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Even & Odd Functions:
Basic Overview
Reflection Symmetry
 Reflection Symmetry (sometimes called Line
Symmetry or Mirror Symmetry) is easy to
recognize, becaus...
Reflection Symmetry
 The reflection in this lake also has
symmetry, but in this case:
 the Line of Symmetry is the horiz...
Line of Symmetry
 The Line of Symmetry (also called the
Mirror Line) does not have to be up-
down or left-right, it can b...
Examples of Lines of Symmetry
Line of Symmetry Sample Artwork Example Shape
Examples of Lines of Symmetry
Line of Symmetry Sample Artwork Example Shape
Even & Odd Functions
 Degree: highest exponent of the
function
 Constants are considered to be even!
 Even degrees:
 O...
Even Functions
 EVEN => All exponents are EVEN
 Example:
 y-axis symmetry
( ) ( )f x f x 
2
( ) 7f x x 
Odd Functions
 ODD => All exponents are ODD
 Example:
 origin symmetry
( ) ( )f x f x  
3
( ) 3f x x x 
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 
Leading Coefficient (LC)
 The coefficient of the term with the
highest exponent
 2 Cases:
 LC > 0
 LC < 0
 Agree?!?!
End Behavior
 What happens to f(x) or y as x
approaches -∞ and +∞
 We can figure this out quickly by
the two things we’v...
Case #1: Even Degree, LC > 0
 Example:
 Both ends go toward +∞
2
( )f x x
Case #2: Even Degree, LC < 0
 Example:
 Both ends go toward -∞
2
( )f x x 
Case #3: Odd Degree, LC > 0
 Example:
3
( )f x x
“match”
, ( )x f x   
, ( )x f x   
Case #4: Odd Degree, LC < 0
 Example:
3
( )f x x 
, ( )x f x   
, ( )x f x   
“opposites”
Show what you know…
1. Determine if the following functions
are even, odd, or neither by
analyzing their graphs.
2. Explai...
#1
Answer:
This function is neither
even nor odd. I chose
this answer because it is
not symmetrical with
respect to the or...
#2
Answer:
This function is neither even nor
odd. I chose this answer because
it is not symmetrical with respect
to the or...
#3
Answer:
This is an even
function. I know this
because it is
symmetrical with
respect to the y-axis.
In other words, I
c...
#4
Answer:
This is an even
function. I know
this because it is
symmetrical with
respect to the y-
axis. In other
words, I ...
Determine if the following are even, odd, or
neither. (Do these on your paper and check your
answers on the next slide)
5....
Answers:
 5. Even
 6. Odd
 7. Neither
 8. Neither
 9. Even
 10. odd
Answer the following:
(submit these answers in the assignment drop box)
11. Explain how you know a function
is even, odd, ...
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Even and off functions basic presentation with questions 2

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Even and off functions basic presentation with questions 2

  1. 1. Even & Odd Functions: Basic Overview
  2. 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. 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. 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. 5. Examples of Lines of Symmetry Line of Symmetry Sample Artwork Example Shape
  6. 6. Examples of Lines of Symmetry Line of Symmetry Sample Artwork Example Shape
  7. 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. 8. Even Functions  EVEN => All exponents are EVEN  Example:  y-axis symmetry ( ) ( )f x f x  2 ( ) 7f x x 
  9. 9. Odd Functions  ODD => All exponents are ODD  Example:  origin symmetry ( ) ( )f x f x   3 ( ) 3f x x x 
  10. 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. 11. Leading Coefficient (LC)  The coefficient of the term with the highest exponent  2 Cases:  LC > 0  LC < 0  Agree?!?!
  12. 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. 13. Case #1: Even Degree, LC > 0  Example:  Both ends go toward +∞ 2 ( )f x x
  14. 14. Case #2: Even Degree, LC < 0  Example:  Both ends go toward -∞ 2 ( )f x x 
  15. 15. Case #3: Odd Degree, LC > 0  Example: 3 ( )f x x “match” , ( )x f x    , ( )x f x   
  16. 16. Case #4: Odd Degree, LC < 0  Example: 3 ( )f x x  , ( )x f x    , ( )x f x    “opposites”
  17. 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. 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. 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. 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. 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. 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. 23. Answers:  5. Even  6. Odd  7. Neither  8. Neither  9. Even  10. odd
  24. 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.

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