Graphical Approach to College Algebra and Trigonometry

This book, intended for a college algebra and trigonomety course, is the culmination of many years of teaching experience with the graphing calculator. In it, the authors treat the standard topics of college algebra and trigonometry by solving analytically, confirming graphically, and motivating through applications. Throughout the first five chapters, the authors present the various classes of functions studied in a standard college algebra and trigonometry text. Chapter One introduces functions and relations, using the linear function as the basis for the presentation. In this chapter, the authors introduce the following approach which is used throughout the next four chapters: after introducing a class of functions, the nature of its graph is examined, then the analytic solution of equations based on that function is discussed. Students are then shown how to provide graphical support for solutions using a graphing calculator. Having established these two methods of solving equations, the authors move on to the analytic methods of solving the associated inequalities. Students then learn how the analytic solutions of these inequalities can also be supported graphically. Finally, once the student has a feel for the particular class of function under consideration, the authors use analytic and graphical methods to solve interesting applications involving that function. By consistently using this approach with all the different classes of functions, students become aware that the authors are always following the same general procedure, and just applying that procedure to a new kind of function. Throughout the text, the authors emphasize the power of technology but provide numerous warnings on its limitations: the authors stress that it is only through the understanding of mathematical concepts that students can fully appreciate the power of graphing calculators and use technology appropriately.

Table of Contents:
(Each chapter (except Chapter R) concludes with a Chapter Summary, Review Exercises, Chapter Test, and Student Project.) 1. Rectangular Coordinates, Functions, and Analysis of Linear Functions. Real Numbers, Logic, and Coordinate Systems. Introduction to Relations and Functions. Linear Functions. Equations of Lines and Geometric Considerations. Solution of Linear Equations; Analytic Method and Graphical Support. Linear Models. Other Applications of Linear Functions. 2. Analysis of Graphs of Functions. Graphs of Elementary Functions and Relations. Vertical and Horizontal Shifts of Graphs of Functions. Stretching, Shrinking, and Reflecting Graphs of Functions. The Absolute Value Function: Graphs, Equations, Inequalities, and Applications. Piecewise Defined Functions Further Topics in the Study of Functions. 3. Polynomial Functions. Complex Numbers. Quadratic Functions and Their Graphs. Solution of Quadratic Equations and Inequalities. Applications of Quadratic Functions and Models. Higher Degree Polynomial Functions and Their Graphs. Topics in the Theory of Polynomial Functions (I). Topics in the Theory of Polynomial Functions (II). Solution of Polynomial Equations and Inequalities and Their Applications. 4. Rational and Root Functions; Inverse Functions. Graphs of Rational Functions. Rational Equations, Inequalities, and Applications. Graphs of Root Functions. Root Equations, Inequalities, and Applications. Inverse Functions. 5. Exponential and Logarithmic Functions. Introduction to Exponential Functions. Logarithms and Their Properties. Introduction to Logarithmic Functions. Exponential and Logarithmic Equations and Inequalities. Applications and Modeling with Exponential and Logarithmic Functions. 6. The Conic Sections and Parametric Equations. Circles and Parabolas. Ellipses and Hyperbolas. Summary of the Conic Sections. Parametric Equations. 7. Matrices and Systems of Equations and Inequalities. Systems of Equations. Solution of Linear Systems by the Echelon Method. Solution of Linear Systems by Row Transformations. Properties of Matrices. Determinants and Cramer's Rule. Solution of Linear Systems by Matrix Inverses. Systems of Inequalities and Linear Programming. Partial Fractions. 8. The Trigonometric Functions and Applications. Angles and Their Measures. The Trigonometric Functions and the Fundamental Identities. Evaluating Trigonometric Functions and the Fundamental Identities. Evaluating Trigonometric Functions. Applications of Right Triangle Trigonometry. The Circular Functions. Analysis of the Sine and Cosine Functions. Analysis of Other Circular Functions 9. Trigonometric Identities and Equations. Identities: Analytic Verification and Graphical Support. Further Identities. The Inverse Circular Functions. Trigonometric Equations and Inequalities. Further Equations and Inequalities. 11. Applications of Trigonometry: Vectors. The Law of Sines. The Law of Cosines and Area Formulas. Vectors and Their Applications. Trigonometric (Polar) Form of Complex Numbers. Powers and Roots of Complex Numbers. Polar Coordinates and Polar Inequalities. 11. Further Topics in Algebra. Sequences and Series. Arithmetic Sequences and Series. Geometric Sequences and Series. The Binomial Theorem. Mathematical Induction. Counting Theory. Probability. R. Reference: Basic Algebraic Concepts. Review of Rules for Exponents and Operations with Polynomials. Review of Factoring. Review of Operations with Rational Expressions. Review of Negative and Rational Exponents. Review of Radicals.