1. Trigonometry Cloud County Community College Spring, 2012 Instructor: Timothy L. Warkentin

2. Course Overview The importance of study and completion of homework. Resources on the Cloud website: syllabus (homework assignments), chapter outlines, homework solutions, handouts, and class notes. The importance of memorization in the study of Trigonometry. Using technology in Calculus I (TI-84 calculators, Graphing Calculator, Internet Resources). Lab: Introduction to Wolfram Alpha

3. Chapter 1: Functions and Graphs 1.1 Equations and Inequalities 1.2 A Two-Dimensional Coordinate System and Graphs 1.3 Introduction to Functions 1.4 Properties of Graphs 1.5 The Algebra of Functions 1.6 Inverse Functions 1.7 Modeling Data Using Regression

4. Chapter 1 Overview Chapter 1 reviews important material needed to begin a study of Trigonometry. A through understanding of functions, function inverses and the notation used in their descriptions is an essential prerequisite to any understanding of Trigonometry.

5. 1.1: Equations and Inequalities 1 The Complex Number System

6. 1.1: Equations and Inequalities 2 Three ways of writing set solutions: –Graphing –Interval Notation –Set Builder Notation The Absolute Value of a Number: the distance the number is from the origin. The Distance Between Two Numbers: the absolute value of the difference between the numbers. Any equation that can be put into the form ax + b = 0 is a Linear Equation. Example 1

7. 1.1: Equations and Inequalities 3 Solving Literal Equations: Example 2 –Clear denominators. –Complete multiplications. –Separate terms with target variable over the equality sign from other terms. –Factor out the target variable. –Divide to isolate the target variable. Solving Quadratic Equations: Examples 3 & 4 –Taking square roots. –Factoring (Zero Product Property). –Completing the Square. –Quadratic Formula.

8. 1.1: Equations and Inequalities 4 Solving Inequalities: Examples 5 & 6 –When an inequality is multiplied or divided by a negative number the direction of the inequality changes. –The Critical Point Method. –The Graphical Method. Solving Absolute Value Inequalities: Examples 7 & 8 –Using sign switches. –Using the distance between two points and a number line.

9. 1.2: A Two-Dimensional Coordinate System and Graphs 1 The Cartesian Coordinate System. The Distance and Midpoint Formulas. Example 1 Graphing by using points. Examples 2-4 Graphing using the TI-84 calculator. Examples 2-4 Finding x (set y = 0) and y (set x = 0) intercepts. Example 5 Finding x intercepts with a TI-84 calculator. Example 5 The Equation of a Circle Example 6 –Switching from Standard Form to General Form (expand the squares) –Switching from General Form to Standard Form (double completion of the square) Example 7

10. 1.3: Introduction to Functions 1 Relations, Functions and 1-to-1 Functions

11. 1.3: Introduction to Functions 2 Domain: pre-images, x-values, independent values, inputs. Range: images, y-values, dependent values, outputs. Function: Every pre-images has exactly one image. Functions can be described using function notation, ordered pairs, Venn diagrams, input/output machine diagrams, and tables. Function notation and dummy variables. Example 1 Piecewise Functions (TI-84 calculators). Example 2 Identifying Functions (The Vertical Line Test - VLT). Examples 3 & 6

12. 1.3: Introduction to Functions 3 Domain Issues: Example 4 –Division by Zero –Even Roots of Negative Numbers –Physical Constraints Graphing functions using points (TI-84 tables) Example 5 Graphing functions using the TI-84 calculator and Graphing Calculator. Increasing and Decreasing Functions. 1-to-1 functions (The Horizontal Line Test – HLT) The Greatest Integer Function (The Floor Function, TI-84 int( command) Example 7 Using functions to solve applications. Examples 8-10

13. 1.4: Properties of Graphs 1 Relation Symmetry: Examples 1 & 2 –y-axis symmetry: -x replacing x yields equivalent equation. –x-axis symmetry: -y replacing y yields equivalent equation. –Origin symmetry: -x replacing x & -y replacing y yields equivalent equation. Function Parity: Example 3 –Even: f[-x] = f[x] –Odd: f[-x] = -f[x] Function Translation: Example 4 –Vertical: g[x] = f[x] + k is f[x] translated vertically by k units. –Horizontal: g[x] = f[x-h] is f[x] translated horizontally by h units.

14. 1.4: Properties of Graphs 2 Function Reflection: Example 5 –-f[x] is f[x] reflected over the x-axis. –f[-x] is f[x] reflected over the y-axis. Function Vertical Elasticity: Example 6 –If a > 1 then g[x] = a·f[x] is f[x] stretched vertically by a factor of a. –If 0 < a < 1 then g[x] = a·f[x] is f[x] compressed vertically by a factor of a.

15. 1.4: Properties of Graphs 3 Function Horizontal Elasticity: Example 7 –If a > 1 then g[x] = f[a·x] is f[x] compressed horizontally by a factor of 1/a. –If 0 < a < 1 then g[x] = f[a·x] is f[x] stretched horizontally by a factor of 1/a. Function Transformation: –a is the vertical scaling factor. –w is the horizontal scaling factor. –h is the horizontal shift. –k is the vertical shift.

16. 1.5: The Algebra of Functions 1 Operations on Functions Examples 2, 5, 6 & 7 Domains of Combined Functions Example 1 The Difference Quotient: Examples 3 & 4

17. 1.6: Inverse Functions 1 The inverse of a Relation is that Relation that switches the order of the ordered pair elements. Every Relation has an Inverse. A Function will have an Inverse Function IFF it is a 1-to-1 Function. Identifying 1-to-1 Functions (The Horizontal Line Test - HLT). Example 1 Proving that a pair of functions are inverses. Example 2 Finding an Inverse (Switch Method). Examples 3 & 4 Restricting the domain of a function (domain surgery). Examples 5 & 6

18. 1.7: Modeling Data Using Regression 1 Linear Regression Models. Example 1 The Correlation Coefficient. The Coefficient of Determination. Quadratic Regression Models. Example 2 Using the TI-84 to model data.