Precalculus, 2nd Edition

Three components contribute to a theme sustained throughout the Coburn Series: that of laying a firm foundation, building a solid framework, and providing strong connections. Not only does Coburn present a sound problem-solving process to teach students to recognize a problem, organize a procedure, and formulate a solution, the text encourages students to see beyond procedures in an effort to gain a greater understanding of the big ideas behind mathematical concepts. Written in a readable, yet mathematically mature manner appropriate for college algebra level students, Coburn's Precalculus uses narrative, extensive examples, and a range of exercises to connect seemingly disparate mathematical topics into a cohesive whole. Coburn's hallmark applications are born out of the author's extensive experiences in and outside the classroom, and appeal to the vast diversity of students and teaching methods in this course area. Benefiting from the feedback of hundreds of instructors and students across the country, Precalculus second edition, continues to emphasize connections in order to improve the level of student engagement in mathematics and increase their chances of success in college algebra. Cover Page......Page 1 Title Page......Page 2 Copyright Page......Page 3 Brief Contents......Page 4 Dedication......Page 5 Preface......Page 7 Acknowledgments......Page 26 Contents......Page 28 Index of Applications......Page 36 CHAPTER 1: Equations and Inequalities......Page 42 1.1 Linear Equations, Formulas, and Problem Solving......Page 43 Technology Highlight: Using a Graphing Calculator as an Investigative Tool......Page 50 1.2 Linear Inequalities in One Variable......Page 55 1.3 Absolute Value Equations and Inequalities......Page 65 Technology Highlight: Absolute Value Equations and Inequalities......Page 69 Mid-Chapter Check......Page 72 Reinforcing Basic Concepts: Using Distance to Understand Absolute Value Equations and Inequalities......Page 73 1.4 Complex Numbers......Page 74 1.5 Solving Quadratic Equations......Page 83 Technology Highlight: The Discriminant......Page 92 1.6 Solving Other Types of Equations......Page 97 Summary and Concept Review......Page 111 Practice Test......Page 116 Calculator Exploration and Discovery: Evaluating Expressions and Looking for Patterns......Page 117 Strengthening Core Skills: An Alternative Method for Checking Solutions to Quadratic Equations......Page 118 Connections to Calculus: Solving Various Types of Equations; Absolute Value Inequalities and Delta/Epsilon form......Page 120 CHAPTER 2: Relations, Functions, and Graphs......Page 124 2.1 Rectangular Coordinates; Graphing Circles and Other Relations......Page 125 Technology Highlight: The Graph of a Circle......Page 133 2.2 Graphs of Linear Equations......Page 138 Technology Highlight: Linear Equations, Window Size, and Friendly Windows......Page 146 2.3 Linear Graphs and Rates of Change......Page 151 2.4 Functions, Function Notation, and the Graph of a Function......Page 163 Mid-Chapter Check......Page 178 2.5 Analyzing the Graph of a Function......Page 179 Technology Highlight: Locating Zeroes, Maximums, and Minimums......Page 190 2.6 The Toolbox Functions and Transformations......Page 198 Technology Highlight: Function Families......Page 207 2.7 Piecewise-Defined Fun......Page 213 Technology Highlight: Piecewise-Defined Functions......Page 220 2.8 The Algebra and Composition of Functions......Page 227 Technology Highlight: Composite Functions......Page 237 Summary and Concept Review......Page 243 Mixed Review......Page 250 Practice Test......Page 252 Calculator Exploration and Discovery: Using a Simple Program to Explore Transformations......Page 253 Strengthening Core Skills: Transformations via Composition......Page 254 Cumulative Review: Chapters 1–2......Page 255 Connections to Calculus: Rates of Change and the Difference Quotient; Transformations and the Area Under a Curve......Page 256 CHAPTER 3: Polynomial and Rational Functions......Page 260 3.1 Quadratic Functions and Applications......Page 261 Technology Highlight: Estimating Irrational Zeroes......Page 266 3.2 Synthetic Division; The Remainder and Factor Theorems......Page 271 3.3 The Zeroes of Polynomial Functions......Page 283 Technology Highlight: The Intermediate Value Theorem and Split Screen Viewing......Page 293 3.4 Graphing Polynomial Functions......Page 298 Reinforcing Basic Concepts: Approximating Real Zeroes......Page 312 3.5 Graphing Rational Functions......Page 313 Technology Highlight: Rational Functions and Appropriate Domains......Page 323 3.6 Additional Insights into Rational Functions......Page 330 Technology Highlight: Removable Discontinuities......Page 338 3.7 Polynomial and Rational Inequalities......Page 344 Technology Highlight: Polynomial and Rational Inequalities......Page 351 3.8 Variation: Function Models in Action......Page 357 Summary and Concept Review......Page 367 Mixed Review......Page 372 Practice Test......Page 373 Strengthening Core Skills: Solving Inequalities Using the Push Principle......Page 375 Cumulative Review: Chapters 1–3......Page 376 Connections to Calculus: Graphing Techniques......Page 378 CHAPTER 4: Exponential and Logarithmic Functions......Page 382 4.1 One-to-One and Inverse Functions......Page 383 Technology Highlight: Investigating Inverse Functions......Page 390 4.2 Exponential Functions......Page 395 Technology Highlight: Solving Exponential Equations Graphically......Page 402 4.3 Logarithms and Logarithmic Functions......Page 407 Mid-Chapter Check......Page 420 Reinforcing Basic Concepts: Linear and Logarithm Scales......Page 421 4.4 Properties of Logarithms; Solving Exponential/Logarithmic Equations......Page 422 4.5 Applications from Business, Finance, and Science......Page 438 Technology Highlight: Exploring Compound Interest......Page 445 Summary and Concept Review......Page 451 Mixed Review......Page 455 Practice Test......Page 456 Calculator Exploration and Discovery: Investigating Logistic Equations......Page 457 Cumulative Review: Chapters 1–4......Page 459 Connections to Calculus: Properties of Logarithms; Area Functions; Expressions Involving e^x......Page 462 CHAPTER 5: An Introduction to Trigonometric Functions......Page 466 5.1 Angle Measure, Special Triangles, and Special Angles......Page 467 5.2 Unit Circles and the Trigonometry of Real Numbers......Page 481 5.3 Graphs of Sine and Cosine Functions; Cosecant and Secant Functions......Page 495 Technology Highlight: Exploring Amplitudes and Periods......Page 506 5.4 Graphs of Tangent and Cotangent Functions......Page 512 Technology Highlight: Zeroes, Asymptotes, and the Tangent/Cotangent Functions......Page 519 Mid-Chapter Check......Page 525 Reinforcing Basic Concepts: Trigonometry of the Real Numbers and the Wrapping Function......Page 526 5.5 Transformations and Applications of Trigonometric Graphs......Page 527 Technology Highlight: Locating Zeroes, Roots, and x-Intercepts......Page 535 5.6 The Trigonometry of Right Triangles......Page 540 5.7 Trigonometry and the Coordinate Plane......Page 553 Summary and Concept Review......Page 563 Mixed Review......Page 571 Practice Test......Page 573 Calculator Exploration and Discovery: Variable Amplitudes and Modeling the Tides......Page 575 Strengthening Core Skills: Standard Angles, Reference Angles, and the Trig Functions......Page 576 Cumulative Review: Chapters 1–5......Page 577 Connections to Calculus: Right Triangle Relationships; Algebraic Form and Polar Form......Page 580 CHAPTER 6: Trigonometric Identities, Inverses, and Equations......Page 584 6.1 Fundamental Identities and Families of Identities......Page 585 6.2 Constructing and Verifying Identities......Page 593 6.3 The Sum and Difference Identities......Page 599 6.4 The Double-Angle, Half-Angle, and Product-to-Sum Identities......Page 609 Mid-Chapter Check......Page 621 Reviewing Basic Concepts: Identities—Connections and Relationships......Page 622 6.5 The Inverse Trig Functions and Their Applications......Page 623 Technology Highlight: More on Inverse Functions......Page 633 6.6 Solving Basic Trig Equations......Page 640 Technology Highlight: Solving Equations Graphically......Page 646 6.7 General Trig Equations and Applications......Page 651 Summary and Concept Review......Page 660 Mixed Review......Page 664 Practice Test......Page 666 Calculator Exploration and Discovery: Seeing the Beats as the Beats Go On......Page 667 Strengthening Core Skills: Trigonometric Equations and Inequalities......Page 668 Cumulative Review: Chapters 1–6......Page 669 Connections to Calculus: Simplifying Expressions Using a Trigonometric Substitution; Trigonometric Identities and Equations......Page 671 CHAPTER 7: Applications of Trigonometry......Page 674 7.1 Oblique Triangles and the Law of Sines......Page 675 7.2 The Law of Cosines; the Area of a Triangle......Page 687 7.3 Vectors and Vector Diagrams......Page 699 Technology Highlight: Vector Components Given the Magnitude and the Angle ......Page 709 Reinforcing Basic Concepts: Scaled Drawings and the Laws of Sine and Cosine......Page 714 7.4 Vector Applications and the Dot Product......Page 715 7.5 Complex Numbers in Trigonometric Form......Page 728 7.6 De Moivre’s Theorem and the Theorem on nth Roots......Page 739 Summary and Concept Review......Page 746 Mixed Review......Page 750 Practice Test......Page 751 Calculator Exploration and Discovery: Investigating Projectile Motion......Page 753 Cumulative Review: Chapters 1–7......Page 754 Connections to Calculus: Trigonometry and Problem Solving; Vectors in Three Dimensions......Page 756 CHAPTER 8: Systems of Equations and Inequalities......Page 760 8.1 Linear Systems in Two Variables with Applications......Page 761 Technology Highlight: Solving Systems Graphically......Page 768 8.2 Linear Systems in Three Variables with Applications......Page 773 Technology Highlight: More on Parameterized Solutions......Page 780 8.3 Partial Fraction Decomposition......Page 784 8.4 Systems of Inequalities and Linear Programming......Page 796 Technology Highlight: Systems of Linear Inequalities......Page 804 Reinforcing Basic Concepts: Window Size and Graphing Technology......Page 809 8.5 Solving Linear Systems Using Matrices and Row Operations......Page 810 Technology Highlight: Solving Systems Using Matrices and Calculating Technology......Page 816 8.6 The Algebra of Matrices......Page 821 8.7 Solving Linear Systems Using Matrix Equations......Page 832 8.8 Applications of Matrices and Determinants: Cramer’s Rule, Geometry, and More......Page 847 Summary and Concept Review......Page 855 Mixed Review......Page 860 Practice Test......Page 861 Calculator Exploration and Discovery: Optimal Solutions and Linear Programming......Page 863 Strengthening Core Skills: Augmented Matrices and Matrix Inverses......Page 864 Cumulative Review: Chapters 1–8......Page 866 Connections to Calculus: More on Partial Fraction Decomposition; The Geometry of Vector and Determinants......Page 868 CHAPTER 9: Analytical Geometry and the Conic Sections......Page 872 9.1 A Brief Introduction to Analytic Geometry......Page 873 9.2 The Circle and the Ellipse......Page 880 9.3 The Hyperbola......Page 893 Technology Highlight: Studying Hyperbolas......Page 902 9.4 The Analytic Parabola......Page 907 Mid-Chapter Check......Page 915 9.5 Nonlinear Systems of Equations and Inequalities......Page 916 9.6 Polar Coordinates, Equations, and Graphs......Page 924 9.7 More on Conic Sections: Rotation of Axes and Polar Form......Page 937 Technology Highlight: Investigating the Eccentricity e......Page 948 9.8 Parametric Equations and Graphs......Page 954 Technology Highlight: Exploring Parametric Graphs......Page 960 Summary and Concept Review......Page 965 Mixed Review......Page 970 Practice Test......Page 971 Calculator Exploration and Discovery: Conic Rotations in Polar Form......Page 972 Strengthening Core Skills: Simplifying and Streamlining Computations for the Rotation of Axes......Page 973 Cumulative Review: Chapters 1–9......Page 975 Connections to Calculus: Polar Graphs and Instantaneous Rates of Change; Systems of Polar Equations......Page 976 CHAPTER 10: Additional Topics in Algebra......Page 980 10.1 Sequences and Series......Page 981 Technology Highlight: Studying Sequences and Series......Page 986 10.2 Arithmetic Sequences......Page 990 10.3 Geometric Sequences......Page 997 10.4 Mathematical Induction......Page 1007 Mid-Chapter Check......Page 1014 Reinforcing Basic Concepts: Applications of Summation......Page 1015 10.5 Counting Techniques......Page 1016 Technology Highlight: Calculating Permutations and Combinations......Page 1022 10.6 Introduction to Probability......Page 1028 Technology Highlight: Principles of Quick-Counting, Combinations, and Probability......Page 1033 10.7 The Binomial Theorem......Page 1040 Summary and Concept Review......Page 1048 Mixed Review......Page 1052 Practice Test......Page 1054 Calculator Exploration and Discovery: Infinite Series, Finite Results......Page 1055 Strengthening Core Skills: Probability, Quick Counting, and Card Games......Page 1056 Cumulative Review: Chapters 1–10......Page 1057 Connections to Calculus: Applications of Summation......Page 1060 CHAPTER 11: Bridges to Calculus–An Introduction to Limits......Page 1064 11.1 An Introduction to Limits Using Tables and Graphs......Page 1065 11.2 The Properties of Limits......Page 1075 Mid-Chapter Check......Page 1084 11.3 Continuity and More on Limits......Page 1085 11.4 Applications of Limits: Instantaneous Rates of Change and the Area Under a Curve......Page 1097 Summary and Concept Review......Page 1107 Mixed Review......Page 1110 Practice Test......Page 1111 Cumulative Review: Chapters 1–11......Page 1112 The Language, Notation, Numbers of Mathematics......Page 1114 Algebraic Expressions and the Properties of Real Numbers......Page 1121 Exponents, Scientific Notation, and a Review of Polynomials......Page 1127 Factoring Polynomials......Page 1137 Rational Expressions......Page 1144 Radicals and Rational Exponents......Page 1153 Appendix II: More on Synthetic Division......Page 1167 Appendix III: More on Matrices......Page 1169 Appendix IV: Deriving the Equation of a Conic......Page 1171 Appendix V: Selected Proofs......Page 1173 Appendix VI: Families of Polar Curves......Page 1176 Student Answer Appendix......Page 1179 Index......Page 1245