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

Characteristics of Graphs of Polynomials

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.

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 Degree: __________ Number of U-Turns/Extrema: ____ Degree: __________ Number of U-Turns/Extrema: ____ Degree: __________ Number of U-Turns/Extrema: ____

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

What is the least possible degree of this function?

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

(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)

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

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

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

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

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

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

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

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

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

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

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

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

Determining Relative Maximum or Minimum. Relative Maximum Relative Minimum

Relative maximum Relative minimum

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

Max and Min: Graph Abs Max: Abs Min: Rel Max: Rel Min:

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

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

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

Zeros? X= -3, -1, 2

y-intercepts Where the graph crosses the y-axis

y-intercepts Analyze the Graph of a Function

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)

WORKSHEET in class Homework: