2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions Tests for Symmetry Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

x-axis symmetry: Not equivalent y-axis symmetry:Origin symmetry: 2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions Tests for Symmetry Example:  No x-axis symmetry. Equivalent  y-axis symmetry. Not equivalent  No origin symmetry.

2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions Tests for Symmetry Example: No x-axis symmetry. y-axis symmetry. No origin symmetry.

2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions Even and Odd Functions y-axis symmetry. origin symmetry. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions Even and Odd Functions Identify the type of symmetry for each graph y-axis symmetry. no symmetry. Even function origin symmetry. Odd function Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. function

2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions Tests for Even and Odd Functions odd function  Identify the type of symmetry for a function function  even function 

Defn: Piecewise Function A function with different equations for different domains. Examples 2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions Piecewise Functions

2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions Piecewise Functions

2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions Piecewise Functions

2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions Piecewise Functions

2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions Piecewise Functions

– Analyzing Graphs of Functions and Piecewise Defined Functions Piecewise Functions 2

2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions Piecewise Functions

2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions For all real numbers x, the greatest integer function returns the largest integer less than or equal to x. The greatest integer function rounds down a real number to the nearest integer.

2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions Intervals of Increasing, Decreasing and Constant Behavior Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Where is the function constant? 2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions Intervals of Increasing, Decreasing and Constant Behavior Where is the function increasing? Where is the function decreasing?

2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions Relative (Local) Maximum and Minimum of a Function

2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions Relative (Local) Maxima and Minima of a Function Relative Maxima Relative Minima Relative Maxima Value Relative Minima Value

2.8 – Algebra of Functions and Function Composition Operations on Functions

2.8 – Algebra of Functions and Function Composition Function Domains Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Find the domain of each of the following functions

The Difference Quotient is used to find the average rate of change between two points. The Difference Quotient also represents to slope of the secant line between two points on a curve.   Secant line 2.8 – Algebra of Functions and Function Composition Difference Quotient

Find the difference quotient 2.8 – Algebra of Functions and Function Composition Difference Quotient

Find the difference quotient 2.8 – Algebra of Functions and Function Composition Difference Quotient

The process of combining two or more functions in order to create another function. One function is evaluated at a value of the independent variable and the result is substituted into the other function as the independent variable. The composition of functions f and g is written as: The composition of functions is a function inside another function. 2.8 – Algebra of Functions and Function Composition Composition of Functions

2.8 – Algebra of Functions and Function Composition Composition of Functions

2.8 – Algebra of Functions and Function Composition Composition of Functions