1. 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

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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

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

11. 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

12.  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!!!

13. 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

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