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# Why It Matters: Linear and Absolute Value Functions

## 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 to 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.  And it would include 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:

$P\left(x\right)=R\left(x\right)-C\left(x\right)$

where $P(x)$ is profit, $R(x)$ is revenue, and $C(x)$ is cost and $x$ equal 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
So if you only sell 2 bikes, you actually lose money.  But 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?
• And how can you determine how many bikes you need to sell to break even?
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.

### Learning Objectives

Linear Functions
• Represent a linear function with an equation, words, a table and a graph
• Determine whether a linear function is increasing, decreasing, or constant
• Write and interpret a linear function

Graphs of Linear Functions

• Graph linear functions by plotting points, using the slope and y-intercept, and by using transformations
• Write the equation of a linear function given it’s graph, including vertical and horizontal lines, match linear equations with their graphs
• Find the equations of vertical and horizontal lines
• Graph an absolute value function, find it’s intercepts

Modeling With Linear Functions

• Identify steps for modeling and solving
• Build linear models from verbal descriptions
• Draw and interpret scatter plots
• Find the line of best fit using the Desmos calculator
• Distinguish between linear and nonlinear relations
• Use a linear model to make predictions