Graphing Factored Polynomials

Graphing Factored Polynomials

Student Outcomes

• Students will use the factored forms of polynomials to find zeros of a function.
• Students will use the factored forms of polynomials to sketch the components of graphs between zeros.

New York State Common Core Math Algebra II, Module 1, Lesson 14

Worksheets for Algebra 2

Classwork

Opening Exercises
An engineer is designing a roller coaster for younger children and has tried some functions to model the height of the roller coaster during the first 300 yards. She came up with the following function to describe what she believes would make a fun start to the ride:
𝐻(𝑥) = −3𝑥⁴ + 21𝑥³ − 48𝑥² + 36𝑥,
where 𝐻(𝑥) is the height of the roller coaster (in yards) when the roller coaster is 100𝑥 yards from the beginning of the ride. Answer the following questions to help determine at which distances from the beginning of the ride the roller coaster is at its lowest height.
a. Does this function describe a roller coaster that would be fun to ride? Explain.
b. Can you see any obvious 𝑥-values from the equation where the roller coaster is at height 0?
c. Using a graphing utility, graph the function 𝐻 on the interval 0 ≤ 𝑥 ≤ 3, and identify when the roller coaster is 0 yards off the ground.
d. What do the 𝑥-values you found in part (c) mean in terms of distance from the beginning of the ride?
e. Why do roller coasters always start with the largest hill first?
f. Verify your answers to part (c) by factoring the polynomial function 𝐻.
g. How do you think the engineer came up with the function for this model?
h. What is wrong with this roller coaster model at distance 0 yards and 300 yards? Why might this not initially bother the engineer when she is first designing the track?

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Example 1

Graph each of the following polynomial functions. What are the function’s zeros (counting multiplicities)? What are the solutions to 𝑓(𝑥) = 0? What are the 𝑥-intercepts to the graph of the function? How does the degree of the polynomial function compare to the 𝑥-intercepts of the graph of the function?
a. 𝑓(𝑥) = 𝑥(𝑥 − 1)(𝑥 + 1)
b. 𝑓(𝑥) = (𝑥 + 3)(𝑥 + 3)(𝑥 + 3)(𝑥 + 3)
c. 𝑓(𝑥) = (𝑥 − 1)(𝑥 − 2)(𝑥 + 3)(𝑥 + 4)(𝑥 + 4)
d. 𝑓(𝑥) = (𝑥² + 1)(𝑥 − 2)(𝑥 − 3)

Example 2

Consider the function 𝑓(𝑥) = 𝑥³ − 13𝑥² + 44𝑥 −32.
a. Use the fact that 𝑥 − 4 is a factor of 𝑓 to factor this polynomial.
b. Find the 𝑥-intercepts for the graph of 𝑓.
c. At which 𝑥-values can the function change from being positive to negative or from negative to positive?
d. To sketch a graph of 𝑓, we need to consider whether the function is positive or negative on the four intervals 𝑥 < 1, 1 < 𝑥 < 4, 4 < 𝑥 < 8, and 𝑥 > 8. Why is that?
e. How can we tell if the function is positive or negative on an interval between 𝑥-intercepts?
f. For 𝑥 < 1, is the graph above or below the 𝑥-axis? How can you tell?
g. For 1 < 𝑥 < 4, is the graph above or below the 𝑥-axis? How can you tell?
h. For 4 < 𝑥 < 8, is the graph above or below the 𝑥-axis? How can you tell?
i. For 𝑥 > 8, is the graph above or below the 𝑥-axis? How can you tell?
j. Use the information generated in parts (f)–(i) to sketch a graph of f
k. Graph 𝑦 = 𝑓(𝑥) on the interval from [0,9] using a graphing utility, and compare your sketch with the graph generated by the graphing utility.

Relevant Vocabulary

INCREASING/DECREASING: Given a function 𝑓 whose domain and range are subsets of the real numbers and 𝐼 is an interval contained within the domain, the function is called increasing on the interval 𝐼 if
𝑓(𝑥1) < 𝑓(𝑥2) whenever 𝑥1 < 𝑥2, in 𝐼.
It is called decreasing on the interval 𝐼 if
𝑓(𝑥1) > 𝑓(𝑥2) whenever 𝑥1 < 𝑥2, in 𝐼.
RELATIVE MAXIMUM: Let 𝑓 be a function whose domain and range are subsets of the real numbers. The function has a relative maximum at 𝑐 if there exists an open interval 𝐼 of the domain that contains 𝑐 such that
𝑓(𝑥) ≤ 𝑓(𝑐) for all 𝑥 in the interval 𝐼.
If 𝑓 has a relative maximum at 𝑐, then the value 𝑓(𝑐) is called the relative maximum value.
RELATIVE MINIMUM: Let 𝑓 be a function whose domain and range are subsets of the real numbers. The function has a relative minimum at 𝑐 if there exists an open interval 𝐼 of the domain that contains 𝑐 such that
𝑓(𝑥) ≥ 𝑓(𝑐) for all 𝑥 in the interval 𝐼.
If 𝑓 has a relative minimum at 𝑐, then the value 𝑓(𝑐) is called the relative minimum value.
GRAPH OF 𝒇: Given a function 𝑓 whose domain 𝐷 and the range are subsets of the real numbers, the graph of 𝑓 is the set of ordered pairs in the Cartesian plane given by
{(𝑥, 𝑓(𝑥)) | 𝑥 ∈ 𝐷}.
GRAPH OF 𝒚 = 𝒇(𝒙): Given a function 𝑓 whose domain 𝐷 and the range are subsets of the real numbers, the graph of 𝑦 = 𝑓(𝑥) is the set of ordered pairs (𝑥, 𝑦) in the Cartesian plane given by
{(𝑥, 𝑦) | 𝑥 ∈ 𝐷 and 𝑦 = 𝑓(𝑥)}.

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