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) = anxn + an-1xn-1 + …..+ a0

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) = -x5 +3x4 – x

f(x) = x4 + 2x2 – 3x

f(x) = 2x3 – 3x2 + 5

Degree: __________ Number of U-Turns/Extrema:

____

Degree: __________ Number of U-Turns/Extrema:

____

Degree: __________ Number of U-Turns/Extrema:

____

3 2( ) 2f x x x x

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?

4 2( ) 2 2f x x x

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. 2 Find the domain and range of f x x( ) 4

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. in

c inc

dec

Decreasing

Incr

easin

g

Constant

Decreasing from ( ,0]

Constant from

[0, 2]Increasing from

[2, )

(1,-2)

(-1,2)

(-, -1] [1, )[-1, 1]increasing increasingdecreasing

Ex. 4b Increasing and decreasing are stated in terms of domain (x-values)

Increasing and Decreasing FunctionsDescribe the increasing and decreasing behavior.

The function is decreasing over the

entire real line.

.7

(2, 1)(0, 1)

(-, 0] [0, 2]increasing decreasing

[2, )constant

Ex. 4c Increasing and decreasing are stated in terms of domain

Increasing and Decreasing FunctionsDescribe the increasing and decreasing behavior.

The function is decreasing on the

interval , 1

increasing on the interval ,1 0

decreasing on the interval ,0 1

increasing on the interval ,1

.8

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-interceptsWhere the graph

crosses the x-axis. Also called zeros.

1,0 & 5,0

Analyze the Graph of a Function

Zeros?

X= -3, -1, 2

y-intercepts

Where the graph crosses the y-axis

y-intercepts

0,5

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

All reals

-2, -2, 1

(0, -4)none

( , 2) (0, ) ( 2,0)

(-2, 0)

(-4, 0)

WORKSHEETin class

Homework: