Table of Contents

Chapter P. Prerequisites: Fundamental Concepts of Algebra.

P.1 Algebraic Expressions and Real Numbers

1. Evaluate algebraic expressions.

2. Use mathematical models.

3. Find the intersection of two sets.

4. Find the union of two sets.

5. Recognize subsets of the real numbers.

6. Use inequality symbols.

7. Evaluate absolute value.

8. Use absolute value to express distance.

9. Identify properties of the real numbers.

10. Simplify algebraic expressions.

P.2 Exponents and Scientific Notation

1. Use the product rule.

2. Use the quotient rule.

3. Use the zero-exponent rule.

4. Use the negative-exponent rule.

5. Use the power rule.

6. Find the power of a product.

7. Find the power of a quotient.

8. Simplify exponential expressions.

9. Use scientific notation.

P.3 Radicals and Rational Exponents

1. Evaluate square roots.

2. Simplify expressions of the form Öa2

3. Use the product rule to simplify square roots.

4. Use the quotient rule to simplify square roots.

5. Add and subtract square roots.

6. Rationalize denominators.

7. Evaluate and perform operations with higher roots.

8. Understand and use rational exponents.

P.4 Polynomials

1. Understand the vocabulary of polynomials.

2. Add and subtract polynomials.

3. Multiply polynomials.

4. Use FOIL in polynomial multiplication.

5. Use special products in polynomial multiplication.

6. Perform operations with polynomials in several variables.

Mid-Chapter Check Point

P.5 Factoring Polynomials

1. Factor out the greatest common factor of a polynomial.

2. Factor by grouping.

3. Factor trinomials.

4. Factor the difference of squares.

5. Factor perfect square trinomials.

6. Factor the sum and difference of two cubes.

7. Use a general strategy for factoring polynomials.

8. Factor algebraic expressions containing fractional and negative exponents.

P.6 Rational Expressions

1. Specify numbers that must be excluded from the domain of rational expressions.

2. Simplify rational expressions.

3. Multiply rational expressions.

4. Divide rational expressions.

5. Add and subtract rational expressions.

6. Simplify complex rational expressions.

Chapter 1. Equations and Inequalities

1.1 Graphs and Graphing Utilities

1. Plot points in the rectangular coordinate system.

2. Graph equations in the rectangular coordinate system.

3. Interpret information about a graphing utility’s viewing rectangle or table.

4. Use a graph to determine intercepts.

5. Interpret information given by graphs.

1.2 Linear Equations and Rational Equations

1. Solve Linear equations in one variable.

2. Solve linear equations containing fractions.

3. Solve rational equations with variables in the denominators.

4. Recognize identities, conditional equations, and inconsistent equations.

1.3 Models and Applications

1. Use linear equations to solve problems.

1.4 Complex Numbers

1. Add and subtract complex numbers.

2. Multiply complex numbers.

3. Divide complex numbers.

4. Perform operations with square roots of negative numbers.

1.5 Quadratic Equations

1. Solve quadratic equations by factoring.

2. Solve quadratic equations by the square root property.

3. Solve quadratic equations by completing the square.

4. Solve quadratic equations using the quadratic formula.

5. Use the discriminant to determine the number and type of solutions.

6. Determine the most efficient method to use when solving a quadratic equation.

Mid-Chapter Check Point

1.6 Other Types of Equations

1. Solve polynomial equations by factoring.

2. Solve radical equations.

3. Solve equations with rational exponents.

4. Solve equations that are quadratic in form.

5. Solve equations involving absolute value.

1.7 Linear Inequalities and Absolute Value Inequalities

1. Use interval notation.

2. Find intersections and unions of intervals.

3. Solve linear inequalities.

4. Recognize inequalities with no solution or all real numbers as solutions.

5. Solve compound inequalities.

6. Solve absolute value inequalities.

Chapter 2. Functions and Graphs.

2.1 Basic Functions and Their Graphs

1. Find the domain and range of a relation.

2. Determine whether an equation is a function.

3. Determine whether an equation represents a function.

4. Evaluate a function.

5. Graph functions by plotting points.

6. Use the vertical line test to identify functions.

7. Obtain information about a function from its graph.

8. Identify the domain and range of a function from its graph.

9. Identify intercepts from a function’s graph.

2.2 More on Functions and Their Graphs

1. Find and simplify a function’s difference quotient.

2. Understand and use piecewise functions.

3. Identify intervals on which a function increases, decreases, or is constant.

4. Use graphs to locate relative maxima or minima.

5. Identify even or odd functions and recognize their symmetries.

6. Graph step functions.

2.3 Linear Functions and Slope

1. Calculate a line’s slope.

2. Write the point-slope form of the equation of a line.

3. Write and graph the slope-intercept form of the equation of a line.

4. Graph horizontal or vertical lines.

5. Recognize and use the general form of a line’s equation.

6. Use intercepts to graph the general form of a line’s equation.

7. Model data with linear functions and make predictions.

2.4 More on Slope

1. Find slopes and equations of parallel and perpendicular line.

2. Interpret slope as rate of change.

3. Find a function’s average rate of change.

Mid-Chapter Check Point

2.5 Transformations of Functions

1. Recognize graphs of common functions.

2. Use vertical shifts to graph functions.

3. Use horizontal shifts to graph functions.

4. Use reflections to graph functions.

5. Use vertical stretching and shrinking to graph functions.

6. Use horizontal stretching to graph functions.

7. Graph functions involving a sequence of transformations.

2.6 Combinations of Functions; Composite Functions

1. Find the domain of a function.

2. Combine functions using the algebra of functions, specifying domains.

3. Form composite functions.

4. Determine domains for composite functions.

5. Write functions as composition.

2.7 Inverse Functions

1. Verify inverse functions.

2. Find the inverse of a function.

3. Use the horizontal line test to determine if a function has an inverse function.

4. Use the graph of a one-to-one function to graph its inverse function.

5. Find the inverse of a function and graph both functions on the same axes.

2.8 Distance and Midpoint Formulas; Circles

1. Find the distance between two points.

2. Find the midpoint of a line segment.

3. Write the standard form of a circle’s equation.

4. Give the center and radius of a circle whose equation is in standard form.

5. Convert the general form of a circle’s equation to standard form.

Chapter 3. Polynomial and Rational Functions.

3.1 Quadratic Function

1. Recognize characteristics of parabolas.

2. Graph parabolas.

3. Determine a quadratic function’s minimum or maximum value.

4. Solve problems involving a quadratic function’s minimum or maximum value.

3.2 Polynomial Functions and Their Graphs

1. Identify polynomial functions.

2. Recognize characteristics of graphs of polynomial functions.

3. Determine end behavior.

4. Use factoring to find zeros of polynomial functions.

5. Identify zeros and their multiplicities.

6. Use the Intermediate Value Theorem.

7. Understand the relationship between degree and turning points.

8. Graph polynomial functions.

3.3 Dividing Polynomials: Remainder and Factor Theorems

1. Use long division to divide polynomials

2. Use synthetic division to divide polynomials.

3. Evaluate a polynomial using the Remainder Theorem.

4. Use the Factor Theorem to solve a polynomial equation.

3.4 Zeros of Polynomial Functions

1. Use the Rational Zero Theorem to find possible rational zeros.

2. Find zeros of a polynomial function.

3. Solve polynomial equations

4. Use the Linear Factorization Theorem to find polynomials with given zeros.

5. Use Descartes’s Rule of Signs.

3.5 Rational Functions and Their Graphs

1. Find the domain of rational functions.

2. Use arrow notation.

3. Identify vertical asymptotes.

4. Identify horizontal asymptotes.

5. Use transformations to graph rational functions.

6. Graph rational functions.

7. Identify slant asymptotes.

8. Solve applied problems involving rational functions.

Mid-Chapter Check Point

3.6 Polynomial and Rational Inequalities

1. Solve Polynomial Inequalities.

2. Solve rational inequalities.

3. Solve problems modeled by polynomial or rational inequalities.

3.7 Modeling Using Variation

1. Solve direct variation problems.

2. Solve inverse variation problems.

3. Solve combined variation problems.

4. Solve problems involving joint variation.

Chapter 4. Exponential and Logarithmic Functions.

4.1 Exponential Functions

1. Evaluate exponential functions.

2. Graph exponential functions.

3. Evaluate functions with base e.

4. Use compound interest formulas.

4.2 Logarithmic Functions

1. Change from logarithmic to exponential form.

2. Change from exponential to logarithmic form.

3. Evaluate logarithms.

4. Use basic logarithmic properties.

5. Graph logarithmic functions.

6. Find the domain of a logarithmic function.

7. Use common logarithms.

8. Use natural logarithms.

4.3 Properties of Logarithms

1. Use the product rule.

2. Use the quotient rule.

3. Use the power rule.

4. Expand logarithmic expressions.

5. Condense logarithmic expressions.

6. Use the change-of-base property.

Mid-Chapter Check Point

4.4 Exponential and Logarithmic Equations

1. Use like bases to solve exponential equations.

2. Use logarithms to solve exponential equations.

3. Use the definition of a logarithm to solve logarithmic equations.

4. Use the one-to-one property of logarithms to solve logarithmic equations.

5. Solve applied problems involving exponential and logarithmic equations.

4.5 Exponential Growth and Decay; Modeling Data

1. Model exponential growth and decay.

2. Use logistic growth models.

3. Model data with exponential and logarithmic functions.

4. Express an exponential model in base e.

Chapter 5. Systems of Equations and Inequalities.

5.1 Systems of Linear Equations in Two Variables.

1. Decide whether an ordered air is a solution of a linear system.

2. Solve linear systems by substitution.

3. Solve linear systems by addition.

4. Identify systems that do not have exactly one ordered-pair solution.

5. Solve problems using systems of linear equations.

5.2 Systems of Linear Equations in Three Variables

1. Verify the solution of a system of linear equations in three variables.

2. Solve systems of linear equations in three variables.

3. Solve problems using systems in three variables.

5.3 Partial Fractions

1. Decompose P/Q, where Q has only distinct linear factors.

2. Decompose P/Q, where Q has only repeated linear factors.

3. Decompose P/Q, where Q has a nonrepeated prime quadratic factor.

4. Decompose P/Q, where Q has a prime, repeated quadratic factor.

5.4 Systems of Nonlinear Equations in Two Variables

1. Recognize systems of nonlinear equations in two variables.

2. Solve nonlinear systems by substitution.

3. Solve nonlinear systems by addition.

4. Solve problems using systems of nonlinear equations.

Mid-Chapter Check Point

5.5 Systems of Inequalities

1. Graph a linear inequality in two variables.

2. Graph a nonlinear inequality in two variables.

3. Graph a system of inequalities.

4. Solve applied problems involving systems of inequalities.

5.6 Linear Programming

1. Write an objective function describing a quantity that must be maximized or minimized.

2. Use inequalities to describe limitations in a situation.

3. Use linear programming to solve problems.

Chapter 6. Matrices and Determinants.

6.1 Matrix Solutions to Linear Systems

1. Write the augmented matrix for a linear system.

2. Perform matrix row operations.

3. Use matrices and Gaussian eliminations to solve systems.

4. Use matrices and Gauss-Jordan elimination to solve systems.

6.2 Inconsistent and Dependent Systems and Their Applications

1. Apply Gaussian elimination to systems without unique solutions.

6.3 Matrix Operations and Their Applications

1. Use matrix notation.

2. Understand what is meant by equal matrices.

3. Add and subtract matrices.

4. Perform scalar multiplication.

5. Solve matrix equations.

6. Multiply matrices.

7. Describe applied situations with matrix operations.

Mid-Chapter Check Point

6.4. Multiplicative Inverses of Matrices and Matrix Equations

1. Find the multiplicative inverse of a square matrix.

2. Use inverses to solve matrix equations.

3. Encode and decode messages.

6.5 Determinants and Cramer's Rule

1. Evaluate a second-order determinant.

2. Solve a system of linear equations in two variables using Cramer’s rule.

3. Evaluate a third-order determinant.

4. Solve a system of linear equations in three variables using Cramer’s rule

5. Use determinants to identify inconsistent systems and systems with dependent equations.

6. Evaluate higher-order determinants.

7. Conic Sections.

7.1 The Ellipse

1. Graph ellipses at the origin.

2. Write equations of ellipses in standard form.

3. Graph ellipses not centered at the origin.

4. Solve applied problems involving ellipses.

7.2 The Hyperbola

1. Locate a hyperbola’s vertices and foci.

2. Write equations of hyperbolas in standard form.

3. Graph hyperbolas centered at the origin.

4. Graph hyperbolas not centered at the origin.

5. Solve applied problems involving hyperbolas.

Mid-Chapter Check Point

7.3 The Parabola

1. Graph parabolas with vertices at the origin.

2. Write equations of parabolas in standard form.

3. Graph parabolas with vertices not at the origin.

4. Solve applied problems involving parabolas.

8. Sequences, Induction, and Probability.

8.1 Sequences and Summation Notation

1. Find particular terms of a sequence from the general term.

2. Use recursion formulas.

3. Use factorial notation.

4. Use summation notation.

8.2 Arithmetic Sequences

1. Find the common difference for an arithmetic sequence.

2. Write terms of an arithmetic sequence.

3. Use the formula for a general term of an arithmetic sequence.

4. Use the formula for the sum of the first n terms of an arithmetic sequence.

8.3 Geometric Sequences and Series

1. Find the common ration of a geometric sequence.

2. Write terms of a geometric sequence.

3. Use the formula for the general term of a geometric sequence.

4. Use the formula for the sum of the first n terms of a geometric sequence.

5. Find the value of annuity.

6. Use the formula for the sum of a infinite geometric series.

Mid-Chapter Check Point

8.4 Mathematical Induction

1. Understand the principle of mathematical induction.

2. Prove statements using mathematical induction.

8.5 The Binomial Theorem

1. Evaluate a binomial coefficient.

2. Expand a binomial raised to a power.

3. Find a particular term in a binomial expansion.

8.6 Counting Principles, Permutations, and Combinations

1. Use the Fundamental Counting Principle.

2. Use the permutations formula.

3. Distinguish between permutation problems and combination problems.

4. Use the combinations formula.

8.7 Probability

1. Compute empirical probability.

2. Compute theoretical probability.

3. Find the probability that an event will not occur.

4. Find the probability of one event or a second event occurring.

5. Find the probability of one event and a second event occurring.

Appendix: Where Did That Come From? Selected Proofs.

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