Skip to ContentGo to accessibility page
OpenStax Logo
Algebra 1

7.7.2 Modeling Real-World Data with Quadratic Functions

Algebra 17.7.2 Modeling Real-World Data with Quadratic Functions

7.7.2 Modeling Real-World Data with Quadratic Functions

7.7.2 • Modeling Real-World Data with Quadratic Functions

Activity

For questions 1 – 6, use the following information to complete the table. Show the predicted number of downloads at each listed price. The first row has been completed for you.

A company that sells movies online is deciding how much to charge customers to download a new movie. Based on data from previous sales, the company predicts that if they charge x x dollars for each download, then the number of downloads, in thousands, is 18 − x 18 − x .
Price (dollars per download) Number of downloads (thousands) Revenue (thousands of dollars)
3 3 15 15 45 45
5 5
  1. _____
  
10 10
  1. _____
  
12 12
  1. _____
  
15 15
  1. _____
  
18 18
  1. _____
  
x x
  1. _____
  
1.

When the price is $ 5 $ 5 per download, what is the number of downloads in thousands?

2.

When the price is $ 10 $ 10 per download, what is the number of downloads in thousands?

3.

When the price is $ 12 $ 12 per download, what is the number of downloads in thousands?

4.

When the price is $ 15 $ 15 per download, what is the number of downloads in thousands?

5.

When the price is $ 18 $ 18 per download, what is the number of downloads in thousands?

6.

When the price is $ x $ x per download, what is the number of downloads in thousands?

For questions 7 – 12 complete the table by finding the revenue at each price. The first row has been completed for you.

Price (dollars per download) Number of downloads (thousands) Revenue (thousands of dollars)
3 3 15 15 45 45
5 5  
  1. _____
10 10  
  1. _____
12 12  
  1. _____
15 15  
  1. _____
18 18  
  1. _____
x x  
  1. _____
7.

When the price is $ 5 $ 5 per download, what is the revenue in thousands of dollars?

8.

When the price is $ 10 $ 10 per download, what is the revenue in thousands of dollars?

9.

When the price is $ 12 $ 12 per download, what is the revenue in thousands of dollars?

10.

When the price is $ 15 $ 15 per download, what is the revenue in thousands of dollars?

11.

When the price is $ 18 $ 18 per download, what is the revenue in thousands of dollars?

12.

When the price is $ x $ x per download, what is the revenue in thousands of dollars?

13.

Is the relationship between the price of the movie and the revenue (in thousands of dollars) linear, quadratic, or exponential? Explain how you know.

14.

Use the graphing tool or technology outside the course. Plot the points that represent the revenue, 𝑟, as a function of the price of one download in dollars, 𝑥, using the Desmos tool below.

Access multimedia content
15.

What price would you recommend the company charge for a new movie? Be prepared to show your reasoning.

Are you ready for more?

Extending Your Thinking
1.

The function that uses the price (in dollars per download) x x to determine the number of downloads (in thousands) 18 − x 18 − x is an example of a demand function, and its graph is known. Economists are interested in factors that can affect the demand function and therefore the price suppliers wish to set.

What are some things that could increase the number of downloads predicted for the same given prices?

2.

If the demand shifted so that we predicted 20 − x 20 − x thousand downloads at a price of x x dollars per download, what do you think would happen to the price that gives the maximum revenue? Check what actually happens.

Self Check

A company that sells music downloads is deciding how much to charge customers to download an album. Based on previous sales, the company predicts that if they charge x dollars for each download, then the number of downloads, in thousands, is 13 − x .

Which of the following represents the revenue for downloads?

  1. 13 − x 2
  2. 13 x − x 2
  3. 13 − x x
  4. 13 − x

Additional Resources

Revenue Represented with Quadratics

Quadratic functions often come up when studying revenue. (Revenue means the money collected when someone sells something.)

Suppose we are selling raffle tickets and deciding how much to charge for each ticket. When the price of the tickets is higher, typically fewer tickets will be sold.

Each ticket has a price of d d dollars, and it is possible to sell 600 − 75 d 600 − 75 d tickets. We can find the revenue by multiplying the price by the number of tickets expected to be sold. A function that models the revenue, r r , collected is r ( d ) = d ( 600 − 75 d ) r ( d ) = d ( 600 − 75 d ) . Here is a graph that represents the function.

It makes sense that the revenue goes down after a certain point, since if the price is too high nobody will buy a ticket. From the graph, we can tell that the greatest revenue, $ 1 , 200 $ 1 , 200 , comes from selling the tickets for $ 4 $ 4 each. So we know, the range of this parabola reaches a maximum value of $1,200.

We can also see that the domain of the function r r is between 0 0 and 8 8 . This makes sense because the cost of the tickets can’t be negative, and if the price were more than $ 8 $ 8 , the model would not work, as the revenue collected cannot be negative. (A negative revenue would mean the number of tickets sold is negative, which is not possible.)

Try it

Revenue Represented with Quadratics

Based on past concerts, a venue expects to sell ( 500 − 10 p ) ( 500 − 10 p ) tickets when each ticket is p p dollars.

Write the function, r ( p ) r ( p ) , to represent the revenue for the concert.

PreviousNext
Order a print copy
Citation/Attribution
Reuse and redistribution of this content in digital or print format:
  • This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's prior written permission.
  • This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License, which means that you can reuse and modify the material only for noncommercial purposes, must attribute OpenStax, and must distribute any derivative works under the same license.
  • Any commercial printing of this textbook, including using a local or custom printer, must be approved by OpenStax, and proper citation provided.
  • OpenStax-copyrighted images, activities, assessments, and similar components of this book are subject to the same licensing – CC-BY-NC-SA. They can be used for noncommercial purposes with attribution. Commercial use requires permission.
  • Permission requests: Anyone who intends to incorporate this content (including text, images, and other components) into large language models, use it in AI offerings, use it commercially (including in print), and/or has questions about another use case is welcome to complete our reuse request form.
Attribution information
  • If you are redistributing all or part of this book in a noncommercial print format, then you must include on every physical page the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

  • If you are redistributing all or part of this book in a noncommercial digital format, then for every page that includes OpenStax content, you must license the derivative work under the same CC-BY-NC-SA license as the original, and include on every digital page view the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

Citation information

The information below includes the information needed to generate citations in most major styles (APA, MLA, etc.); you must reformat and organize the information as needed to fit the requirements of the style. Use the information below to generate a citation. We recommend using a citation tool such as this one.

© Apr 23, 2026 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo, and Rice University name, and Rice University logo trademarks, or wordmarks are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.