Lesson 7

This warm-up prompts students to analyze and compare the features of the graphs of four functions. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminologies students know and how they talk about characteristics of linear, exponential, and quadratic functions and their graphs.

Arrange students in groups of 2–4. Display the graphs for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as discrete, increasing, \(x\)-intercept, and \(y\)-intercept. Also, press students on unsubstantiated claims.

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).

As students work, monitor how they decide (without graphing) whether the relationships being examined are quadratic. Because the input values in the table do not increase by equal amounts, just looking at the the output values from row to row would not help.

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

• The relationship between the two quantities can be defined by \(x(18-x)\), which looks like some quadratic expressions that represented the visual patterns in earlier lessons.
• \(x(18-x)\) can be written as \(18x - x^2\), which has a squared term.

Ask students to read the opening paragraph of the activity statement. Then, ask students to make some predictions:

• “Suppose the price per download increases. What do you think would happen to the number of downloads as the price goes up?” (Students are likely to predict that as the price increases, the number of downloads decreases, as fewer people are willing to pay the higher price.)
• “What happens to the number of downloads as the price decreases?” (As the price decreases, the number of downloads increases. Lower prices tend to encourage more people to purchase a product.)
• “Would a business make more money when it sells a product at a higher price or when it sells at a lower price?” (Allow students to make some hypotheses, but it’s not necessary to confirm one way or another.)

Explain that price and the number of sales affect the revenue of a business, and that the term revenue means the money collected when someone sells something. For example, if the price of a movie download is $3 and there are 10 downloads, the revenue is $30.

Consider arranging students in groups of 2 so that they can discuss their thinking (especially the question about whether price and revenue have a quadratic relationship). 

Some students may choose to use a spreadsheet tool to complete the table, and subsequently to use graphing technology to plot the data. Make these tools accessible, in case requested.

Display the completed table for all to see. 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 that 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.)

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?” (It depends. If 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 it decides to pay customers when downloading a movie (or setting a negative dollar value for the price), then it would lose money, but this isn’t likely.)

This activity serves two main goals. It prompts students to look closely at appropriate domains for quadratic functions given the situations they represent. It also gives them an opportunity to identify or estimate the vertex of the graph and the zeros of the functions, and then to interpret them in context.

In the previous activity, we saw a function representing the revenue, in thousands of dollars, from selling new movies at \(x\) dollars. If we graph the equation \(r = x (18- x)\) using graphing technology, it would produce a graph like this:

• “What domain is appropriate for the price of a new movie?” (The price can't be negative and it can't be more than \$18, or else the revenue is negative, which is not possible assuming they are selling a non-negative number of downloads for a non-negative price. So an appropriate domain would be \(0 \leq x \leq 18\). The part of the graph to the left of the vertical axis has no meaning because we can’t have a negative price. The part representing a price above \$18 would mean the company is losing money, so it has meaning, but is unlikely to be considered if the company is trying to maximize revenue.)
• “What are the zeros of the function? What do they tell us in this situation?” (0 and 18. They tell us the prices at which the company would earn no revenue.)
• “What is the vertex of the graph representing the function? What does it tell us in this situation?” (It appears to be at \((9, 81)\). It tells us the price that would produce the greatest revenue.)

Tell students they will now think about the domain, vertex, and zeros of a few quadratic functions we have seen so far.

Arrange students in groups of 2–4. Consider asking each group to work on only 1–2 functions and then to share their findings with the class, or choosing only a couple of functions for the class to investigate. If the activity is divided amongst groups and if time permits, consider asking each group to prepare a presentation or to display their work for a gallery walk.

Some students may confuse zeros and horizontal intercepts (\(x\)-intercepts). Watch for students that write the zeros as ordered pairs such as \((25,0)\) 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 \(x\)-coordinate of an \(x\)-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 graphs of quadratic functions may or may not show the vertex, depending on the situation it represents.

Here are graphs representing the functions defined by \(f(n) = n^2 + 4\) and \(h(t) = 576 - 16t^2\) (in the second and fourth questions), each adjusted for the domain appropriate in the situation. In each graph, the \(y\)-intercept is the vertex, but because a negative domain is not applicable, we don’t see a “turn” in the graph (where the output changes from increasing to decreasing, or vice versa).

• The function \(f\) represents the number of squares at step \(n\) 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.
• The function \(h\) models the height of an object \(t\) 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 \(0 \leq t \leq 6\).