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Warm up

Warm up. The domain of a function is its y-values b) x-values c) intercepts The range of a function is its a) y-values b) x-values c) intercepts. Characteristics of Graphs of Polynomials. Extrema …. f(x) = a n x n + a n-1 x n-1 + …..+ a 0.

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Warm up

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

  2. Characteristics of Graphs of Polynomials

  3. Extrema….. f(x) = anxn + an-1xn-1 + …..+ a0 The function of f has at most n – 1 relative extrema (relative minimums or maximums) • 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.

  4. What if you didn’t have a graph? f(x) = x4 + 2x2 – 3x Degree: __________ Number of U-Turns/Extrema: ____ Degree: __________ Number of U-Turns/Extrema: ____ f(x) = -x5 +3x4 – x Degree: __________ Number of U-Turns/Extrema: ____ f(x) = 2x3 – 3x2 + 5

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

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

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

  8. (2,4) (4,0) (-1,-5) What is the domain of f(x)? Ex. 1 y = f(x) [-1,4) Must be written in interval notation Domain is [-1,4)

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

  10. (2,4) y = f(x) (4,0) Range (-1,-5) Domain

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

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

  13. 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. dec inc inc

  14. Increasing Decreasing Constant Decreasing from Constant from [0, 2] Increasing from

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

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

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

  18. 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

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

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

  21. Determining Relative Maximum or Minimum. Relative Maximum Relative Minimum

  22. Relative maximum Relative minimum

  23. AbsoluteMinimum & Maximum Absolute Minimum: the lowest point Absolute Maximum: the highest point

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

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

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

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

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

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

  30. Analyze the Graph of a Function y-intercepts

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

  32. WORKSHEETin classHomework:

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