1. WARM UP
Zeros:
Domain:
Range:
Relative Maximum:
Relative Minimum:
Intervals of Increase:
Intervals of Decrease:

2. Symmetry
Essential Question: How do you determine the shape and symmetry of the graph by the polynomial equation?

3. Even, Odd, or Neither Functions
Not to be confused with End behavior
To determine End Behavior, we check to see if the leading degree is even or odd
With Functions, we are determining symmetry (if the entire function is even, odd, or neither)

4. Even and Odd Functions (algebraically)
A function is even if f(-x) = f(x)
If you plug in x and -x and get the same solution, then it’s even.
Also: It is symmetrical over the y-axis.
The easiest thing to do is to plug in 1 and -1 (or 2 and -2)
if you get the same y, then it’s Even.
If you get the opposite y, then it’s Odd.
If you get different y’s, then it’s Neither.
A function is odd if f(-x) = -f(x)
If you plug in x and -x and get opposite solutions, then it’s odd.
Also: It is symmetrical over the origin

5. Y – Axis Symmetry Fold the y-axis
Even Function
(x, y)  (-x, y) -5 1 -4 2 -1 3 4 4 11 -1 -4 -2 -3
(x, y)  (-x, y)

6. Test for an Even Function
A function y = f(x) is even if , for each x in the domain of f.
f(-x) = f(x)
Symmetry with respect to the y-axis

7. Symmetry with respect to the origin
(x, y)  (-x, -y)
(2, 2)  (-2, -2)
(1, -2)  (-1, 2)
Odd Function

8. Test for an Odd Function
A function y = f(x) is odd if , for each x in the domain of f.
f(-x) = -f(x)
Symmetry with respect to the Origin

9. Ex. 1
Even, Odd or Neither?
Graphically
Algebraically
EVEN

10. Ex. 2
Even, Odd or Neither?
Graphically
Algebraically
ODD

11. Ex. 3
Even, Odd or Neither?
Graphically
Algebraically
EVEN

12. Ex. 4
Even, Odd or Neither?
Graphically
Algebraically
Neither

13. Even, Odd or Neither?
EVEN
ODD

14. Even functions are symmetric about the y-axis
What do you notice about the graphs of even functions?
Even functions are symmetric about the y-axis

15. Odd functions are symmetric about the origin
What do you notice about the graphs of odd functions?
Odd functions are symmetric about the origin

16. EVEN

17. ODD

18. Neither

19. Neither

20. EVEN

21. ODD

22. Neither

23. EVEN