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Trigonometry

- Cloud County Community College
- Spring, 2012
- Instructor Timothy L. Warkentin

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

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

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.

1.1 Equations and Inequalities 1

- The Complex Number System

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

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.

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.

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

1.3 Introduction to Functions 1

- Relations, Functions and 1-to-1 Functions

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

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

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 fx
- Odd f-x -fx
- Function Translation Example 4
- Vertical gx fx k is fx translated

vertically by k units. - Horizontal gx fx-h is fx translated

horizontally by h units.

1.4 Properties of Graphs 2

- Function Reflection Example 5
- -fx is fx reflected over the x-axis.
- f-x is fx reflected over the y-axis.
- Function Vertical Elasticity Example 6
- If a gt 1 then gx afx is fx stretched

vertically by a factor of a. - If 0 lt a lt 1 then gx afx is fx

compressed vertically by a factor of a.

1.4 Properties of Graphs 3

- Function Horizontal Elasticity Example 7
- If a gt 1 then gx fax is fx compressed

horizontally by a factor of 1/a. - If 0 lt a lt 1 then gx fax is fx 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.

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

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

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.