Intermediate Algebra TO LEARN MORE ABOUT ALL OUR OFFERINGS VISIT KNEW TON.COM/HIGHERED 1 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 2 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 3 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 4 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 5 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 6 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 7 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 8 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 9 TO LEAN MORE, VISIT KNEW TON.COM/HIGHERED 10
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