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 $x$ dollars for each download, then the number of downloads, in thousands, is $18 - x$ .

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

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

There are $13$ thousand downloads when the price is $$ 5$ per download.

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

There are $8, 000$ downloads when the price is $$ 10$ per download.

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

There are $6, 000$ downloads when the price is $$ 12$ per download.

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

There are $3, 000$ downloads when the price is $$ 15$ per download.

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

There are $0$ downloads when the price is $$ 18$ per download.

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

Compare your answer:

$18 - x$

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)
$3$	$15$	$45$
$5$		_____
$10$		_____
$12$		_____
$15$		_____
$18$		_____
$x$		_____

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

The revenue is $$ 65, 000$ when the price is $$ 5$ per download.

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

The revenue is $$ 80, 000$ when the price is $$ 10$ per download.

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

The revenue is $$ 72, 000$ when the price is $$ 12$ per download.

10.

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

The revenue is $$ 45, 000$ when the price is $$ 15$ per download.

11.

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

The revenue is $0$ when the price is $$ 18$ per download.

12.

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

Compare your answer:

$x (18 - x)$

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.

Compare your answer: Your answer may vary, but here is a sample.

It is quadratic. The output $x (18 - x)$ can be written as $18 x - x^{2}$ , which is a quadratic expression.

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.

Compare your answer:

A scatterplot on a coordinate grid. The x-axis represents the price of one download in dollars and the y-axis represents the revenue in thousands of dollars. The x-axis scale is 5 and extends from 0 to 25. The y-axis scale is 50 and extends from 0 to 200.

Access multimedia content

15.

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

Compare your answer: Your answer may vary, but here is a sample.

Based on the graph, the largest revenue comes when the price is between $$ 5$ and $$ 10$ . It is hard to tell exactly, but perhaps $$ 9$ , halfway between $$ 0$ and $$ 18$ .

Are you ready for more?

Extending Your Thinking

The function that uses the price (in dollars per download) $x$ to determine the number of downloads (in thousands) $18 - x$ 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?

Compare your answer: Your answer may vary, but here is a sample.

The quality of the movies could improve. The number of people who like watching movies could increase. The price of watching a movie in a theater could go up.

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

Compare your answer: Your answer may vary, but here is a sample.

The price that gives the maximum revenue would increase from $$ 9$ to $$ 10$ .

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?

$13-x^2$
$13x-x^2$
$\frac {13-x}{x}$
$13-x$

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 $d$ dollars, and it is possible to sell $600 - 75 d$ 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, $r$ , collected is $r (d) = d (600 - 75 d)$ . Here is a graph that represents the function.

A parabola on a coordinate grid. The x-axis represents the ticket price in dollars and the y-axis represents the revenue in thousands of dollars. The x-axis scale is 1 and extends from 0 to 9. The y-axis scale is 200 and extends from 0 to 1,400.

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, $$ 1, 200$ , comes from selling the tickets for $$ 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 $r$ is between $0$ and $8$ . This makes sense because the cost of the tickets can’t be negative, and if the price were more than $$ 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 $(500 - 10 p)$ tickets when each ticket is $p$ dollars.

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

Compare your answer: Your answer may vary, here is a sample.

Step 1 - Multiply the tickets sold by the price of each ticket, $p$ .

$p (500 - 10 p)$

Step 2 - Distribute the $p$ .

$500 p - 10 p^{2}$

Step 3 - Write as a function, $r (p)$ .

$r (p) = 500 p - 10 p^{2}$