Algebra II
Unit #3: Polynomial Functions
5.2 Notes: Graphing Polynomial Functions
Name: ________________________
Block: __________
**GOAL: BE ABLE TO SKETCH AND DESCRIBE A GRAPH OF A POLYNOMIAL FUNCTION
WITHOUT A CALCULATOR USING PROPERTIES of the equation to find KEY FEATURES of the
graph: (degree, lead coefficient, end-behavior, zeros/x-intercepts, y-intercept, and turning points
(max/min))
Part I: Multiple Zeros and the Graph Behavior
When a factor is repeated in a function, this results in a “___________________ zero.”
In other words: the zero has multiplicity of n, where n is the number of times that zero appears as a factor.
Choose the zero’s behavior:
ODD multiplicity: graph will
touch / cross @ the zero
EVEN multiplicity: graph
will touch / cross @ the zero
5.2 Polynomials, Linear Factors, and Zeros
Practice: Identify the zeros of the function, the multiplicity of each, and whether the graph will touch or
cross at the zero.
a) 𝑓(𝑥) = (𝑥 − 3)2 (𝑥 + 4)
b) 𝑓(𝑥) = 2𝑥 4 (𝑥 + 6)3 (𝑥 − 1)
c) 𝑓(𝑥) = −𝑥(𝑥 − 2)3 (𝑥 + 7)4
Part II: Finding Zeros by Factoring
Strategy: Find the zeros (x-intercepts) of each function by FACTORING. List the multiplicity of each
zero. **Remember to factor the GCF first, then set EVERY factor equal to ____ and solve for x!**
a) 𝑓(𝑥) = 𝑥 3 − 2𝑥 2 − 15𝑥
Zeros & Multiplicities:
GCF: ______
b) 𝑓(𝑥) = −5𝑥 3 + 45𝑥
GCF: _______
Zeros & Multiplicities:
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c) 𝑓(𝑥) = −2𝑥 4 + 10𝑥 3 + 12𝑥 2
Zeros & Multiplicities:
GCF: _______
d) 𝑓(𝑥) = 3𝑥 3 + 18𝑥 2 + 27𝑥
GCF: ________
Zeros & Multiplicities:
Key Features Review: (Unit #1-2 and 5.1)
1. How can you determine the end behavior of each polynomial function?
2. How can you find the y-intercept of each function?
3. The maximum number of turning points is always __________________ than the degree. How can
you find the actual turning points with your graphing calculator?
4. How can you find extra points on the graph to get a more accurate shape?
*5. How do you define increasing or decreasing intervals of the function using the turning points?
Part III: Graphing Functions (Put it all together!) Graph the functions (a-d) on pg. 1-2 using all key
features. Use your answers to 1-4 above to remind you how to find the key features! Label!
a)
b)
c)
d)
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More Practice: - you don’t need exact max / min values (just make it turn!)
Without a calculator, sketch a graph of the function 𝑓(𝑥) = (𝑥 + 2)2 (𝑥 − 2)(𝑥 − 3). Label key features.
Identify the DEGREE: ________ Lead Coefficient: __________End Behavior: __________________
Y-INTERCEPT: f(0) = ______
Without a calculator, sketch a graph of the function 𝑓(𝑥) = −2𝑥 2 (𝑥 + 4)(𝑥 − 1)3 . Label key features.
Identify the DEGREE: ________ Lead Coefficient: __________End Behavior: __________________
Y-INTERCEPT: f(0) = ______
Practice: Find the zeros of the function. State the multiplicity of all zeros. Sketch a graph of the
function using the key features (zeros, y-intercept, end behavior) and multiplicity rules.
1. y = x(x  8)2
2. y = x4  8x3 + 16x2
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Practice: Graph the function and label the key features. List the maximum/minimum as ordered pairs.
Example: 𝑓(𝑥) = 𝑥 3 + 2𝑥 2 − 24𝑥
Example: 𝑓(𝑥) = (𝑥 − 2)2 (𝑥 + 1)2 (𝑥 + 4)
zeros: __________________
y-intercept: ______________
End Behavior: ____________
Maximum: ______________
Minimum: _______________
Increasing Interval(s):
zeros: __________________
y-intercept: ______________
End Behavior: ____________
Maximum: ______________
Minimum: _______________
Increasing Interval(s):
Decreasing Interval(s):
Decreasing Interval(s):
1. f(x) = x3  7x2 + 10x
2. f(x) = x4  6x3+ 9x2
zeros: __________________
y-intercept: ______________
End Behavior: ____________
Maximum: ______________
Minimum: _______________
Increasing Interval(s):
zeros: __________________
y-intercept: ______________
End Behavior: ____________
Maximum: ______________
Minimum: _______________
Increasing Interval(s):
Decreasing Interval(s):
Decreasing Interval(s):
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Writing: Explain how the seeing the graph of a polynomial function can help you factor the polynomial.
Writing: How can you use the zeros (x-intercepts) of a function to write an equation for the polynomial
in intercept form? How can you expand/convert this to standard form?
Now Try It! Write a polynomial function in standard form with the given zeros (x-intercepts). Show
your work.
a.
𝑥 = 3, −4 , 0
b. 𝑥 = −5, 1 (𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦 𝑜𝑓 2)
c. 𝑥 = −1 (𝑚𝑢𝑙𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦 𝑜𝑓 3)
Application:
A metalworker wants to make an open box from a sheet of metal by cutting equal squares
with side length x from each corner as shown in the picture.
a. Write expressions for length, width, and height of the open box.
b. Use your expressions to write a function for the volume of the box in factored form.
c. Graph the function. Label the axes and any critical points you use to
answer the questions.
d. What is the maximum volume of the box the metalworker can make?
e. What are the dimensions of the box that will give the maximum volume?
Show your work.
Do MathXL 5.2
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5-2 Puzzle “Made in the Shade”
Find the zeros of each polynomial below. For each corresponding row, shade in each number that is a zero. The
illustration made from shading the squares suggests the answer to the riddle below.
A. P(x) = x(x2  1)
B. P(x) = x(x + 2)(x + 1)(x2 + 2x  3)
__________________________
___________________________
C. P(x) = x(x + 4)(x + 3)(x + 1)(x  1)
D. P(x) = x(x2  25)(x2 + 4x + 3)
___________________________
____________________________
E. P(x) = (x2 + x  20)(x + 2)(x2 + 4x + 3)
F . P(x) = (x2  9)(x2  25)
_____________________________
_____________________________
G. P(x) = (x2 + 9x + 20)(x2  5x + 6)(x  5)
H. P(x) = (x2  5x + 6)(x2  9x + 20)
______________________________
______________________________
I. P(x) = x2  6x + 9
J. P(x) = (x2  4x + 4)(x2  4x + 4)
________________________________
_______________________________
K. P(x) = x(x2  2x + 1)(x  2)
_______________________________
A
–5
–4
–3
–2
–1
0
1
2
3
4
5
B
5
–5
–4
–3
–2
–1
0
1
2
3
4
C
4
5
–5
–4
–3
–2
–1
0
1
2
3
D
3
4
5
–5
–4
–3
–2
–1
0
1
2
E
2
3
4
5
–5
–4
–3
–2
–1
0
1
F
1
2
3
4
5
–5
–4
–3
–2
–1
0
G
0
1
2
3
4
5
–5
–4
–3
–2
–1
H
–1
0
1
2
3
4
5
–5
–4
–3
–2
I
–2
–1
0
1
2
3
4
5
–5
–4
–3
J
–3
–2
–1
0
1
2
3
4
5
–5
–4
K
–4
–3
–2
–1
0
1
2
3
4
5
–5
Riddle: This grows above the ground, but the solutions to the polynomials above lie beneath.
And as it grows, it provides shade to those underneath. What is it?
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