## Why Use Linear and Absolute Value Functions?

You have a great idea for a small business. You and a friend have developed a battery-powered bike. It’s perfect for getting around a college campus or even local stops in town. You enjoy making the bikes, but would it be a worthwhile business—one from which you can earn a profit?

The profit your business can earn depends on two main factors. First, it depends on how much it costs you to make the bikes. These costs include the parts you buy to make each bike as well as any rent and utilities you pay for the location where you make the bikes. It also includes any salaries you pay people to help you.

Second, profit depends on revenue, which is the amount of money you take in by selling the bikes.

Profit = Revenue – Costs

Both revenue and costs are linear functions. They depend on the number of bikes you sell. You can then rewrite the profit equation as a function:

[latex]P\left(x\right)=R\left(x\right)-C\left(x\right)[/latex]

where [latex]P(x)[/latex] is profit, [latex]R(x)[/latex] is revenue, [latex]C(x)[/latex] is cost and [latex]x[/latex] is equal to the number of bikes produced and sold.

You and your business partner determine that your fixed costs, those you can’t change such as the room you rent for the business, are $1,600, and your variable costs, those associated with each bike, are $200. If you sell each bike for $600, the table shows your profits for different numbers of bikes.

Number of bikes | Profit ($) |

2 | –800 |

5 | 400 |

10 | 2,400 |

As seen in the table above, if you only sell 2 bikes, you actually lose money. However, if you sell 5 or more bikes, you earn a profit.

- How can you figure out whether you will have a profit or a loss?
- How can you determine how many bikes you need to sell to break even?
- How will shifting your price affect your profits?

In this module you’ll find out how to answer all of these questions. Read on to learn how you might get your business up and running. At the end of the module, we’ll revisit your bike business to find out the very point at which you’ll start to earn a profit.