1. Symmetry of Functions Even, Odd, or Neither?

2. Even Functions All exponents are even. May contain a constant. f(x) = f(-x) Symmetric about the y-axis

3. All even exponents Example: Both exponents are even. It does not matter what the coefficients are.

4. May Contain a Constant Example Even exponents (coefficients don’t matter) Constant does not affect even function.

5. f(x) = f(-x) Given f(x) = 5x² - 7, find f(-x) to determine if f(x) is even, odd, or neither. 1)Substitute –x for x. 2)f(-x) = 5(-x)² - 7 = 5x² -7 3)Because f(x) = f(-x), f(x) = 5x² - 7 is an even function.

6. f(x) = f(-x) Given f(x) = 4x² - 2x + 1, determine if f(x) is even, odd, or neither. 1)Substitute –x for x. 2)f(-x) = 4(-x)² - 2(-x) + 1 = 4x² +2x + 1 3)f(-x) ≠ f(x) 4)Therefore, f(x) is NOT an even function. (We will revisit to determine what it is.)

7. Symmetric About the y-axis The following are symmetric about the y- axis.

8. Odd Functions Only odd exponents. NO constants! f(-x) = -f(x) Symmetric about the origin.

9. All Odd Exponents Example All odd exponents. Understood 1 exponent

10. NO Constants Example: Odd exponents NO constants in odd functions!

11. f(-x) = -f(x) Given f(x) = 4x³ + 2x, find f(-x) and f(- x) to determine if f(x) is even, odd, or neither. f(-x) = 4(-x)³ + 2(-x) = -4x³ - 2x -f(x) = -4x³ - 2x Because f(-x) = -f(x), f(x) is an odd function.

12. f(-x) = -f(x) Given f(x) = 5x³ + 7x², find f(-x) and f(-x) to determine if f(x) is even, odd, or neither. f(-x) = 5(-x)³ + 7(-x)² = -5x³ + 7x² -f(x) = -5x³ - 7x² f(-x) ≠ -f(x), therefore f(x) is NOT an odd function.

13. Symmetric About the Origin These graphs are symmetric about the origin.

14. Neither? Mixture of even and odd exponents. All odd exponents with a constant. f(x) ≠ f(-x) AND f(-x) ≠ -f(x)

15. Examples of Neither f(x) = 4x³ - 5x² f(x) = 5x³ + 7 Mixture of odd and even exponents. Odd exponents with a constant.

16. Examples of Neither If f(x) = -3x³ + 2x², determine if f(x) is even, odd, or neither. 1)Find f(-x). 2)f(-x) = -3(-x)³ + 2(-x)² = 3x³ + 2x² 3)Find –f(x). 4)-f(x) = 3x³ - 2x² 5)Because f(-x) ≠ f(x) and f(-x) ≠ -f(x), f(x) is neither even nor odd.