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Study Guides > College Algebra CoRequisite Course

Why It Matters: Linear and Absolute Value Functions

A bike chain wrapped around a rear-tire gear of a stationary red bike.

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.

Learning Outcomes

Review Topics for Success
  • Graph linear functions using tables
  • Define slope for a linear function
  • Calculate slope given two points
  • Interpret the slope of a linear function that models a real-world situation
Linear Functions
  • Represent a linear function with an equation, words, table, and a graph
  • Determine whether a linear function is increasing, decreasing, or constant
  • Calculate slope for a linear function given two points
  • Write and interpret a linear function
Graphing and Writing Equations of Linear Functions
  • Graph linear functions by plotting points, using the slope and y-intercept, and using transformations
  • Write the equation of a linear function given its graph
  • Match linear functions with their graphs
  • Find the x-intercept of a function given its equation
  • Find the equations of vertical and horizontal lines
  • Determine whether lines are parallel or perpendicular given their equations
  • Find equations of lines that are parallel or perpendicular to a given line
  • Graph an absolute value function
  • Find the intercepts of an absolute value function
Modeling with Linear Functions
  • Use prescribed strategies to build linear models
  • Use intercepts and data points to build a linear model
  • Use a diagram to build a model
  • Draw and interpret scatter plots
  • Find the line of best fit using a calculator
  • Distinguish between linear and nonlinear relations
  • Use a linear model to make predictions
 

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  • Why It Matters: Linear and Absolute Value Functions. Authored by: Lumen Learning. License: CC BY: Attribution.

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