of a quadratic function
• Find the root of a polynomial function
• Use factoring to solve polynomial functions in
applications
RADICAL EXPRESSIONS
Evaluate nth Roots
OPERATIONS ON RATIONAL FUNCTIONS
• Identify and evaluate square and cube roots
Simplify and Find the Domain of Rational Functions
• Identify and evaluate nth roots
• Find the domain of rational functions
• Simplify radicals with numerical radicands
• Simplify rational functions composed of quadratic
functions
Simplify Radical Expressions
• Understand even and odd indices when
simplifying radicals with variable radicands
• Simplify rational functions composed of
polynomial functions
• Simplify radical expressions with square and cube
roots
• Use the difference quotient
• Simplify radical expressions with nth roots
Multiply and Divide Rational Functions
• Multiply rational functions
• Divide rational functions
• Simplify rational expressions using both
multiplication and division
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INT E RME DIATE A LG EB R A : TA B L E OF CON T E N T S
Add and Subtract Radical Expressions
RADICAL EQUATIONS
• Perform addition and subtraction with like radicals
Solve Radical Equations
• Add and subtract radical expressions with
numerical radicands
• Solve equations including one square root
• Solve equations including two square roots
• Add and subtract radical expressions with variable
radicands
• Solve equations including cube roots
COMPLEX NUMBERS
The Distance Formula and Applications
Operations on Complex Numbers
• Use the distance formula
• Rewrite the square root of a negative number as
an imaginary number
• Determine if three given points form a right
triangle
• Add and subtract complex numbers
• Determine the perimeter of a triangle given three
points
• Perform multiplication with complex numbers
• Divide complex numbers
Multiply Radical Expressions
• Multiply the square root of negative numbers
• Multiply radical expressions with numerical
radicands
Chapter 5: Solving Equations and
Inequalities
• Multiply radical expressions with variable
radicands
EXTRACTING SQUARE ROOTS AND COMPLETING
• Multiply two binomials that contain radical
expressions
THE SQUARE
The Square Root Property
Divide Radical Expressions
• Solve a quadratic equation using the square root
property
• Divide radical expressions with the quotient rule
for radicals
• Solve a quadratic equation with a perfect square
binomial using the square root property
• Rationalize a denominator with a monomial
square root expression in the denominator
Completing the Square
• Rationalize a denominator with a monomial
higher index radical expression in the
denominator
• Convert a quadratic in standard form to a
quadratic with a perfect square binomial
• Rationalize a denominator with a binomial in the
denominator
• Solve a quadratic equation by completing the
square
• Solve a quadratic equation by completing the
square with an odd coefficient of x
RULES OF EXPONENTS AND RATIONAL
EXPONENTS
• Solve a quadratic equation using the quadratic
formula
Rules of Exponents
• Simplify expressions using the product and
quotient rules of exponents
• Solve a quadratic equation not given in standard
form using the quadratic formula
• Simplify expressions using the power rule of
exponents
• Determine the type and number of solutions of a
quadratic equation
• Simplify expressions with negative exponents
Solve Equations Quadratic in Form
Rational Exponents
• Solve an equation with rational or negative
exponents by writing in quadratic form
• Rewrite an expression with a rational exponent as
a radical
• Solve other equations by writing in quadratic form
• Rewrote a radical expression as an expression
with rational exponents
• Solve a cubic polynomial and understand the
fundamental theorem of algebra
• Evaluate expressions with a rational exponent that
have a negative base
• Simplify expressions including rational exponents
• Simplify radical expressions with different indices
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INT E RME DIATE A LG EB R A : TA B L E OF CON T E N T S
Inverse Functions
QUADRATIC FUNCTIONS AND THEIR GRAPHS
• Verify that two functions are inverses of each
other
Quadratic Functions and the Parabola
• Determine the x and y intercepts of a parabola
• Determine if a function is a one-to-one function
• Find the vertex and the axis of symmetry of a
parabola
• Find the inverse of a linear or quadratic function
algebraically
• Graph a parabola from a quadratic function given
in standard form
• Find the inverse of other types of functions
algebraically
• Find the maximum or minimum of a quadratic
function
EXPONENTIAL FUNCTIONS
Vertex Form of a Quadratic Function
Graphs of Exponential Functions
• Find the vertex and graph a parabola from a
quadratic function given in vertex form
• Determine the domain and range of an
exponential function and recognize its graph
• Write a quadratic function in vertex form by
completing the square
• Graph an exponential function with fractional
bases or involving reflections
• Graph an exponential function with the natural
base
QUADRATIC INEQUALITIES
Find the Solution Set of a Quadratic Inequality
Compound Interest
• Determine the solution set of a quadratic
inequality from a graph
• Use the compound interest formula
• Use the continuously compounding interest
formula
• Solve a quadratic inequality in one variable by
factoring
• Solve a quadratic inequality in one variable with
the quadratic formula
LOGARITHMIC FUNCTIONS
Definition of the Logarithm
• Find the domain of a radical function with a
quadratic radicand
• Understand the definition of the logarithm and
convert between exponential and logarithmic
form
POLYNOMIAL AND RATIONAL INEQUALITIES
• Evaluate common or natural logarithms without
the use of a calculator
Solve Polynomial Inequalities
• Solve a polynomial inequality given in factored
form
• Solve a simple common or natural logarithmic
equation by rewriting in exponential form
• Solve a polynomial inequality requiring factoring
to solve
Graphs of Logarithmic Functions
• Determine the domain and range of a logarithmic
function with a base greater than 1 and recognize
its graph
Solve Rational Inequalities
• Solve a rational inequality with only one rational
expression
• Graph a logarithmic function with a fractional
base
• Solve a rational inequality with two rational
expressions
Chapter 6: Exponential and
Logarithmic Functions
COMPOSITE FUNCTIONS AND INVERSE
FUNCTIONS
Composite Functions and Inverse Functions
• Perform a composition of functions with two
functions
• Compose a function with itself
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INT E RME DIATE A LG EB R A : TA B L E OF CON T E N T S
Chapter 7: Conic Sections and
Nonlinear Systems
PROPERTIES OF THE LOGARITHM
The Inverse Properties of Logarithms and Expanding
and Contracting Logarithms
CONIC SECTIONS
• Apply the inverse properties of the logarithm
Parabolas
• Apply the product or quotient property of
logarithms to simplify a logarithmic expression
• Convert the equation of a parabola from general
form to standard form and find the vertex of a
parabola
• Apply the power property of logarithms to
simplify a logarithmic expression
• Use the properties of logarithms to expand a
logarithmic expression
• Use both general and standard forms of a
quadratic equation to graph and find key points of
a parabola
• Use the properties of logarithms to condense a
logarithmic expression
• Graph a parabola that opens to the left or the
right
• Use the distance and midpoint formulas to find
the center and radius of circles
EXPONENTIAL AND LOGARITHMIC EQUATIONS
Solving Exponential Equations and the Change of
Base Formula
Circles
• Solve an exponential equation with the one-toone property of exponential functions
• Graph a circle and determine the radius and
center given the equation in standard form
• Solve an exponential equation with logarithms
• Find the intercepts of a circle given the equation
in standard form
• Evaluate a logarithm using the change of base
formula
• Write the equation of a circle in standard form
given the center and radius or points on the circle
Solving Logarithmic Equations
• Solve a logarithmic equation with the one-to-one
property of logarithmic functions
• Write the equation of a circle given in general
form as an equation in standard form
• Solve a logarithmic equation requiring the
properties of logarithms
• Graph a circle given the equation in general form
Ellipses
• Find the inverse of a logarithmic function
• Write the equation for an ellipse in standard form
given key points
APPLICATIONS OF EXPONENTIAL AND
• Graph an ellipse given an equation in standard
form
LOGARITHMIC FUNCTIONS
Exponential Growth and Decay
• Find the intercepts of an ellipse given an equation
in standard form
• Use the compound interest formula to determine
the time for an investment to yield a given
amount
• Write the equation of an ellipse given in general
form as an equation in standard form
• Use the continuous compound interest formula
to determine the time for an investment to yield a
given amount
• Graph an ellipse given an equation in general form
Hyperbolas
• Use the exponential growth formula to investigate
population growth and other applications
• Write the equation for a hyperbola in standard
form given key points
• Use the exponential decay formula to investigate
radioactive decay and other applications
• Graph a hyperbola given an equation in standard
form
• Find the intercepts of a hyperbola given an
equation in standard form
• Graph a hyperbola given an equation in general
form
• Identify a conic section given its equation in
general form
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INT E RME DIATE A LG EB R A : TA B L E OF CON T E N T S
SYSTEMS OF EQUATIONS WITH CONIC SECTIONS
THE BINOMIAL THEOREM
Nonlinear Systems of Equations
Factorials and the Binomial Theorem
• Solve a nonlinear system of equations given a
circle and a line
• Evaluate an expression involving factorials
• Solve a nonlinear system of equations given a
circle and a parabola
• Expand powers of binomials using the binomial
theorem or Pascal’s triangle
• Calculate a binomial coefficient
Chapter 8: The Binomial Theorem and
Sequences and Series
• Expand powers of binomials with negative terms
or a coefficient greater than 1
SEQUENCES AND SERIES
Introduction to Sequences and Series
• Find terms of a sequence given the general term
of a sequence
• Find terms of a sequence given a recurrence
relation
• Determine the partial sum of a sequence
• Write an infinite series in expanded form
Arithmetic Sequences
• Find an equation for the general term of an
arithmetic sequence
• Find a term or an equation for an arithmetic
sequence given two terms in the sequence
Arithmetic Series
• Find a partial sum of an arithmetic sequence
• Use the partial sum of an arithmetic sequence in
applications
Geometric Sequences
• Find an equation for the general term of a
geometric sequence
• Find a term or an equation for a geometric
sequence given two terms in the sequence
Geometric Series
• Find a partial sum of a finite geometric sequence
• Find the sum of an infinite geometric series
• Write a repeating decimal as a fraction using
infinite series or use geometric series in
applications
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