MATH133 UNIT 2: Quadratic Equations
Individual Project Assignment: Version 2A
Name (Required): __________________________________________________
Show all of your work details for these calculations. Please review this Web site to see how to
type mathematics using the keyboard symbols. Handwritten scanned work is not acceptable for
AIU Online.
Problem 1: Modeling Profit for a Business
IMPORTANT: See Question 3 below. This is mandatory.
Remember that the standard form for the quadratic function equation is y = f (x) = ax2 + bx + c
and the vertex form is y = f (x) = a(x – h)2 + k, where (h, k) are the coordinates of the vertex of
this quadratic function’s graph.
You will use P(x) = -0.2x2 + bx – c where (-0.2x2 + bx) represents the business’s variable profit
and c is the business’s fixed costs.
So, P(x) is the store’s total annual profit (in $1,000) based on the number of items sold, x.
1. (List your chosen value for 𝑏 between 100 and 200.)
2. (List what the fixed costs might represent for your fictitious business, and be creative;
also list your chosen value for c from the table below).
If your last name begins with the letter
Choose a fixed cost between
A–E
$5,000–$5,700
F–I
$5,800–$6,400
J–L
$6,500–$7,100
M–O
$7,200–$7,800
P–R
$7,800–$8,500
S–T
$8,600–$9,200
U–Z
$9,300–$10,000
MATH133 Unit 2 IP 2A Answer Form
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3. Important: By Wednesday night at midnight, submit a Word document with only
your name and your chosen values for b and c above in Parts 1 and 2. Submit this in
the Unit 2 IP submissions area. This submitted Word document will be used to
determine the Last Day of Attendance for government reporting purposes.
4. (State that quadratic profit model function’s equation by replacing 𝑏 and 𝑐 with your
chosen values.)
5. (Choose five values of 𝑥 (number of items sold) between 500 and 1000. Insert those 𝑥values in the table.)
𝑥
𝑃(𝑥)
6. Plug these five values into your model for 𝑃(𝑥) and evaluate the annual business profit
given those sales volumes. (Be sure to show all your work for these calculations;
complete the table below.)
𝑥
𝑃(𝑥)
7. Use the five ordered pairs of numbers from 5 and 6, and Excel or another graphing utility,
to graph your quadratic profit model and insert the graph into your Word answer
document. The graph of a quadratic function is called a parabola. (Insert graph below.)
MATH133 Unit 2 IP 2A Answer Form
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8. (Show work details or explain how you found the vertex. Write the vertex in ordered-pair
form: (𝑥, 𝑦).)
9. (Write the explanation and the equation of the line of symmetry.)
10. (Write your quadratic profit function in vertex form, 𝑦 = 𝑃(𝑥) = 𝑎(𝑥 – ℎ)2 + 𝑘,
where (ℎ, 𝑘) is the vertex of this quadratic function’s graph. Show the details of how you
found this equation.)
11. (State the maximum profit (if any), and show how you determined how many items must
be sold to give the maximum profit.)
12. (State how knowing the number of items sold that produces the maximum profit help you
to run business more effectively.)
MATH133 Unit 2 IP 2A Answer Form
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13. (Give an analysis of the results of these profit calculations, and give some specific
examples of how these calculations could influence your business decisions.)
14. (State which intellipath Learning Nodes seemed to be most helpful in completing this
assignment.)
Problem 2: Fencing a Backyard
Suppose that you need to fence a rectangular play area in your backyard for your child or pet.
Further, suppose that you know the length must be 8 feet longer than the width. The back of your
house will serve as one side of the fenced area. Note: The perimeter (distance around) of a
general rectangle is P = 2L + 2W, and its area is A = L x W. In this situation, P = L + 2W.
L feet
W = L – 8 feet
House
1. (Write the value of area chosen.)
MATH133 Unit 2 IP 2A Answer Form
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If your last name begins with the letter
Choose an area that must be fenced in this range
(in square feet)
A–E
3,000–3,999
F–I
4,000–4,999
J–L
5,000–5,999
M–O
6,000–6,999
P–R
7,000–7,999
S–T
8,000–8,999
U–Z
9,000–9,999
2. (Write the equation of the perimeter in terms of the length, L, only.)
3. (Write the area equation in terms of the length, L, only.)
4. (What can you observe about the characteristics of that quadratic area function? Will this
quadratic function’s graph cross the horizontal axis? How do you know?)
MATH133 Unit 2 IP 2A Answer Form
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5. (Show all your work for finding both the length and the width of this rectangular fenced
area.)
6. (Show all your work for calculating the cost of the fence.)
7. (Show all your work for calculating the cost per square foot of the fenced area.)
8. (What observations and conclusions can you make about the results of them?)
9. (List the intellipath Learning Nodes that were helpful with this assignment,)
Reference
Formatting math as text. (n.d.). Retrieved from the Purple Math Web site:
http://www.purplemath.com/modules/mathtext.htm
MATH133 Unit 2 IP 2A Answer Form
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MATH133 UNIT 2: Quadratic Equations
Individual Project Assignment: Version 2A
Show all of your work details for these calculations. Please review this Web site to see how to
type mathematics using the keyboard symbols.
Problem 1: Modeling Profit for a Business
IMPORTANT: See Question 3 below for special IP instructions. This is mandatory.
Remember that the standard form for the quadratic function equation is y = f (x) = ax2 + bx + c
and the vertex form is y = f (x) = a(x – h)2 + k, where (h, k) are the coordinates of the vertex of
this quadratic function’s graph.
You will use P(x) = −0.2x2 + bx – c where (−0.2x2 + bx) represents the business’ variable profit
and c is the business’s fixed costs.
So, P(x) is the store’s total annual profit (in $1,000) based on the number of items sold, x.
1. Choose a value between 100 and 200 for b. That value does not have to be a whole
number.
2. Think about and list what the fixed costs might represent for your fictitious business (be
creative). Start by choosing a fixed cost, c, between $5,000 and $10,000, according to the
first letter of your last name from the values listed in the following chart:
If your last name begins with the letter
Choose a fixed cost between
A–E
$5,000–$5,700
F–I
$5,800–$6,400
J–L
$6,500–$7,100
M–O
$7,200–$7,800
P–R
$7,800–$8,500
S–T
$8,600–$9,200
U–Z
$9,300–$10,000
Page 1 of 4
3. Important: By Wednesday night at midnight, submit a Word document with only
your name and your chosen values for b and c above in Parts 1 and 2. Submit this in
the Unit 2 IP submissions area. This submitted Word document will be used to
determine the Last Day of Attendance for government reporting purposes.
4. Replace b and c with your chosen values in Parts 1 and 2 in P(x) = −0.2x2 + bx − c. This
is your quadratic profit model function. State that quadratic profit model functions
equation.
5. Next, choose 5 values of x (number of items sold) between 500 and 1,000. Think about
the general characteristics of quadratic function graphs (parabolas) to help you with
choosing these 5 values of x.
6. Plug these 5 values into your model for P(x), and evaluate the annual business profit
given those sales volumes. (Be sure to show all of your work for these calculations.)
7. Use the 5 ordered pairs of numbers from 5 and 6 and Excel or another graphing utility to
graph your quadratic profit model, and insert the graph into your Word answer document.
The graph of the quadratic function is called a parabola.
8. What is the vertex of the quadratic function graph? (Show your work details, or explain
how you found the vertex.)
9. What is the equation of the line of symmetry? Explain how you found this equation.
10. Write the vertex form for your quadratic profit function.
11. Is there a maximum profit for your business? If so, how many items must be sold to
produce the maximum profit, and what is that maximum profit? If your quadratic profit
function has a maximum, show your work or explain how the maximum profit figure was
obtained.
12. How would knowing the number of items sold that produces the maximum profit help
you to run your business more effectively.
13. Analyze the results of these profit calculations and give some specific examples of how
these calculations could influence your business decisions.
14. Which of the intellipath Learning Nodes seemed to be most helpful in completing this
assignment?
Page 2 of 4
Problem 2: Fencing a Backyard
Suppose that you need to fence a rectangular play area in your backyard for your child or pet.
Further, suppose that you know the length must be 8 feet longer than the width. The back of your
house will serve as one side of the fenced area. Note: The perimeter (distance around) of a
general rectangle is P = 2L + 2W, and its area is A = L x W. In this situation, P = L + 2W.
L feet
W = L – 8 feet
House
1. Based on the first letter of your last name, choose a value for your backyard area that
must be fenced from the range corresponding to the first letter of your last name indicated
in the following table.
If your last name begins with the letter
Choose an area that must be fenced in this range
(in square feet)
A–E
3,000–3,999
F–I
4,000–4,999
J–L
5,000–5,999
M–O
6,000–6,999
P–R
7,000–7,999
S–T
8,000–8,999
U–Z
9,000–9,999
2. Using the relationship between the length and the width above, write the equation of the
perimeter in terms of the length, L, only.
Page 3 of 4
3. Using the relationship between the length and width above, write the area equation in
terms of the length, L, only.
4. If you have written the area equation correctly in Question 3, then the area will be a
quadratic function in terms of the length, L, only. What can you observe about the
characteristics of that quadratic area function? (Hint: Think about the values of a, b, and
c in this quadratic function and what those values tell you about the graph of this
quadratic function.) Will this quadratic function’s graph cross the horizontal axis? How
do you know?
5. What are the length and width of this rectangle that will give the chosen area for your
backyard? (Show all of your work.)
6. If fencing materials cost an average of $19.30 per linear foot (including installation,
gates, and other accessories), how much will fencing your 3-sided backyard cost? (Show
all of your work.)
7. What is the cost per square foot of fencing your backyard using this 3-sided fence?
(Show all of your work.)
8. Based on these calculated values, what observations and conclusions can you make about
the results of them?
9. Which of the intellipath Learning Nodes do you think were helpful with this assignment?
Reference
Formatting math as text. (n.d.). Retrieved from the Purple Math Web site:
http://www.purplemath.com/modules/mathtext.htm
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