warm up the domain of a function is its a)y-values b) x-values c) intercepts the range of a...
TRANSCRIPT
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: