In this activity, students encounter a quadratic function in a business context. They study the relationship between the price of downloading a movie and the number of downloads, and see that the relationship can be described with a quadratic function.

Students engage in aspects of modeling as they use a table of values to build a model, create a graph to further understand the relationship, and use their model to make a business recommendation (MP4). They practice looking for and expressing regularity in repeated reasoning (MP8) as they calculate the number of downloads and the expected revenue at various prices. If students opt to use spreadsheet or graphing technology, they practice choosing appropriate tools strategically (MP5).

Monitor for students who reason that the relationship between price and revenue is quadratic because:

• The relationship between the two quantities can be defined by , which looks like some quadratic expressions that represented the visual patterns in earlier lessons.
• can be written as , which has a squared term.

Display the completed table. Select previously identified students to share how they decided whether the relationships between the quantities in the situation are quadratic. If students suggest that the U-shape graph shows the relationship, clarify that we can’t rely on the general shape of a few plotted points to tell us if the relationship is quadratic.

Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do the approaches have in common? How are they different?”
• “Did anyone solve the problem the same way, but would explain it differently?”
• “Why do the different approaches lead to the same conclusion?”

Then, discuss questions such as:

• “Is it possible for the company to make no money? How do you know?” (Yes, it is possible to make no money. We can tell by looking at the graph: when the price is $0 (or the movie is free), the business won’t collect anything for the downloads, and when the price is $18, no one will download the movie.)
• “How do we know at what price they’d make the most money?” (We can calculate the predicted revenue at some other prices between 0 and 18 and see which price gives the greatest revenue. The graph also gives a hint, but we may need to plot a few more points to estimate the maximum point from the graph.)

If time permits, ask students: “Is it possible for the company to lose money?” (Not by this model. If there are additional considerations, like it costs the company money to buy the rights to the movie from the producer, then not collecting any revenue could be seen as losing money. Or if the company decides to pay customers when downloading a movie, then it would lose money, but this isn’t likely.)

Some students may confuse zeros and horizontal intercepts (-intercepts). Watch for students that write the zeros as ordered pairs such as rather than 25. Emphasize that while these two terms are related there is difference. A zero is an input value that makes the function's output 0, and the horizontal intercept is the point where the graph of the function meets the horizontal axis. A zero of a function is the -coordinate of an -intercept of its graph.

Invite groups to share their responses and explanations. If not already discussed or displayed by students, show examples of graphs that are each adjusted for a domain appropriate for the function represented.

Explain that the graph of a quadratic function may or may not show the vertex, depending on the situation that it represents.

Here are graphs representing the functions defined by and , each adjusted for the domain appropriate in the situation. In each graph, the -intercept is the vertex, but because a negative domain is not applicable, we don’t see a turn in the graph.

• Function represents the number of squares at step of a geometric pattern. Because partial step numbers are not possible, it makes sense for the graph to show discrete points at whole-number input values, rather than a continuous curve that includes all rational numbers for the input.
• Function models the height of an object seconds after being dropped. Because a negative number of seconds is not meaningful here, assuming that object stops once it hits the ground at 6 seconds, an appropriate domain for the function would be .