Lesson 7

Building Quadratic Functions to Describe Situations (Part 3)

7.1: Which One Doesn’t Belong: Graphs of Four Functions (5 minutes)

Warm-up

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.

Launch

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.

Student Facing

Which one doesn’t belong?

A

Graph of a line, origin O, with grid. Horizontal axis from negative 2 to 2 by 1’s. Vertical axis from negative 2 to 6 by ones. Line passes through negative 2 comma 2, 0 comma 2, and 2 comma 2.

B

Graph of a line, origin O, with grid. Horizontal axis from negative 2 to 2 by 1’s. Vertical axis from negative 2 to 6 by ones. Line passes through negative 1 comma negative 2, 0 comma 2, and 1 comma 6.

C

Graph of 13 data points.

D

Graph of curved line, origin O, with grid. Horizontal axis from negative 2 to 2 by 1’s. Vertical axis from negative 2 to 6 by 1’s. Line passes through negative 1 comma point 5, 0 comma 3 point 5, 1 comma 7.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.

7.2: What Price to Charge? (15 minutes)

Activity

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.

Launch

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

  • “Suppose the price per download increasesWhat 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.

Representation: Access for Perception. Read the opening task statement and questions aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language; Conceptual processing

Student Facing

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 \(18-x\).

  1. Complete the table to show the predicted number of downloads at each listed price. Then, find 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    
    \(x\)    
  2. Is the relationship between the price of the movie and the revenue (in thousands of dollars) quadratic? Explain how you know.
  3. Plot the points that represent the revenue, \(r\), as a function of the price of one download in dollars, \(x\).

    Blank coordinate plane with grid, origin O.
  4. What price would you recommend the company charge for a new movie? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

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.

  1. 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 \(20-x\) thousand downloads at a price of \(x\) dollars per download, what do you think will happen to the price that gives the maximum revenue? Check what actually happens.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.)

Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion of the prompts listed in the synthesis. After a student shares their thinking with the class, provide the class with the following sentence frames to help them respond: "I agree because….” or "I disagree because….” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. 
Design Principle(s): Support sense-making

7.3: Domain, Vertex, and Zeros (15 minutes)

Activity

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.

Launch

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:

Graph with x-axis labeled with price in dollars and y-axis labeled with revenue in thousands of dollars.
  • “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.

Representation: Internalize Comprehension. Represent the same information through different modalities by using diagrams. To support students in thinking of an appropriate domain for a given situation, suggest that they draw a diagram (or series of diagrams) to help visualize some initial values that makes sense. 
Supports accessibility for: Conceptual processing; Visual-spatial processing

Student Facing

Here are 4 sets of descriptions and equations that represent some familiar quadratic functions. The graphs show what a graphing technology may produce when the equations are graphed. For each function:

  • Describe a domain that is appropriate for the situation. Think about any upper or lower limits for the input, as well as whether all numbers make sense as the input. Then, describe how the graph should be modified to show the domain that makes sense.
  • Identify or estimate the vertex on the graph. Describe what it means in the situation.
  • Identify or estimate the zeros of the function. Describe what it means in the situation.

1. The area of rectangle with a perimeter of 25 meters and a side length \(x\): \(A (x)= x \boldcdot \frac{(25- 2x)}{2}\)

Graph of the quadratic function \(A(x)=x\boldcdot \frac{(25-2x)}{2}\) on a coordinate plane, origin \(O\).
  • Domain:

  • Vertex:

  • Zeros:

2. The number of squares as a function of step number \(n\): \(f(n) = n^2 + 4\)

Graph of the quadratic function f of n = n squared + 4. Horizontal axis scale negative 12 to 12 by 6’s, labeled “step number”. Vertical axis scale 0 to 200 by 40’s, labeled “number of squares”. 
  • Domain:

  • Vertex:

  • Zeros:​​​

3. The distance in feet that an object has fallen \(t\) seconds after being dropped: \(g(t) =16t^2\)

Graph of the quadratic function \(g(t)=16t^2 \) on a coordinate plane, origin \(O\).
  • Domain:

  • Vertex:

  • Zeros:

4. The height in feet of an object \(t\) seconds after being dropped: \(h(t) = 576 - 16t^2\)

Graph of the quadratic function h(t)=576 - 16t^2 on a coordinate plane, origin O.
  • Domain:

  • Vertex:

  • Zeros:

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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).

Points on coordinate plane. X-axis labeled with step number and y-axis labeled with number of squares.
Graph of line. X-axis labeled with time in seconds and y-axis labeled with height in feet.
  • 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 \(0 \leq t \leq 6\).
Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion. Ask students to select one of the functions and to prepare a visual display of their work. Students should consider how to display their reasoning so that another student can interpret what they see. Suggest that students should add notes and details to the graphs or functions to help communicate their thinking. Arrange students in groups of 2, and provide 2–3 minutes of quiet think time for students to read and interpret each other's work before a whole-class discussion.
Design Principle(s): Cultivate conversation; Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

Students have looked at the zeros and domains of functions in different situations. They also thought about what the vertices of the graphs of functions mean in the given contexts. To help students consolidate the key ideas in this lesson, consider asking them to complete one or more of the following sentence stems:

  • In general, the zeros of a function tell us . . . For example: . . .
  • In general, the vertex of a graph that represents a function tells us . . . For example: . . .
  • To determine an appropriate domain for a function, we need to consider . . .

7.4: Cool-down - Making the Greatest Revenue (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

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.

Let’s say that with a price of \(d\) dollars, it is possible to sell \(600 - 75d\) 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 \(r(d) = d(600-75d)\). Here is a graph that represents the function.

Graph of non linear 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, comes from selling the tickets for \$4 each.

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 does 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.)