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Even & Odd Functions: Basic Overview. Reflection Symmetry  Reflection Symmetry (sometimes called Line Symmetry or Mirror Symmetry) is easy to recognize,

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Presentation on theme: "Even & Odd Functions: Basic Overview. Reflection Symmetry  Reflection Symmetry (sometimes called Line Symmetry or Mirror Symmetry) is easy to recognize,"— Presentation transcript:

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:

8 Even Functions  EVEN => All exponents are EVEN Example:  y-axis symmetry

9 Odd Functions  ODD => All exponents are ODD Example:  origin symmetry

10 NEITHER even nor odd  NEITHER => Mix of even and odd exponents Examples:

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 +∞

14 Case #2: Even Degree, LC < 0  Example:  Both ends go toward -∞

15 Case #3: Odd Degree, LC > 0  Example: “match”

16 Case #4: Odd Degree, LC < 0  Example: “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)

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.


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