Polynomial Functions
November 4, 2011
Notes:
Review:
1. Sketch the graphs of the following power functions. Assume that the power is positive.
𝒚 = 𝒌𝒙𝒆𝒗𝒆𝒏 𝒑𝒐𝒘𝒆𝒓
𝒚 = −𝒌𝒙𝒆𝒗𝒆𝒏 𝒑𝒐𝒘𝒆𝒓
y
𝒚 = 𝒌𝒙𝒐𝒅𝒅 𝒑𝒐𝒘𝒆𝒓
y
x
𝒚 = −𝒌𝒙𝒐𝒅𝒅 𝒑𝒐𝒘𝒆𝒓
y
x
y
x
x
2. The function, 𝑦 = 1𝑥 4 + 2𝑥 3 − 14𝑥 2 + 40, is called a __________________ function.
3. Graph the polynomial function from question 2 on your graphing calculator using the following
window: Xmin=−10, Xmax=10, Xscl=1, Ymin=−100, Ymax=100, Yscl=20.
a. Write down the end behavior for this graph.
b. Which power function in question 1 has the same end behavior as the polynomial function
graphed? Why do you think that this might be?
c. List at least two differences in the graph of the polynomial function and the power function
you listed in part (b).
Vocabulary:
 Polynomial Function:

General Form of a Polynomial Function:

Degree:

Leading Term:
1
Polynomial Functions
November 4, 2011
Concepts:
 Global Behavior:


Vertical Intercept:


Horizontal Intercepts (Zeros):


Turning Points:

4. For each of the following polynomials, identify the degree, the leading term, the global behavior,
and the vertical intercept.
a. 𝑦 = 1𝑥 4 + 2𝑥 3 − 14𝑥 2 + 40
b. 𝑓(𝑥) = 4𝑥 3 + 11𝑥 5 − 11
c. 𝑔(𝑥) = −2𝑥 4 − 20
d. 𝑦 = 1 + 7𝑥 2 − 5𝑥 3
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Polynomial Functions
5. Given the polynomial function:
November 4, 2011
𝑦 = −2(𝑥 − 3)2 (𝑥 + 4),
a. What is the leading term?
b. Sketch a graph of how the power function looks for very large positive or negative values of x
(global behavior)?
c. What is the degree of the function?
d. What is the maximum number of turning points for this function?
e. What is the maximum number of horizontal intercepts?
f.
Determine the vertical intercept.
g. Determine the horizontal intercepts.
h. Sketch a graph of the function.
y
x
6. Find the leading term and the vertical intercept of the polynomial function without multiplying out.
a. 𝑓(𝑥) = 𝑥(3𝑥 − 2)(5𝑥 2 + 1)
b. 𝑔(𝑥) = (𝑥 2 − 1)(𝑥 3 + 𝑥 − 5)4
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Polynomial Functions
November 4, 2011
7. Identify the leading term and the horizontal intercepts of each of the polynomial functions. Using
technology, graph each function to verify your work.
a. 𝑦 = 3𝑥 + 6
b. 𝑔(𝑥) = (𝑥 + 5)(𝑥 − 3)(2𝑥 + 5)
In Class Activity/Homework -- Graphing Polynomial Functions
For each of the following polynomials:
a)
b)
c)
d)
Remember that an intercept
is a point so should be given
as an ordered pair (x, y).
Identify the leading term
Identify the vertical intercept
Identify the horizontal intercepts (zeros)
Match each polynomial function with its graph (there will be one extra graph)
1) 𝑓(𝑥) = (𝑥 − 1)(𝑥 − 3)(𝑥 + 3)
a)
b)
c)
d)
2) 𝑓(𝑥) = −𝑥 3 − 𝑥 2 + 4𝑥 + 4
a)
b)
c)
d)
3) 𝑓(𝑥) = (𝑥 − 1)(𝑥 2 + 4)
a)
b)
c)
d)
4) 𝑓(𝑥) = (𝑥 2 − 1)(𝑥 2 − 9)
a)
b)
c)
d)
5) 𝑓(𝑥) = (𝑥 − 2)(𝑥 − 4)(𝑥 + 3)2
a)
b)
c)
d)
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