Even & Odd Functions:Basic Overview

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.

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.

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.

Examples of Lines of SymmetryLine of Symmetry Sample Artwork Example Shape

Examples of Lines of SymmetryLine of Symmetry Sample Artwork Example Shape

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

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( ) 53

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’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!

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. Explain why you chose your answer.

#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.

#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.

#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.

#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.

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( ) 43

f x x x

2( ) 5 9f x x

3( ) 2f x x x

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, 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.