1. END BEHAVIOR & SYMMETRY Objective: describe the end behavior of a function; determine whether a function is “even, odd, or neither” How do the exponents of a polynomial change the shape of its graph?

2. Symmetry  EVEN FUNCTIONS  Symmetric about the y-axis  f(x) = f(-x)  All exponents are even numbers

3.  ODD FUNCTIONS  Symmetric about the origin  f( - x) = - f(x)  All exponents are odd numbers

4. Example 1 – “Even”

5. Example 2 – “Odd”

6. Example 3 – “Even” f(x) = 2x 4 – 5x 2

7. Example 4 – “Odd” f(x) = 2x 3 – 5x

8. Example 5 – “Even” XY -27 5 03 15 27

9. Example 6 – “Odd” XY -2-7 -5 03 15 27

10. Examples of “neither” Ex 1) f(x) = x 3 + 5 Ex 2)

11. Even, Odd, or Neither

16. XY -21 5 08 15 21

17. Even, Odd, or Neither XY -24 8 012 116 220

18. Even, Odd, or Neither XY -21 5 08 1-5 2

19. Even, Odd, or Neither f(x) = 3x 4 – x 2

20. Even, Odd, or Neither f(x) = -x 6 + 5x 2

21. Even, Odd, or Neither f(x) = -2x 3 + x

22. Even, Odd, or Neither f(x) = 4x 5 + 2x 3 + x

23. Even, Odd, or Neither f(x) = 4x 2 + 2x

24. Even, Odd, or Neither f(x) = 4x 2 + 2

25. End Behavior  The direction the graph is going on the “left end” and “right end”  What does “y” do as “x” gets really BIG?  What does “y” do as “x” gets really SMALL?

26. UP ( ∞) Down (- ∞)

27. UP ( ∞) Down (- ∞) UP ( ∞) Down (- ∞)

28. Example 1 – describe the end behavior of the function

29. Example 2 – describe the end behavior of the function

30. Example 3 – describe the end behavior of the function

31. Example 4 – describe the end behavior of the function