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Characteristics of Polynomials:
Domain, Range, & Intercepts
1. What is interval notation?
2. What is the domain & range of a function?
3. How do I find the intercepts of a functions graphically and algebraically?
Daily Questions…….
How do we write in interval notation?
x < 2…. when you want to include use a bracket [ when you want to exclude use a
parenthesis (
( , 2) Draw a number line first if needed….
Let’s do another type….
4 9x Draw a number line first if needed….
4,9
Domain
all the x-values Read the graph from left to right
all the y-values Read the graph from bottom to top
Range
(2,4)
(-1,-5)
(4,0)
What is the domain of f(x)?
y = f(x)
Ex. 1
Domain
¿
(2,4)
(-1,-5)
(4,0)
y = f(x)
Ex. 2: What is the range of f(x)?
Range
[−𝟓 ,𝟒]
With polynomials….
The DOMAIN is always All Reals,
The RANGE will be one of the following: All Reals, Lower Boundary to infinity, Negative infinity to Upper Boundary,
,
, LB,
,UB
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: 1, 5
X-Intercepts: (-1,0)(1,0)(2,0)
Zeros, Roots
x = -1, 1, 2
X-Intercepts: (-2, 0) (-2, 0) (3,0)
Zeros, Roots:
x = -2, -2, 3
y-intercepts
Where the graph crosses the y-axis
y-Intercept: (0,-12)
y-Intercept: (0,2)
Find the y-intercepts & number of zeros:
a)
b)
4 2f x 3x 5x 1
2f x 2x 3x 15
Y- int: (0, -1)
Y- int: (0, 15)
# of zeros: 4
# of zeros: 2
Find the following
1.Domain:
2.Range:
3. x-intercepts:
4. y-intercepts:
All reals
All reals
(-2,0)(-2,0)(1,0)
(0, -4)
Find the following
1.Domain:
2.Range:
3. Zeros:
4. y-intercepts:
All reals
[-4, ∞)
-2, 2
(0, -4)
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
More About Polynomials…
1. When is a function increasing, decreasing, & constant?
2. Where are the max & min of a function?
Decreasing
Incr
easin
g
Constant
(1,-2)
(-1,2)
(-, -1) (1, )(-1, 1)
increasing increasingdecreasing
Ex.Increasing and decreasing are
stated in terms of domain
(2, 1)(0, 1)
(-, 0) (0, 2)increasing decreasing
(2, )constant
Ex. Increasing and decreasing are stated in terms of domain
Determine the intervals over which the function is increasing and decreasing…
decreasing (- ,0) 2,
Increasing (0,2)
Relative (Local) Minimum & Maximum Values
Relative Minimum: all of the lowest points
Relative Maximum: all of the highest points
Absolute (Global) Minimum & Maximum
Absolute Minimum: the lowest point
Absolute Maximum: the highest point
Relative maximum
Relative minimum
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)max (0, -4)min
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
[-4, ∞)
-2, 2(0, -4)
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
(0, -4) (min)
(0, -4) (min)
(0, ∞)(-∞, 0)
End Behavior, Extrema & Sketching
End Behavior
You look left and right to figure out what’s happening up and down!
( ) _______
( ) _______
x f x
x f x
right
left
up, down, or flat
up, down, or flat
End Behavior: From a Graph
( ) _______
( ) _______
x f x
x f x
( ) _______
( ) _______
x f x
x f x
1.
2.2.
End Behavior: From a Graph3.
4.
3.
( ) _______
( ) _______
x f x
x f x
( ) _______
( ) _______
x f x
x f x
Determine the left and right behavior based on the equation.
6. f(x) = -x5 +3x4 – x
5. f(x) = x4 + 2x2 – 3x
7. f(x) = 2x3 – 3x2 + 5
( ) _______
( ) _______
x f x
x f x
( ) _______
( ) _______
x f x
x f x
( ) _______
( ) _______
x f x
x f x
Tell me what you know about the equation…
Odd exponent
Positive leading coefficient
Tell me what you know about the equation…
Even exponent
Positive leading coefficient
Tell me what you know about the equation…
Odd exponent
Positive leading coefficient
Extrema are turns in the graph.
• If you are given a graph take the turns and add 1 to get the least possible degree of the polynomial.
• If you are given the function, take the degree and subtract 1 to get the max possible number of extrema.
f(x) = 2x3 – 3x2 + 5
3
8. What is the least possible degree of this function?
9. What is the least possible degree of this function?
4
What if you didn’t have a graph?
11. f(x) = -x5 +3x4 – x
10. f(x) = x4 + 2x2 – 3x
12. f(x) = 2x3 – 3x2 + 5
Number of Extrema: ____
Number of Extrema: ____
Number of Extrema: ____
Sketching:
# of Zeros: _________ Y-Int: ________
( ) _______
( ) _______ _______
x f x
x f x extrema
213. 8 20
: 10, 2
f x x x
given zeros
Sketching:
( ) _______
( ) _______ _______
x f x
x f x extrema
3 214. 2 2
: 2, 1,1
f x x x x
given zeros
# of Zeros: _________ Y-Int: ________