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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

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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. 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?

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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.

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.

House

L feet

W = L – 8 feet

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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