Algebra II Items to Support Formative Assessment
Unit 4: Polynomial Functions
Perform arithmetic operations on polynomials.
A.APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they
are closed under the operations of addition, subtraction, and multiplication; add, subtract, and
multiply polynomials. (Cross-cutting)
A.APR.A.1 Task
Complete the operations below using the polynomials 𝑓(𝑥) = 3𝑥 + 2, 𝑔(𝑥) = 4𝑥 2 ,
ℎ(𝑥) = 𝑥 3 − 2𝑥, and 𝑘(𝑥) = 𝑥 4 . Be sure to state any domain restrictions if applicable and
answers should be written in standard form.
a. 𝑓(𝑥) + ℎ(𝑥)
b. 𝑘(𝑥) − ℎ(𝑥)
c. 𝑓(𝑥) ∙ 𝑔(𝑥)
d. 𝑓(𝑥)/𝑔(𝑥)
e. [𝑓(𝑥) + 𝑘(𝑥)] ∙ ℎ(𝑥)
1. State the name of the new functions created by performing the above operations for each
example (use function toolbox if necessary).
2. Would 𝑘(𝑥)/𝑔(𝑥) produce a polynomial function? Explain.
3. Extension Question: Can addition, subtraction, or multiplication be used to combine the
functions given to create a function that is not a member of the polynomial function family?
Explain.
Answers:
a. x3 + x + 2
b. x4- x3 + 2x
c. 12x3 + 8x2
d. (3x + 2)/(4x2); x ≠ 0
e. 𝑥 7 − 2𝑥 5 + 3𝑥 4 + 2𝑥 3 − 6𝑥 2 − 4𝑥
1. Student may classify a. as cubic polynomial, b. as quartic polynomial, etc. Clearly d is not a
polynomial, as it is a rational function.
2. No, dividing creates a function that is not polynomial and although the result appears to be
quadratic, it is not a continuous function at x = 0.
3. No. This should lead to a discussion about how the polynomials are closed under the
operations of addition, subtraction and multiplication
.
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A.APR.A.1 Item 1
Given the functions 𝑓(𝑥) = 3𝑥 2 + 1 and 𝑔(𝑥) = 𝑥 4 − 𝑥 2 , select all function operations that
would result in a polynomial function.
a. 𝑓(𝑥) + 𝑔(𝑥)
b. 𝑓(𝑥) − 𝑔(𝑥)
2
c. (𝑓(𝑥))
d. 𝑓(𝑥)/𝑔(𝑥)
Answers: a, b, c
A.APR.A.1 Item 2
Given the functions f(𝑥) = 𝑥 3 − 4, and g (𝑥) = 𝑥 2 − 2x, use function operations to combine
𝑓(𝑥) and 𝑔(𝑥) in two different ways so that the results will remain a new polynomial function.
Then combine f(x) and g(x) in one way so that the result will no longer be a polynomial function.
Answers:
Answers will vary, but part 1 can only include adding, subtracting, and multiplying. Part 2 could
be division, square root, use of negative exponents, etc.
A.APR.A.1 Item 3
Given the functions 𝑓(𝑥) = 5𝑥 4 , and 𝑔(𝑥) = 𝑥 2 − 3 , use function operations to combine f(x)
and g(x) in two different ways so that the results will be a polynomial function. Then combine
𝑓(𝑥) and 𝑔(𝑥) in one way so that the result will no longer be a polynomial.
Answers:
Answers will vary, but part 1 can only include adding, subtracting, and multiplying. Part 2 could
be division, square root, use of negative exponents, etc.
A.APR.A.1 Item 4
Given the functions 𝑓(𝑥) = 𝑥 2 + 𝑥 − 2, and 𝑔(𝑥) = 𝑥 + 5, select all that function operations
that would result in producing a polynomial function.
a. 𝑓(𝑥) + 𝑔(𝑥)
b. 𝑔(𝑥) / 𝑓(𝑥)
c. 5𝑓(𝑥)
d. 2 𝑓(𝑥) − 3 𝑔(𝑥)
Answers: a, c, d
Build a function that models a relationship between two quantities.
F.BF.A.1 Write a function that describes a relationship between two quantities. (Cross-cutting)
b. Combine standard function types using arithmetic operations.
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F.BF.A.1b Task
Given f (x) = 4x 3 - 5x 2 + 9x - 3 and g(x) = 7x 3 - 2x find the following:
A. ( f + g)(x)
B. ( f - g)(x)
C. (g - f )(x)
D. ( f × g)(x)
æ fö
E. ç ÷ (x) . Be sure to state any domain restrictions.
è gø
Answer:
A. ( f + g)(x) = 11x 3 - 5x 2 + 7x - 3
B. ( f - g)(x) = 3x 3 - 5x 2 + 11x - 3
C. (g - f )(x) = 3x 3 + 5x 2 - 11x + 3
D. ( f × g)(x) = 28x 6 - 35x 5 - 8x 4 + 52x 3 - 18x 2 + 6x
æ fö
E. ç ÷ (x) =
è gø
4x 3 - 5x 2 + 9x - 3
2
, x ¹ 0, ±
3
7
7x - 2x
F.BF.A.1b Item 1
Based on the diagram of the rectangular box shown below, find an expression for the volume of
the box V(x). If the volume of the box is 56cm3, find the length of each side.
Answer:
V = (x - 1)(2x)(x - 4)
56 = (x - 1)(2x)(x - 4), x = 5.253, so the dimensions are 4.253cm x 10.506cm x 1.253cm.
Domain
F.BF.A.1b Item 2
Given f (x) = 4x 2 - 5x + 1 and g(x) = 2x - 7 , find f (x) × g(x) .
Answer: f (x) × g(x) = 8x 3 - 38x 2 + 37x - 7
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F.BF.A.1b Item 3
Given 𝑓(𝑥) = 2𝑥 4 − 3𝑥 3 + 2𝑥 − 8 and 𝑔(𝑥) = 9𝑥 4 + 7𝑥 3 − 15𝑥 2 + 10𝑥 − 8, find
(𝑓 + 𝑔)(𝑥) and (𝑓 − 𝑔)(𝑥).
Answer:
( f + g)(x) = -7x 4 + 4x 3 - 15x 2 + 12x - 16
( f - g)(x) = 11x 4 - 10x 3 + 15x 2 - 8x
F.BF.A.1b Item 4
Given 𝑓(𝑥) = 2𝑥 − 3, find (𝑓(𝑥))3.
Answer: 8𝑥 3 − 36𝑥 2 + 54𝑥 − 27
F.BF.A.1b Item 5
The two congruent legs of an isosceles triangle each measure 2𝑥 3 + 𝑥 − 1 units and the
perimeter is 7𝑥 3 − 4𝑥 2 + 2𝑥 − 5 𝑢𝑛𝑖𝑡𝑠. Write a polynomial that represents the measure of the
third side (base) of the triangle.
Answer: 3𝑥 3 + 4𝑥 2 + 7
F.BF.A.1b Item 6
The table below represents values for polynomial functions 𝑓(𝑥) and 𝑔(𝑥). Let a new function
ℎ(𝑥) be defined as ℎ(𝑥) = 𝑓(𝑥) + 𝑔(𝑥). Complete the table for ℎ(𝑥).
x
f(x)
-1
7
0
9
1
11
2
13
3
15
x
g(x)
-1
-1
0
-3
1
7
2
29
3
63
x
h(x)
-1
0
1
2
3
Answer: 6, 6, 18, 42, 78
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F.BF.A.1b Item7
Let 𝑓(𝑥) = 𝑥 2 − 1, 𝑔(𝑥) = 3𝑥 2 − 3𝑥 + 2 and ℎ(𝑥) = 4𝑥 − 4 perform the indicated operation,
simplify, and express your final answer in standard form.
a. 𝑔(2𝑥)
b. 𝑓(ℎ(𝑥))
c. ℎ(𝑓(𝑥))
Answers:
a. 12𝑥 2 − 6𝑥 + 2
b. 16𝑥 2 − 16𝑥 + 15
c. 4𝑥 2 − 8
F.BF.A.1b Item8
Let ℎ(𝑥) = (2𝑥 − 1)3 − 3, find two functions 𝑓(𝑥) and 𝑔(𝑥) such that ℎ(𝑥) = 𝑓(𝑔(𝑥)).
Answers: See student solutions as answers may vary.
F.BF.A.1b Item9
𝑥
Use composition of functions to verify that 𝑓(𝑥) = 3 − 2 and 𝑔(𝑥) = 3𝑥 + 6 are inverses of
each other.
Answers: Students should show that 𝑓(𝑔(𝑥)) = 𝑔(𝑓(𝑥)) = 𝑥
F.BF.A.1b Item10
http://frontenacss.limestone.on.ca/teachers/dcasey/0F7D449A00870BC8.62/Composite%20Functions.pdf
See question #13
Add the following question: Let 𝑚(𝑥) = ℎ(𝑓(𝑔(𝑥))) evaluate 𝑚(1).
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F.BF.A.1b Item10
A cell phone company’s vice president determines that company revenue in D millions of dollars accrued
from p number of phones sold can be expressed by the function 𝐷(𝑝) = √𝑝 + 0.075𝑝 and that the
company’s cell phone sales, in thousands, can be expressed by the formula 𝑝(𝑡) = .5𝑡 2 + 3𝑡 + 85, where
t is the number of years after 2005. Find a formula for company revenue as a function of time.
Answer:
𝐷(𝑡) = √. 5𝑡 2 + 3𝑡 + 85 + 0.075(.5𝑡 2 + 3𝑡 + 85)
F.BF.A.1b Item11
Using the graphs of 𝑓(𝑥) and 𝑔(𝑥), find 𝑔(𝑓(1.5)).
𝑓(𝑥)
𝑔(𝑥)
Answer: 7
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