Do Now from 1.2b Find all values of x algebraically for which the given algebraic expression is not defined. Support your answer graphically. and

More Properties of Functions Sec. 1.2c: Symmetry, Asymptotes, End Behavior, Even and Odd Functions…

We will look at three types of symmetry: 1. Symmetry with respect to the y-axis Ex: f(x) = x 2 For all x in the domain of f, f(–x) = f(x) Functions with this property are called even functions

We will look at three types of symmetry: 2. Symmetry with respect to the x-axis Ex: x = y 2 Graphs with this symmetry are not functions, but we can say that (x, –y) is on the graph whenever (x, y) is on the graph

We will look at three types of symmetry: 3. Symmetry with respect to the origin Ex: f(x) = x 3 For all x in the domain of f, f(–x) = – f(x) Functions with this property are called odd functions

Guided Practice  Even  Neither Tell whether each of the following functions is odd, even, or neither (solve graphically and algebraically): First, check the graph, then verify algebraically…  Even  Neither

Guided Practice Tell whether each of the following functions is odd, even, or neither (solve graphically and algebraically): First, check the graph, then verify algebraically…  Odd

Definition: Horizontal and Vertical Asymptotes The line y = b is a horizontal asymptote of the graph of a function y = f(x) if f(x) approaches a limit of b as x approaches + or – . In limit notation: 8 8 or

Definition: Horizontal and Vertical Asymptotes The line x = a is a vertical asymptote of the graph of a function y = f(x) if f(x) approaches a limit of + or – as x approaches a from either direction. In limit notation: 8 8 or

Consider the graph of: Do the definitions of asymptotes make sense in this graph?

Vertical Asymptotes at x = –1 and x = 2, Horizontal Asymptote at y = 0 Guided Practice Identify any horizontal or vertical asymptotes of the graph of: Vertical Asymptotes at x = –1 and x = 2, Horizontal Asymptote at y = 0

 Take a look at Example 11 on p.97!!! End Behavior Sometimes, it is useful to consider the “end behavior” of a function. That is, what does the function look like as the dependent variable (x) approaches infinity?  Take a look at Example 11 on p.97!!!

Whiteboard practice: State whether the function is odd, even, or neither. Support graphically and confirm algebraically. g(-x)=-g(x) therefore…ODD State whether the function is odd, even, or neither. Support graphically and confirm algebraically. g(-x)=g(x) therefore…EVEN

Whiteboard practice: Homework: p. 99 47-61 odd Using any method, find all asymptotes. Homework: p. 99 47-61 odd