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Algebra and Trigonometry With Modeling and Visualizationby Gary K. Rockswold
Synopses & ReviewsPublisher Comments:Gary Rockswold teaches algebra in context, answering the question, “Why am I learning this?” By experiencing math through applications, students see how it fits into their lives, and they become motivated to succeed. Rockswold’s focus on conceptual understanding helps students make connections between the concepts and as a result, students see the bigger picture of math and are prepared for future courses.
Introduction to Functions and Graphs; Linear Functions and Equations; Quadratic Functions and Equations; More Nonlinear Functions and Equations; Exponential and Logarithmic Functions; Trigonometric Functions; Trigonometric Identities and Equations; Further Topics in Trigonometry; Systems of Equations and Inequalities; Conic Sections; Further Topics in Algebra
For all readers interested in college algebra and trigonometry. Synopsis:Gary Rockswold focuses on teaching algebra in context, answering the question, “Why am I learning this?” and ultimately motivating the reader to succeed.
Introduction to Functions and Graphs. Linear Functions and Equations. Quadratic Functions and Equations. Nonlinear Functions and Equations. Exponential and Logarithmic Functions. Trigonometric Functions. Trigonometric Identities and Equations. Further Topics in Trigonometry. Systems of Equations and Inequalities. Conic Sections. Further Topics in Algebra. Basic Concepts From Algebra and Geometry.
For all readers interested in algebra and trigonometry. About the AuthorDr. Gary Rockswold has been teaching mathematics for 25 years at all levels from seventh grade to graduate school, including junior high and high school students, talented youth, vocational, undergraduate and graduate students, and adult education classes. He is currently employed at Minnesota State University, Mankato, where he is a fullprofessor of mathematics and the chair of the mathematics department. He graduated with majors in mathematics and physics from St. Olaf College in Northfield, Minnesota, where he was elected to Phi Beta Kappa. He received his Ph.D. in applied mathematics from Iowa State University. He has an interdisciplinary background and has also taught physical science, astronomy, and computer science. Outside of mathematics, he enjoys spending time with his wife and two children. Table of ContentsChapter 1: INTRODUCTION TO FUNCTIONS AND GRAPHS
1.1 Numbers, Data, and Problem Solving Sets of Numbers Scientific Notation Problem Solving 1.2 Visualization of Data OneVariable Data Two Variable Data The Distance Formula The Midpoint Formula Graphing with a Calculator (Optional) Checking Basic Concepts for Sections 1.1 and 1.2 1.3 Functions and Their Representations Basic Concepts Representations of Functions Formal Definition of a Function Graphing Calculators and Functions (Optional) Identifying Functions 1.4 Types of Functions and Their Rates of Change Constant Functions Linear Functions Slope as a Rate of Change Nonlinear Functions Average Rate of Change The Difference Quotient Checking Basic Concepts for Sections 1.3 and 1.4 Chapter 1 Summary Chapter 1 Review Exercises Chapter 1 Extended and Discovery Exercises Chapter 2: LINEAR FUNCTIONS AND EQUATIONS 2.1 Linear Functions and Models Exact and Approximate Models Representations of Linear Functions Modeling with Linear Functions Linear Regression (Optional) 2.2 Equations of Lines Forms for Equations of Lines Determining Intercepts Horizontal, Vertical, Parallel, and Perpendicular Lines Modeling Data (Optional) Interpolation and Extrapolation Direct VariationChecking Basic Concepts for Sections 2.1 and 2.2 2.3 Linear Equations Equations Symbolic Solutions Graphical and Numerical Solutions ProblemSolving Strategies 2.4 Linear Inequalities Inequalities Interval Notion Techniques for Solving Inequalities Compound InequalitiesChecking Basic Concepts for Sections 2.3 and 2.4 2.5 PiecewiseDefined Functions Evaluating and Graphing PiecewiseDefined Functions The Greatest Integer Function The Absolute Value Function Equations and Inequalities Involving Absolute ValuesChecking Basic Concepts for Section 2.5 Chapter 2 Summary Chapter 2 Review Exercises Chapter 2 Extended and Discovery Exercises
Chapter 12 Cumulative Review Exercises
Chapter 3: QUADRATIC FUNCTIONS AND EQUATIONS 3.1 Quadratic Functions and Models Basic Concepts Completing the Square and the Vertex Formula Applications and Models Quadratic Regression (Optional) 3.2 Quadratic Equations and Problem Solving Basic Concepts Solving Quadratic Equations Problem Solving Checking Basic Concepts for Sections 3.1 and 3.2 3.3 Quadratic Inequalities Graphical Solutions Symbolic Solutions 3.4 Transformations of Graphs Vertical and Horizontal Translations Stretching and Shrinking Reflection of Graphs Combining Transformations Modeling with Transformations (Optional) Checking Basic Concepts for Sections 3.3 and 3.4 Chapter 3 Summary Chapter 3 Review Exercises Chapter 3 Extended and Discovery Exercises Chapter 4: NONLINEAR FUNCTIONS AND EQUATIONS 4.1 Nonlinear Functions and Their Graphs Polynomial Functions Increasing and Decreasing Functions Extrema of Nonlinear Functions Symmetry 4.2 Polynomial Functions and Models Graphs of Polynomial Functions PiecewiseDefined Polynomial Functions Polynomial Regression (Optional)Checking Basic Concepts for Sections 4.1 and 4.2 4.3 Real Zeros of Polynomial Functions Division of Polynomials Factoring Polynomials Graphs and Multiple Zeros Rational Zeros Polynomial Equations 4.4 The Fundamental Theorem of Algebra Complex Numbers Quadratic Equations with Complex Solutions Fundamental Theorem of Algebra Polynomial Equations with Complex SolutionsChecking Basic Concepts for Sections 4.3 and 4.4 4.5 Rational Functions and Models Rational Functions Vertical Asymptotes Horizontal Asymptotes Identifying Asymptotes Rational Equations Variation 4.6 Polynomial and Rational Inequalities Polynomial Inequalities Rational InequalitiesChecking Basic Concepts for Sections 4.5 and 4.6 4.7 Power Functions and Radical Equations Rational Exponents and Radical Notation Power Functions and Models Equations Involving Rational Exponents Equations Involving Radicals Power Regression (Optional)Checking Basic Concepts for Section 4.7 Chapter 4 Summary Chapter 4 Review Exercises Chapter 4 Extended and Discovery Exercises
Chapters 14 Cumulative Review Exercises Chapter 5: EXPONENTIAL AND LOGARITHMIC FUNCTIONS 5.1 Combining Functions Arithmetic Operations on Functions Composition of Functions 5.2 Inverse Functions and Their Representations Inverse Operations OnetoOne Functions Symbolic Representations of Inverse Functions Other Representations of Inverse Functions Checking Basic Concepts for Sections 5.1 and 5.2 5.3 Exponential Functions and Models Linear and Exponential Growth Exponential Models Compound Interest The Natural Exponential Function 5.4 Logarithmic Functions and Models The Common Logarithmic Function Basic Equations Logarithms with Other Bases General Logarithmic Equations Checking Basic Concepts for Sections 5.3 and 5.4 5.5 Properties of Logarithms Basic Properties of Logarithms Change of Base Formula 5.6 Exponential and Logarithmic Equations Exponential Equations Logarithmic Equations Checking Basic Concepts for 5.5 and 5.6 5.7 Constructing Nonlinear Models Exponential Model Logarithmic Model Logistic ModelChecking Basic Concepts for Section 5.7 Chapter 5 Summary Chapter 5 Review Exercises Chapter 5 Extended and Discovery Exercises
Chapter 6: TRIGONOMETRIC FUNCTIONS 6.1 Angles and Their Measure Angles Degree Measure Radian Measure Arc Length Area of a Sector 6.2 Right Triangle Trigonometry Basic Concepts of Trigonometric Functions Applications of Right Triangle Trigonometry Complementary Angles and CofunctionsChecking Basic Concepts for 6.1 and 6.2 6.3 The Sine and Cosine Functions and Their Graphs Definitions The Unit Circle Representations of the Sine and the Cosine Functions Applications of the Sine and Cosine Functions Modeling with the Sine Function (Optional) 6.4 Other Trigonometric Functions and Their Graphs Definitions and Basic Identities Representations of Other Trigonometric Functions Applications of Trigonometric FunctionsChecking Basic Concepts for Sections 6.3 and 6.4 6.5 Graphing Trigonometric Functions Transformations of Trigonometric Graphs Graphing Trigonometric Functions by Hand Simple Harmonic Motion Models Involving Trigonometric Functions (Optional) 6.6 Inverse Trigonometric Functions Review of Inverses The Inverse Sine Function The Inverse Cosine Function The Inverse Tangent Function Solving Triangles and Equations Checking Basic Concepts for Sections 6.5 and 6.6 Chapter 6 Summary Chapter 6 Review Exercises Chapter 6 Extended and Discovery Exercises Chapters 16 Cumulative Review Exercises Chapter 7: TRIGONOMETRIC IDENTITIES AND EQUATIONS 7.1 Fundamental Identities Reciprocal and Quotient Identities Pythagorean Identities NegativeAngle Identities 7.2 Verifying Identities Simplifying Trigonometric Expressions Verification of IdentitiesChecking Basic Concepts for Section 7.1 and 7.2 7.3 Trigonometric Equations Reference Angles Solving Trigonometric Equations Solving Inverse Trigonometric Equations 7.4 Sum and Difference Identities Sum and Difference Identities for Cosine Other Sum and Difference Identities Checking Basic Concepts for Section 7.3 and 7.4 7.5 MultipleAngle Identities DoubleAngle Identities HalfAngle Formulas Solving Equations ProducttoSum and SumtoProduct Identities Checking Basic Concepts for Section 7.5 Chapter 7 Summary Chapter 7 Review Exercises Chapter 7 Extended and Discovery Exercises
Chapter 8: FURTHER TOPICS IN TRIGONOMETRY 8.1 Law of Sines Oblique Triangles Solving Triangles The Ambiguous Case 8.2 Law of Cosines Derivation of the Law of Cosines Solving Triangles Area FormulasChecking Basic Concepts for Sections 8.1 and 8.2 8.3 Vectors Basic Concepts Operations on Vectors The Dot Product Work 8.4 Parametric Equations Basic Concepts Applications of Parametric EquationsChecking Basic Concepts for Sections 8.3 and 8.4 8.5 Polar Equations The Polar Coordinate System Graphs of Polar Equations Graphing Calculators and Polar Equations (Optional) Solving Polar Equations 8.6 Trigonometric Form and Roots of Complex Numbers Trigonometric Form Products and Quotients of Complex Numbers De Moivre’s Theorem Roots of Complex NumbersChecking Basic Concepts for Sections 8.5 and 8.6 Chapter 8 Summary Chapter 8 Review Exercises Chapter 8 Extended and Discovery Exercises Chapters 18 Cumulative Review Exercises Chapter 9: SYSTEMS OF EQUATIONS AND INEQUALITIES 9.1 Functions and Equations in Two Variables Functions of Two Variables Systems of Equations The Method of Substitution Graphical and Numerical Methods Joint Variation 9.2 Systems of Equations and Inequalities in Two Variables Types of Linear Systems in Two Variables The Elimination Method Systems of Linear and Nonlinear Inequalities Linear ProgrammingChecking Basic Concepts for 9.1 and 9.2 9.3 Systems of Linear Equations in Three Variables Basic Concepts Solving with Elimination and Substitution Systems with No Solutions Systems with Infinitely Many Solutions 9.4 Solutions to Linear Systems Using Matrices Representing Systems of Linear Equations with Matrices RowEchelon Form Gaussian Elimination Solving Systems of Linear Equations with Technology (Optional)Checking Basic Concepts for Sections 9.3 and 9.4 9.5 Properties and Applications of Matrices Matrix Notation Sums, Differences, and Scalar Multiples of Matrices Matrix Products Technology and Matrices (Optional) 9.6 Inverses of Matrices Matrix Inverses Finding Inverses Symbolically Representing Linear Systems with Matrix Equations Solving Linear Systems with InversesChecking Basic Concepts for Sections 9.5 and 9.6 9.7 Determinants Definition and Calculation of Determinants Area of Regions Cramer’s Rule Limitations on the Method of Cofactors and Cramer’s Rule Checking Basic Concepts for Section 9.7 Chapter 9 Summary Chapter 9 Review Exercises Chapter 9 Extended and Discovery Exercise Chapter 10: CONIC SECTIONS 10.1 Parabolas Equations and Graphs of Parabolas Reflective Property of Parabolas Translations of Parabolas 10.2 Ellipses Equations and Graphs of Ellipses Reflective Property of Ellipses Translations of Ellipses Circles Solving Systems of Equations and InequalitiesChecking Basic Concepts for Section 10.1 and 10.2 10.3 Hyperbolas Equations and Graphs of Hyperbolas Reflective Property of Hyperbolas Translations of Hyperbolas Solving Systems of Nonlinear Equations Checking Basic Concepts for Section 10.3 Chapter 10 Summary Chapter 10 Review Exercises Chapter 10 Extended and Discovery Exercises Chapter 11: FURTHER TOPICS IN ALGEBRA 11.1 Sequences Basic Concepts Representations of Sequences Arithmetic Sequences Geometric Sequences 11.2 Series Basic Concepts Arithmetic Series Geometric Series Summation NotationChecking Basic Concepts for Sections 11.1 and 11.2 11.3 Counting Fundamental Counting Principle Permutations Combinations 11.4 The Binomial Theorem Derivation of the Binomial Theorem Pascal’s TriangleChecking Basic Concepts for Sections 11.3 and 11.4 11.5 Mathematical Induction Mathematical Induction Proving Statements Generalized Principle of Mathematical Induction 11.6 Probability Definition of Probability Compound Events Independent and Dependent EventsChecking Basic Concepts for Sections 11.5 and 11.6 Chapter 11 Summary Chapter 11 Review Exercises Chapter 11 Extended and Discovery Exercises Chapters 111 Cumulative Review Exercises
Chapter R: REFERENCE BASIC CONCEPTS FROM ALGEBRA AND GEOMETRY R.1 Formulas from Geometry Geometric Shapes in a Plane The Pythagorean Theorem ThreeDimensional Objects Similar Triangles A Summary of Geometric Formulas R.2 Circles Equations and Graphs of Circles Finding the Center and Radius of a Circle R.3 Integer Exponents Bases and Positive Exponents Zero and Negative Exponents Product, Quotient, and Power Rules R.4 Polynomial Expressions Addition and Subtraction of Monomials Addition and Subtraction of Polynomials Distributive Properties Multiplying Polynomials Some Special Products R.5 Factoring Polynomials Common Factors Factoring by Grouping Factoring x^{2 }+ bx + c Factoring Trinomials by Grouping Factoring Trinomials with FOIL Difference of Two Squares Perfect Square Trinomials Sum and Difference of Two Cubes R.6 Rational Expressions Simplifying Rational Expressions Multiplication and Division of Rational Expressions Least Common Multiples Common Denominators Addition and Subtraction of Rational Expressions Clearing Fractions Complex Fractions R.7 Radical Notation and Rational Exponents Radical Notation Rational Exponents Properties of Rational Exponents R.8 Radical Expressions Product Rule for Radical Expressions Quotient Rule for Radical Expressions Addition and Subtraction Multiplication Rationalizing the Denominator
APPENDIX A: A Library of Functions APPENDIX B: Using the Graphing Calculator APPENDIX C: Partial Fractions APPENDIX D: Rotation of Axes
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