Odd, Even

1. Even & Odd Functions:
Basic Overview

2. Reflection Symmetry
 Reflection Symmetry (sometimes called Line
Symmetry or Mirror Symmetry) is easy to
recognize, because one half is the reflection of
the other half.
 Here is a dog. Her face
made perfectly
symmetrical with a bit
of photo magic.
 The white line down
the center is the Line
of Symmetry.

3. Reflection Symmetry
 The reflection in this lake also has
symmetry, but in this case:
 the Line of Symmetry is the horizon
 it is not perfect symmetry, because the
image is changed a little by the lake surface.

4. Line of Symmetry
 The Line of Symmetry (also called the
Mirror Line) does not have to be up-
down or left-right, it can be in any
direction.
~But there are four
common directions, and
they are named for the
line they make on the
standard XY graph.

5. Examples of Lines of Symmetry
Line of Symmetry Sample Artwork Example Shape

6. Examples of Lines of Symmetry
Line of Symmetry Sample Artwork Example Shape

7. Even & Odd Functions
 Degree: highest exponent of the
function
 Constants are considered to be even!
 Even degrees:
 Odd degrees:
( )f x x 3
( ) 2f x x
2
( ) 5f x x 0
( ) 4 4*1 4f x x  

8. Even Functions
 EVEN => All exponents are EVEN
 Example:
 y-axis symmetry
( ) ( )f x f x 
2
( ) 7f x x 

9. Odd Functions
 ODD => All exponents are ODD
 Example:
 origin symmetry
( ) ( )f x f x  
3
( ) 3f x x x 

10. NEITHER even nor odd
 NEITHER => Mix of even and odd
exponents
 Examples:
4 32
( ) 5
3
f x x x 
3
( ) 6 2f x x 

11. Leading Coefficient (LC)
 The coefficient of the term with the
highest exponent
 2 Cases:
 LC > 0
 LC < 0
 Agree?!?!

12. End Behavior
 What happens to f(x) or y as x
approaches -∞ and +∞
 We can figure this out quickly by
the two things we’ve already
discussed
 Degree of function (even or odd)
 Leading coefficient (LC)
 Let’s look at our 4 cases…jot these
down in your graphic organizer!

13. Case #1: Even Degree, LC > 0
 Example:
 Both ends go toward +∞
2
( )f x x

14. Case #2: Even Degree, LC < 0
 Example:
 Both ends go toward -∞
2
( )f x x 

15. Case #3: Odd Degree, LC > 0
 Example:
3
( )f x x
“match”
, ( )x f x   
, ( )x f x   

16. Case #4: Odd Degree, LC < 0
 Example:
3
( )f x x 
, ( )x f x   
, ( )x f x   
“opposites”

17. Show what you know…
1. Determine if the following functions
are even, odd, or neither by
analyzing their graphs.
2. Explain why you chose your answer.

18. #1
Answer:
This function is neither
even nor odd. I chose
this answer because it is
not symmetrical with
respect to the origin or
the y-axis.

19. #2
Answer:
This function is neither even nor
odd. I chose this answer because
it is not symmetrical with respect
to the origin or the y-axis.

20. #3
Answer:
This is an even
function. I know this
because it is
symmetrical with
respect to the y-axis.
In other words, I
could fold it at the y-
axis and it is
symmetrical.

21. #4
Answer:
This is an even
function. I know
this because it is
symmetrical with
respect to the y-
axis. In other
words, I could
fold it at the y-
axis and it is
symmetrical.

22. Determine if the following are even, odd, or
neither. (Do these on your paper and check your
answers on the next slide)
5.
6.
7.
8.
9.
10.
2
( ) 3 4f x x 
3
( ) 2 4f x x x 
2 3
( ) 3 2 4 4f x x x x    
2 32
( ) 4
3
f x x x  
2
( ) 5 9f x x  
3
( ) 2f x x x 

23. Answers:
 5. Even
 6. Odd
 7. Neither
 8. Neither
 9. Even
 10. odd

24. Answer the following:
(submit these answers in the assignment drop box)
11. Explain how you know a function
is even, odd, or neither when you are
looking at the graph? (like in questions 1-4)
12. Explain how you know a function
is even, odd, or neither when you are
looking at the equation? (like in questions 5-10)
13. Write an even function.
14. Write an odd function.
15. Write a function that is neither.