A Graphical Approach to Algebra and Trigonometry

Hornsby/Lial/Rockswold’s Graphical Approach covers functions through a consistent four part analytical process that asks students to 1) Examine the nature of the graph 2) Solve a typical equation analytically and graphically 3) Solve the related inequality analytically and graphically, and finally, 4) Apply analytic and graphical methods to solve an application of that class of function. KEY TOPICS: Linear Functions, Equations, and Inequalities; Analysis of Graphs of Functions; Polynomial Functions; Rational, Power, and Root Functions; Inverse, Exponential, and Logarithmic Functions; Systems and Matrices; Analytic Geometry and Nonlinear Systems; Trigonometric Functions and Applications; Trigonometric Identities and Equations; Applications of Trigonometry and Vectors; Reference: Basic Algebraic Concepts MARKET: For all readers interested in algebra. Cover Title Page Copyright Page Acknowledgments Contents Preface Resources for Success Photo Credits 1 Linear Functions, Equations, and Inequalities 1.1 Real Numbers and the Rectangular Coordinate System Sets of Real Numbers The Rectangular Coordinate System Viewing Windows Approximations of Real Numbers Distance and Midpoint Formulas 1.2 Introduction to Relations and Functions Set-Builder Notation and Interval Notation Relations, Domain, and Range Functions Tables and Graphing Calculators Function Notation Reviewing Basic Concepts (Sections 1.1–1.2) 1.3 Linear Functions Basic Concepts about Linear Functions Slope of a Line and Average Rate of Change Slope–Intercept Form of the Equation of a Line 1.4 Equations of Lines and Linear Models Point–Slope Form of the Equation of a Line Standard Form of the Equation of a Line Parallel and Perpendicular Lines Linear Models and Regression Reviewing Basic Concepts (Sections 1.3–1.4) 1.5 Linear Equations and Inequalities Solving Linear Equations in One Variable Graphical Approaches to Solving Linear Equations Identities and Contradictions Solving Linear Inequalities in One Variable Graphical Approaches to Solving Linear Inequalities Three-Part Inequalities 1.6 Applications of Linear Functions Problem-Solving Strategies Applications of Linear Equations Break-Even Analysis Direct Variation Formulas Reviewing Basic Concepts (Sections 1.5–1.6) Summary Review Exercises Test 2 Analysis of Graphs of Functions 2.1 Graphs of Basic Functions and Relations; Symmetry Continuity Increasing, Decreasing, and Constant Functions The Identity Function The Squaring Function and Symmetry with Respect to the y-Axis The Cubing Function and Symmetry with Respect to the Origin The Square Root and Cube Root Functions The Absolute Value Function The Relation x = y 2 and Symmetry with Respect to the x-Axis Even and Odd Functions 2.2 Vertical and Horizontal Shifts of Graphs Vertical Shifts Horizontal Shifts Combinations of Vertical and Horizontal Shifts Effects of Shifts on Domain and Range Horizontal Shifts Applied to Equations for Modeling 2.3 Stretching, Shrinking, and Reflecting Graphs Vertical Stretching Vertical Shrinking Horizontal Stretching and Shrinking Reflecting across an Axis Combining Transformations of Graphs Reviewing Basic Concepts (Sections 2.1–2.3) 2.4 Absolute Value Functions The Graph of y = 0ƒ(x) 0 Properties of Absolute Value Equations and Inequalities Involving Absolute Value 2.5 Piecewise-defined Functions Graphing Piecewise-Defined Functions The Greatest Integer Function Applications of Piecewise-Defined Functions 2.6 operations and Composition Operations on Functions The Difference Quotient Composition of Functions Applications of Operations and Composition Reviewing Basic Concepts (Sections 2.4–2.6) Summary Review Exercises Test 3 Polynomial Functions 3.1 Complex Numbers The Imaginary Unit i Operations with Complex Numbers 3.2 Quadratic Functions and Graphs Completing the Square Graphs of Quadratic Functions Vertex Formula Extreme Values Applications and Quadratic Models 3.3 Quadratic Equations and Inequalities Zero-Product Property Square Root Property and Completing the Square Quadratic Formula and the Discriminant Solving Quadratic Equations Solving Quadratic Inequalities Formulas Involving Quadratics Reviewing Basic Concepts (Sections 3.1–3.3) 3.4 Applications of Quadratic Functions and Models Applications of Quadratic Functions A Quadratic Model 3.5 Higher-degree Polynomial Functions and Graphs Cubic Functions Quartic Functions Extrema End Behavior x-Intercepts (Real Zeros) Comprehensive Graphs Curve Fitting and Polynomial Models Reviewing Basic Concepts (Sections 3.4–3.5) 3.6 Topics in the Theory of Polynomial Functions (I) Intermediate Value Theorem Division of Polynomials by x - k and Synthetic Division Remainder and Factor Theorems Division of Any Two Polynomials 3.7 Topics in the Theory of Polynomial Functions (II) Complex Zeros and the Fundamental Theorem of Algebra Number of Zeros Rational Zeros Theorem Descartes’ Rule of Signs Boundedness Theorem 3.8 Polynomial Equations and inequalities; Fur ther Applications and Models Polynomial Equations and Inequalities Complex nth Roots Applications and Polynomial Models Reviewing Basic Concepts (Sections 3.6–3.8) Summary Review Exercises Test 4 Rational, Power, and Root Functions 4.1 Rational Functions and Graphs (I) The Reciprocal Function (Omitted) The Function (Omitted) 4.2 Rational Functions and Graphs (II) Vertical and Horizontal Asymptotes Graphing Techniques Oblique Asymptotes Graphs with Points of Discontinuity Graphs with No Vertical Asymptotes 4.3 Rational Equations, Inequalities, Models, and Applications Solving Rational Equations and Inequalities Models and Applications of Rational Functions Inverse Variation Combined and Joint Variation Rate of Work Reviewing Basic Concepts (Sections 4.1–4.3) 4.4 Functions Defined by Powers and Roots Power and Root Functions Modeling Using Power Functions Groups of (Omitted) Graphing Circles and Horizontal Parabolas Using Root Functions 4.5 Equations, Inequalities, and Applications Involving Root Functions Equations and Inequalities An Application of Root Functions Reviewing Basic Concepts (Sections 4.4–4.5) Summary Review Exercises Test 5 Inverse, Exponential, and Logarithmic Functions 5.1 Inverse Functions Inverse Operations One-to-One Functions Inverse Functions and Their Graphs Equations of Inverse Functions An Application of Inverse Functions to Cryptography 5.2 Exponential Functions Real-Number Exponents Graphs of Exponential Functions Exponential Equations (Type 1) Compound Interest The Number e and Continuous Compounding An Application of Exponential Functions 5.3 Logarithms and Their Properties Definition of a Logarithm Common Logarithms Natural Logarithms Properties of Logarithms Change-of-Base Rule Reviewing Basic Concepts (Sections 5.1–5.3) 5.4 Logarithmic Functions Graphs of Logarithmic Functions Finding an Inverse of an Exponential Function A Logarithmic Model 5.5 Exponential and Logarithmic Equations and Inequalities Exponential Equations and Inequalities (Type 2) Logarithmic Equations and Inequalities Equations Involving Exponentials and Logarithms Formulas Involving Exponentials and Logarithms Reviewing Basic Concepts (Sections 5.4–5.5) 5.6 Further Applications and Modeling with Exponential and Logarithmic Functions Physical Science Applications Financial and Other Applications Modeling Data with Exponential and Logarithmic Functions Summary Exercises on Functions: Domains, Defining Equations, and Composition Finding the Domain of a Function: A Summary Determining Whether an Equation Defines y as a Function of x Composite Functions and Their Domains Summary Review Exercises Test 6 Systems and Matrices 6.1 Systems of Equations Linear Systems Substitution Method Elimination Method Special Systems Nonlinear Systems Applications of Systems 6.2 Solution of Linear Systems in Three Variables Geometric Considerations Analytic Solution of Systems in Three Variables Applications of Systems Fitting Data Using a System 6.3 Solution of Linear Systems by Row Transformations Matrix Row Transformations Row Echelon Method Reduced Row Echelon Method Special Cases An Application of Matrices Reviewing Basic Concepts (Sections 6.1–6.3) 6.4 Matrix Properties and Operations Terminology of Matrices Operations on Matrices Applying Matrix Algebra 6.5 Determinants and Cramer’s Rule Determinants of 2 X 2 Matrices Determinants of Larger Matrices Derivation of Cramer’s Rule Using Cramer’s Rule to Solve Systems 6.6 Solution of Linear Systems by Matrix Inverses Identity Matrices Multiplicative Inverses of Square Matrices Using Determinants to Find Inverses Solving Linear Systems Using Inverse Matrices Fitting Data Using a System Reviewing Basic Concepts (Sections 6.4–6.6) 6.7 Systems of Inequalities and Linear Programming Solving Linear Inequalities Solving Systems of Inequalities Linear Programming 6.8 Partial Fractions Decomposition of Rational Expressions Distinct Linear Factors Repeated Linear Factors Distinct Linear and Quadratic Factors Repeated Quadratic Factors Reviewing Basic Concepts (Sections 6.7–6.8) Summary Review Exercises Test 7 Analytic Geometry and Nonlinear Systems 7.1 Circles and Parabolas Conic Sections Equations and Graphs of Circles Equations and Graphs of Parabolas Translations of Parabolas An Application of Parabolas 7.2 Ellipses and Hyperbolas Equations and Graphs of Ellipses Translations of Ellipses An Application of Ellipses Equations and Graphs of Hyperbolas Translations of Hyperbolas Reviewing Basic Concepts (Sections 7.1–7.2) 7.3 The Conic Sections and Nonlinear Systems Characteristics Identifying Conic Sections Eccentricity Nonlinear Systems 7.4 Parametric Equations Graphs of Parametric Equations and Their Rectangular Equivalents Alternative Forms of Parametric Equations An Application of Parametric Equations Reviewing Basic Concepts (Sections 7.3–7.4) Summary Review Exercises Test 8 Trigonometric Functions and Applications 8.1 Angles and Their Measures Basic Terminology Degree Measure Standard Position and Coterminal Angles Radian Measure Arc Lengths and Areas of Sectors Linear and Angular Speed 8.2 Trigonometric Functions and Fundamental Identities Trigonometric Functions Function Values of Quadrantal Angles Reciprocal Identities Signs and Ranges of Function Values Pythagorean Identities Quotient Identities An Application of Trigonometric Functions Reviewing Basic Concepts (Sections 8.1–8.2) 8.3 Right Angles and Evaluating Trigonometric Functions Right-Triangle Definitions of the Trigonometric Functions Trigonometric Function Values of Special Angles Cofunction Identities Reference Angles Special Angles as Reference Angles Finding Function Values with a Calculator Finding Angle Measures 8.4 Applications of Right Triangles Significant Digits Solving Triangles Angles of Elevation or Depression Bearing Further Applications of Trigonometric Functions Reviewing Basic Concepts (Sections 8.3–8.4) 8.5 The Circular Functions Circular Functions Applications of Circular Functions 8.6 Graphs of the sine and Cosine Functions Functions Periodic Functions Graph of the Sine Function Graph of the Cosine Function Graphing Techniques, Amplitude, and Period Translations and Transformations Determining a Trigonometric Model Using Curve Fitting Reviewing Basic Concepts (Sections 8.5–8.6) 8.7 Graphs of the Other Circular Functions Graphs of the Secant and Cosecant Functions Graphs of the Tangent and Cotangent Functions 8.8 Harmonic Motion Simple Harmonic Motion Damped Oscillatory Motion Reviewing Basic Concepts (Sections 8.7–8.8) Summary Review Exercises Test 9 Trigonometric Identities and Equations 9.1 Trigonometric Identities Fundamental Identities Using the Fundamental Identities Verifying Identities 9.2 Sum and Difference Identities Cosine Sum and Difference Identities Sine and Tangent Sum and Difference Identities Reviewing Basic Concepts (Sections 9.1–9.2) 9.3 Further Identities Double-Number Identities Product-to-Sum and Sum-to-Product Identities Half-Number Identities 9.4 The Inverse Circular Functions Review of Inverse Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Other Inverse Trigonometric Functions Inverse Function Values Reviewing Basic Concepts (Sections 9.3–9.4) 9.5 Trigonometric Equations and Inequalities (I) Equations Solvable by Linear Methods Equations Solvable by the Zero-Product Property and Quadratic Formula Methods Using Trigonometric Identities to Solve Equations 9.6 Trigonometric Equations and Inequalities (II) Equations and Inequalities Involving Multiple-Number Identities Equations and Inequalities Involving Half-Number Identities Applications of Trigonometric Equations Reviewing Basic Concepts (Sections 9.5–9.6) Summary Review Exercises Test 10 Applications of Trigonometry and Vectors 10.1 The Law of Sines Congruency and Oblique Triangles Derivation of the Law of Sines Using the Law of Sines Ambiguous Case 10.2 The Law of Cosines and Area Formulas Derivation of the Law of Cosines Using the Law of Cosines Area Formulas 10.3 Vectors and Their Applications Basic Terminology Interpretations of Vectors Operations with Vectors Dot Product and the Angle between Vectors Applications of Vectors Reviewing Basic Concepts (Sections 10.1–10.3) 10.4 Trigonometric (Polar) Form of Complex Numbers The Complex Plane and Vector Representation Trigonometric (Polar) Form Products of Complex Numbers in Trigonometric Form Quotients of Complex Numbers in Trigonometric Form 10.5 Powers and Roots of Complex Numbers Powers of Complex Numbers (De Moivre’s Theorem) Roots of Complex Numbers Reviewing Basic Concepts (Sections 10.4–10.5) 10.6 Polar Equations and Graphs Polar Coordinate System Graphs of Polar Equations Classifying Polar Equations Converting Equations 10.7 More Parametric Equations Parametric Graphing Revisited Parametric Equations with Trigonometric Functions The Cycloid Applications of Parametric Equations Reviewing Basic Concepts (Sections 10.6–10.7) Summary Review Exercises Test 11 Further Topics in Algebra 11.1 Sequences and Series Sequences Series and Summation Notation Summation Properties 11.2 Arithmetic Sequences and Series Arithmetic Sequences Arithmetic Series 11.3 Geometric Sequences and Series Geometric Sequences Geometric Series Infinite Geometric Series Annuities Reviewing Basic Concepts (Sections 11.1–11.3) 11.4 Counting Theory Fundamental Principle of Counting n-Factorial Permutations Combinations Distinguishing between Permutations and Combinations 11.5 The Binomial Theorem A Binomial Expansion Pattern Pascal’s Triangle Binomial Coefficients The Binomial Theorem rth Term of a Binomial Expansion Reviewing Basic Concepts (Sections 11.4–11.5) 11.6 Mathematical Induction Proof by Mathematical Induction Proving Statements Generalized Principle of Mathematical Induction Proof of the Binomial Theorem 11.7 Probability Basic Concepts Complements and Venn Diagrams Odds Union of Two Events Binomial Probability Reviewing Basic Concepts (Sections 11.6–11.7) Summary Review Exercises Test R: Reference: Basic Algebraic Concepts R.1 Review of Exponents and Polynomials Rules for Exponents Terminology for Polynomials Adding and Subtracting Polynomials Multiplying Polynomials R.2 Review of Factoring Factoring Out the Greatest Common Factor Factoring by Grouping Factoring Trinomials Factoring Special Products Factoring by Substitution R.3 Review of Rational Expressions Domain of a Rational Expression Lowest Terms of a Rational Expression Multiplying and Dividing Rational Expressions Adding and Subtracting Rational Expressions Complex Fractions R.4 Review of Negative and Rational Exponents Negative Exponents and the Quotient Rule Rational Exponents R.5 Review of Radicals Radical Notation Rules for Radicals Simplifying Radicals Operations with Radicals Rationalizing Denominators Test Appendix A: Geometry Formulas Appendix B: Vectors in Space Appendix C: Polar Form of Conic Sections Appendix D: Rotation of Axes Answers to Selected Exercises Index A B C D E F G H I J K L M N O P Q R S T U V W X Y Z