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Even & Odd Functions: Basic Overview

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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.

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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.

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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.

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

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

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Even & Odd Functions Degree: highest exponent of the function Constants are considered to be even! Even degrees: Odd degrees:

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

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

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

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

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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!

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Case #1: Even Degree, LC > 0 Example: Both ends go toward +∞

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

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

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Case #4: Odd Degree, LC < 0 Example: “opposites”

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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.

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#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.

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#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.

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#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.

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#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.

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Determine if the following are even, odd, or neither. (Do these on your paper and check your answers on the next slide)

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

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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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