Graphing Polynomial Functions (1)
Objective: To analyze graphs of polynomial function using information about zeros, multiplicity, and end behavior.
Do Now:
1. Given f  x   x 3  6x 2  11x  6 , find all the factors and zeros of the function by choosing one
x-intercept(also called a “zero” or a root) from the graph and performing synthetic division.
y
3
2
1
x
-4
-3
-2
-1
1
2
3
4
5
-1
-2
Let’s Explore: Graph each function on a graphing calculator and identify the x-intercepts.
2. f  x    x  2 x  2
1. f  x   x  2
“zero”: ________
“zero”: ________
3. f  x    x  2 x  2 x  2
“zero”: ________
Class Work:
Multiplicity of the zero Describe how it appears
1
2
3
Example of a Graph
*Discuss the multiplicity of each zero from the Do Now*
Examples:
1. Analyze f  x   x 3  10x 2  8x  192 by filling in the information below.
y
Degree: _____
Name: ___________
Leading Coefficient: _____
y-intercept ( x  0 ):
x
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
11
Zeros & Multiplicity:
End Behavior:
As x   , f  x   ______
Factors:
As x   , f  x   ______
2. Analyze y  x 3  12x 2  48x  64 .
Degree: _____
y
Name: ___________
5
Leading Coefficient: _____
y-intercept ( x  0 ):
x
-10
-8
-6
-4
-2
2
-5
Zeros & Multiplicity:
Factors:
End Behavior:
As x   , f  x   ______
As x   , f  x   ______
4
6
8
3. Analyze f  x   x 4  4x 3  16x  16 .
Degree: _____
y
Name: ___________
Leading Coefficient: _____
x
y-intercept ( x  0 ):
-5
-4
-3
-2
-1
1
2
3
4
Zeros & Multiplicity:
End Behavior:
As x   , f  x   ______
Factors:
As x   , f  x   ______
4. Analyze f  x   4x 4  28x 3  33x 2  56x  16 .
Degree: _____
Name: ___________
Leading Coefficient: _____
y-intercept ( x  0 ):
Zeros & Multiplicity:
End Behavior:
As x   , f  x   ______
Factors:
As x   , f  x   ______
y
5. Analyze f  x   x 4  9x 3  27x 2  27x .
Degree: _____
Name: ___________
x
-3
-2
-1
1
2
3
4
5
Leading Coefficient: _____
y-intercept ( x  0 ):
Zeros & Multiplicity:
Factors:
End Behavior:
As x   , f  x   ______
As x   , f  x   ______
6. Analyze f  x   x 3  x 2  25x  25 .
y
Degree: _____
Name: ___________
Leading Coefficient: _____
y-intercept ( x  0 ):
x
-2
-1
1
2
3
4
5
6
Zeros & Multiplicity:
End Behavior:
As x   , f  x   ______
As x   , f  x   ______
Factors:
7. Analyze y  x 2  10x  25 .
Degree: _____
Name: ___________
Leading Coefficient: _____
y-intercept ( x  0 ):
Zeros & Multiplicity:
End Behavior:
As x   , f  x   ______
Factors:
As x   , f  x   ______
7