Unit: Properties of Functions

This activity will help you meet these educational goals:

Mathematical Practices You will make sense of problems and solve them, use mathematics to model real-world situations, and use appropriate tools strategically.

Introduction

Imagine that you are planning to open a small clothing store that sells designer jeans. You are anxious to make your first sale, but as a person who has strong business sense, you know there are management tasks to deal with long before you greet your first customer. Many of these tasks involve the financial health of the business. To better understand and plan for your financial future, it will be important for you to interpret different kinds of algebraic functions.

In this activity, you will graph revenue and cost curves for your small business and determine your break-even point, the point when revenue equals cost. You will also decide on an optimal commission to pay your sales employees, one that will give them incentive to sell jeans but not break the bank. Finally, as acting manager, you will investigate several different loan options that will put money in your pocket and help get your business off the ground.

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Directions and Analysis

Task 1: Break-Even Point

Business owners must clearly understand the concepts of revenue, expense, and profit. These simple business ideas act together to determine the financial success of a company.

a.Look up the definitions of revenue, expense, and profit, and write them below. Write a simple equation that relates the three concepts.

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b.Open the spreadsheet tool and click the Break-Even Point tab. The worksheet shows profit (P(x)), revenue(R(x)), and costs(C(x)) as a function of units sold. Throughout this activity, the word unit refers to one pair of designer jeans. According to the profit curve P(x) for the data shown, how many units must you sell before you begin to make a profit? In other words, at what point does the profit curve cross the $0-axis? Make sure the cells for m, b, and d work out to m = $40, b = $8,000, and d = $125.

Spreadsheet: (copy and paste in browser)

https://contentstore.ple.platoweb.com/content/sharedmedia/Documents/Algebra1_B2-UA_data.xls

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c.The break-even point of your small business is the point at which your revenues equal your costs. This means that the amount of money that you make is equal to the amount of money that you are spending. Profit is zero at the break-even point. Study the curves for revenue, R(x), and costs, C(x). Based on the current data, how many units represent the break-even point of your business? How does the value compare with the value you found in part b? What are the values for R(x) and C(x) at the break-even point? Assume m = $40, b = $8,000, and d = $125.

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d.Revenue, R(x), is the amount of money that your business makes. In this case, revenue is entirely dependent on your unit sale price, d, and how many units, x, you sell. The relationship is linear. In the current spreadsheet, d = $125. Take some time to gradually increase and decrease the value of d. As you do, see how the cost, revenue, and profit curves change. What happens as d increases? What happens to the profit curve if d drops to $60 or below? Describe the situation in terms of the slope of the curves and the profitability of the business. Assume m = $40 and b = $8,000.

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e.A business owner should understand the total costs, or expenses, of running the business. A cost is anything that the company spends money on. Costs can be fixed or variable. A variable cost is a cost that changes relative to the number of units sold. For example, the cost of the materials used to make jeans is a variable cost. It goes up or down based on your sales. (Commission paid to salespeople is also a variable cost that will be discussed later.) The business is presently spending m = $40 to buy raw materials (e.g., denim, stitching, beads, and sequins) to make each pair of designer jeans. Take some time to vary the value of m. What happens if m climbs to $105 or higher? What happens if the raw materials are free of charge? Describe the situation in terms of the slope of the curves and the profitability of the business. Assume b = $8,000 and d = $125.

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f.A fixed cost is a cost that does not change with sales volume. Here, the fixed costs of the business are insurance, rent, utilities, and base pay for employees. The sum of the fixed costs, b, and the commission represents the point where the cost function crosses the vertical axis ($). Why would a business owner want to minimize the fixed costs? Vary the fixed costs in the spreadsheet to see what happens. Explain your answer in terms of the equation P(x) = R(x) - C(x).

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g.People operate small businesses for a variety of reasons. Some owners enjoy their craft, and others appreciate working with customers. Additionally, most business owners aim to operate a profitable business for themselves and their employees. How can you, the seller of designer jeans, maximize the profits of your business? Explain in terms of fixed costs, variable costs, and your unit sale price. What parameters put an upper limit on your profits? Why is not reasonable to expect never-ending profits in the real world? Modify the inputs in the spreadsheet to help you decide. Describe what happens to the cost, revenue, and profit functions. Use this equation to guide you: P(x) = R(x) - C(x).

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h.Suppose that the fixed costs for your clothing store are b = $10,000. Materials required to make each pair of jeans cost $24. What unit price must you establish for each pair of jeans to create a break-even point at 250 units? Leave the commission function N(x) as is.

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Task 2: Sales Commission

As the store owner, you have decided to pay your sales employees commission to bolster your sales of denim. A commission is a fee paid to your employees based on how many units they sell. By agreeing to pay commission in addition to base pay, you give your employees incentive to sell more product and increase the profitability of your business. In this example, commission is a variable cost because it depends on units sold. Sales commissions can lead to satisfied employees and benefit the bottom line of your business.

Open the spreadsheet tool and click the Sales Commission tab. This sheet displays the same profit function that you saw on the break-even point sheet. Changing the profit function on the break-even point sheet will also change the profit function here.

a.For simplicity, assume that you have one salesperson who sells jeans in your store. Here is how your sales commission works. If your employee sells u units or fewer, you will pay her a flat monthly commission of $800. This means the employee is guaranteed $800 in commission each month. However, if she sells more than u units, you will pay her an extra f dollars for each additional unit. For your store, N(x) is called the commission function. Based on the assumptions, what type of function is N(x): linear, quadratic, cubic, logarithmic, exponential, piecewise, or absolute value? What clues helped you identify the function? Use the graph and the data to help you.

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b.In the current spreadsheet, u = 30, f = $20, and $800 is the flat commission for sales equal to or smaller than u. This means that your salesperson will collect a flat $800 for selling u = 30 pairs of jeans. Once she sells 31 pairs or more, she will collect f = $20 for each additional pair sold. Which two linear equations describe the two intervals of N(x)? Express your answer in terms of u, f, and x, where x is the number of units sold. You may study the formulas in the spreadsheet if you find them helpful.

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Interval

Equation

first {x u}

N(x) =

second {u < x}

N(x) =

c.You know that N(x) is a piecewise function that describes sales commission. Since N(x) is a piecewise function, which other functions in the spreadsheet must also be piecewise? Explain. Look to the general equation P(x) = R(x) - C(x) for the answer. Refer to the legend in the break-even point graph for help.

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d.As a business owner, you should try to identify a sales commission that is balanced. You want a commission that is fair to employees and gives them incentive to sell, however, you can't pay too much commission or it will cut into profits. Change the values of u and f in the spreadsheet to see how they affect the profit function. Toggle between the Sales Commission tab and the Break-Even Point tab to see the results. To help narrow your search, assume that your break-even point is around 200 units a month. Which values of u and f give a monthly commission that seems fair to workers, yet seem financially reasonable? Which values of u and f result in too high a commission with little profit? Describe the changes to the curves. Support your answer using data from the spreadsheet when you can.

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Task 3: Business Loans

You are infatuated with designer clothing. It has been your passion since you were young. That is why you have decided to open a clothing store that sells exclusive designer blue jeans. Unfortunately, having passion and desire to do something you love is just not enough sometimes. In the business world, you also need something called capital. Capital is money used to start a company. It can come from several different sources, including your personal bank account, outside investors, and sales of stock, or it can come from lenders, such as banks. This activity will focus on capital borrowed from banks. For your clothing store, you will use the acquired capital to pay for things like hiring lawyers, advertisements, interior decorating, inventory, and initial fixed and variable costs.

Open the spreadsheet tool and click the Business Loan tab. The graph displays the value A, which is the interest and principal required to pay back a t-year loan from the bank. The graph also displays 12J(t), a portion of your company's profits that you intend to use to repay the loan at the end of the loan period using one large payment. (Companies typically try to make monthly payments against a loan before the term is up. For the sake of this activity, assume you will make a lump-sum payment when the loan terminates.) Remember, you can predict your company's total monthly profits using data from the Break-Even Point spreadsheet. As the owner, you will decide how much of that profit to use to pay back your loan.

a.Study the two functions shown, A(t) and 12J(t). Based on the graph and the data, what kinds of functions are they? Choose among linear, quadratic, cubic, logarithmic, exponential, piecewise, and absolute value. Describe the features of each function that tipped you off.

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b.The equation A = Pe^rt describes a bank loan that compounds continuously. The variables in the equation are described in the table:

Variable

Definition

A

This is the principal plus interest on the loan. Principal is how much money you borrowed. Interest is a graduated fee that you pay to the bank for the privilege of borrowing their money.

P

This is the principal, or how much money you borrowed. Do not confuse P in the compounding interest equation with P in the profit equation. One is principal, the other is profit.

e

This is Euler's number: e 2.7.

r

This is the interest rate expressed as a percentage.

t

This is the time allotted to repay the loan. It's also called the life of the loan.

For the sake of this activity, assume that you will collect profit from sales for a number of months and then use a portion of that profit to pay off the entire loan in one lump-sum payment once the loan terminates. Based on this assumption, what does the intersection of the 12J(t) curve and the A(t) curve represent? Explain using your own words.

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c.Take some time to gradually increase P in increments of $100,000 while keeping r and J(t) constant. What happens to the relationship between the two curves? What does this mean with respect to the bank loan? Why is this a dangerous situation with respect to the financial health of your business? Why would banks put safeguards in place to prevent this from occurring?

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d.Now gradually increase the interest rate, r, in increments of 1% while keeping principal, P, and profits, J(t), constant. What happens to the relationship between the two curves? Why is it important to secure a low interest rate? Explain in terms of the graph and the data.

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e.Let's say that you take out a loan in the amount of $100,000. You plan to set aside $1,450 of your monthly profits for the next 10 years to repay the loan. After 10 years have passed, you will make a lump-sum payment with that money to clear the loan. What is the maximum interest rate that you should agree to in order to accomplish this? How did you arrive at your answer?

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Task 4: Taking Charge of Your Business

You've worked with three different spreadsheets to evaluate components of your small business: Break-Even Point, Sales Commission, and Business Loan. It's now time for you to choose your own parameters regarding a designer jeans store. This part of the activity will require some research. Go online to investigate real-world small business costs for things like raw materials used to make blue jeans, liability insurance, rent, and utilities. Also, find out what retail salespeople earn for base pay and commission. Finally, investigate the current annual interest rates for small business loans. (These are only some of the parameters you'll need to look into.)

Once you've gathered enough information, enter the information into the three spreadsheets to design your own financial scenario. There are no right or wrong answers. Your goal is to design a scenario that mimics the real world and challenges you to make decisions along the way. You will have to decide how much money to borrow from the bank, what unit sale price to establish, and how much commission to pay salespeople. You will also need to predict how many unit sales you'll make each month, depending on the size and location of your store. You'll want to design a financial scenario that generates a profit, keeps salespeople satisfied, keeps you out of perpetual debt, and has customers coming back.

Open the spreadsheet tool to get started. One you're through with your design, answer the question below. For simplicity, assume that you employ one salesperson.

a.Summarize the financial scenario for your designer jeans store. Describe the data and graphs that you created. Be sure to explain how you arrived at your unit sale price, how you estimated the sales volume, and how you determined the size of the loan that you needed. Expect to write two to three paragraphs . If you found valuable information on websites, list those resources in the Resources section at the end of this document.

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