2. Warm up The domain of a function is its a)y-values b) x-values c) intercepts  The range of a function is its a) y-values b) x-values c) intercepts

3. Characteristics of Graphs of Polynomials

4. Extrema….. The function of f has at most n – 1 relative extrema (relative minimums or maximums) f(x) = a n x n + a n-1 x n-1 + …..+ a 0 Extrema are turns in the graph. If you are given a graph take the turns and add 1 to get the degree. If you are given the function, take the degree and subtract 1 to get the turns.

5. What if you didn’t have a graph? f(x) = -x 5 +3x 4 – x f(x) = x 4 + 2x 2 – 3x f(x) = 2x 3 – 3x 2 + 5 Degree: __________ Number of U-Turns/Extrema: ____ Degree: __________ Number of U-Turns/Extrema: ____ Degree: __________ Number of U-Turns/Extrema: ____

6. What is the least possible degree of this function? What is the domain and range of this function?

7. What is the least possible degree of this function?

8. Domain and Range Remember that domain is all the x-values (the input). Remember that range is all the y-values (the output).

9. (2,4) (-1,-5) (4,0) What is the domain of f(x)? y = f(x) Ex. 1 Must be written in interval notation Domain is [-1,4) [-1,4)

10. (2,4) (-1,-5) (4,0) y = f(x) What is the range of f(x)? Range [-5,4]

11. (2,4) (-1,-5) (4,0) y = f(x) Domain Range

12. Ex. 2Find the domain and range of Graphically Domain: [4,  )Range: [0,  )

13. Increasing, Decreasing, and Constant How can you tell whether a graph is increasing, degreasing, or constant?

14. A function is increasing when its graph rises as it goes from left to right. A function is decreasing when its graph falls as it goes from left to right. inc dec

15. Decreasing Increasing Constant Decreasing fromConstant from [0, 2] Increasing from

16. (1,-2) (-1,2) (- , -1] [1,  ) [-1, 1] increasing decreasing Ex. 4b Increasing and decreasing are stated in terms of domain (x-values)

17. Increasing and Decreasing Functions Describe the increasing and decreasing behavior. The function is decreasing over the entire real line.

18. (2, 1)(0, 1) (- , 0] [0, 2] increasing decreasing [2,  ) constant Ex. 4cIncreasing and decreasing are stated in terms of domain

19. Increasing and Decreasing Functions Describe the increasing and decreasing behavior. The function is decreasing on the interval increasing on the interval decreasing on the interval increasing on the interval

20. Domain (-8, 4] [3,∞) Range (-3, ∞) Increasing (-8, -4] [3, ∞) Decreasing na Constant na Two Part Graphs

21. Relative Minimum & Maximum Values (direction change) Relative Minimum: all of the lowest points Relative Maximum: all of the highest points

22. Determining Relative Maximum or Minimum. Relative Maximum Relative Minimum

23. Relative maximum Relative minimum

24. Absolute Minimum & Maximum Absolute Minimum: the lowest point Absolute Maximum: the highest point

25. Max and Min: Graph Abs Max: Abs Min: Rel Max: Rel Min:

26. Analyze the Graph of a Function Abs Max: Abs Min: Rel Max: Rel Min:

27. Zeros/x-intercepts/Solutions/Roots Where the graph crosses the x-axis What’s a zero?

28. x-intercepts Where the graph crosses the x-axis. Also called zeros. Analyze the Graph of a Function

29. Zeros? X= -3, -1, 2

30. y-intercepts Where the graph crosses the y-axis

31. y-intercepts Analyze the Graph of a Function

32. Find the following 1.Domain: 2.Range: 3. Zeros: 4. y-intercepts: 5. Absolute Max/Min: 6. Relative Max /Min: 7. Increasing: 8. Decreasing: All reals -2, -2, 1 (0, -4) none (-2, 0) (-4, 0)

33. WORKSHEET in class Homework: