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Unit 4, Lesson 7

Building Quadratic Functions to Describe Situations (Part 3)

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Integrated Math 2

7.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Which Three Go Together?

Materials

None

Activity Narrative

This Warm-up prompts students to compare four graphs. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

Launch

Arrange students in groups of 2–4. Display the graphs for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three graphs that go together and can explain why. Next, tell students to share their response with their group, and then together to find as many sets of three as they can.

Activity

None

Which three go together? Why do they go together?

Student Response

to access Student Response.

Building on Student Thinking

Activity Synthesis

Invite each group to share one reason why a particular set of three go together. Record and display the responses. After each response, ask the class if they agree or disagree. Because there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure that the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as “discrete,” “increasing,” “-intercept,” and “-intercept.” Also, press students on unsubstantiated claims.

7.2

Activity

Instructional Routines

Aspects of Mathematical Modeling MLR7: Compare and Connect

Materials

None

Activity Narrative

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.

This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.

7.3

Activity

Instructional Routines

None

Materials

None

Activity Narrative

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

Display the graph. Remind students that this graph represents the revenue from selling new movies at dollars. Tell students that this graph was produced with graphing technology, but that it requires some additional understanding, from people, about the situation.

<p>Graph with x-axis labeled with price in dollars and y-axis labeled with revenue in thousands of dollars.</p>

“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 . 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 that the company is losing money, so it has meaning, but it 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 that represents the function? What does it tell us in this situation?” (It appears to be at . 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 that 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 consider choosing only a couple of functions for the class to investigate. If the activity is divided among 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 make sense.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

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 . . . ."

to access Cool-down.

Student Lesson Summary

Quadratic functions often come up when studying revenue, which is the amount of money collected when selling 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 dollars, it is possible to sell 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, , collected is . Here is a graph that represents the function.

When ticket prices are low, a lot of tickets may be sold, but the total revenue is still low because the tickets are cheap. When the ticket prices get close to \$8, not many tickets are sold so the revenue is low again. From the graph, we can tell that the greatest revenue comes when there is a balance between ticket price and number of tickets sold. In this situation, that is \$1,200 of revenue when tickets are sold for \$4 each.

We can also see that for function , the domain is between 0 and 8. This makes sense because the cost of the tickets can’t be negative. If the price is more than \$8, the model doesn’t work because the revenue collected can’t be negative. A negative revenue (based on a non-negative ticket price) could occur only if the number of tickets sold is negative, which is not possible.

Launch

Arrange students in groups of 2.

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 it at a lower price?” (Allow students to make some hypotheses, but it’s not necessary to confirm one way or another.)

Explain that the 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.

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 they are requested.

Select work from students who use different strategies, such as those described in the Activity Narrative, to share later.

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

Activity

None

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 dollars for each download, then the number of downloads, in thousands, is .

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

Is the relationship between the price of the movie and the revenue (in thousands of dollars) quadratic? Explain how you know.
Plot the points that represent the revenue, , as a function of the price of one download in dollars, .

Blank coordinate plane with grid, origin O. Horizontal axis scale 0 to 25 by 5’s, labeled “price (dollars per download)”. Vertical axis scale 0 to 200 by 50’s, labeled “revenue (thousands of dollars)”.
What price would you recommend that the company charge for a new movie? Explain your reasoning.

Student Response

to access Student Response.

Building on Student Thinking

Here are four sets of descriptions and equations that represent some familiar quadratic functions. The graphs show what 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 they meant in the situation.

The area of a rectangle with a perimeter of 25 meters and a side length of :

Graph of the quadratic function on a coordinate plane, origin . Horizontal axis scale negative 10 to 15 by 5s, labeled “length (meters)”. Vertical axis scale 0 to 60 by 10’s, labeled “area (square meters)”. Some of the points of this function are (0 comma 0), (1 comma 11 point 5) and (5 comma 37 point 5) to a maximum at (6 point 2 5 comma 39 point 0 6 3) then decreasing through (10 comma 25), (12 comma 6) and (12 point 5 comma 0).
- Domain:
- Vertex:
- Zeros:
The number of squares as a function of step number :

Graph of the quadratic function on a coordinate plane, origin . 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”. Some of the points of this function are (negative 10 comma 104), (negative 6 comma 40) and (negative 3 comma 13) to a minimum at (0 comma 4) then increasing through (3 comma 13), (6 comma 40) and (10 comma 104).
- Domain:
- Vertex:
- Zeros:
The distance, in feet, that an object has fallen seconds after being dropped:

Graph of the quadratic function on a coordinate plane, origin . Horizontal axis scale negative 8 to 8 by 4’s, labeled “time (seconds)”. Vertical axis scale 0 to 600 by 200’s, labeled “distance fallen (feet)”. Some of the points of this function are (negative 8 comma 1024), (negative 5 comma 400) and (negative 1 comma 16) to a minimum at (0 comma 0) then increasing through (1 comma 16), (5 comma 400) and (8 comma 1024).
- Domain:
- Vertex:
- Zeros:
The height, in feet, of an object seconds after being dropped:

Graph of the quadratic function h(t)=576 - 16t^2 on a coordinate plane, origin O. Horizontal axis scale negative 8 to 8 by 4’s, labeled “time (seconds)”. Vertical axis scale 0 to 800 by 200’s, labeled “height (feet)”. Some of the points of this function are (negative 6 comma 0), (negative 4 comma 320) and (negative 2 comma 512) to a maximum at (0 comma 576) then decreasing through (2 comma 512), (4 comma 320) and (6 comma 0).
- Domain:
- Vertex:
- Zeros:

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Lesson 7

Building Quadratic Functions to Describe Situations (Part 3)

Warm-up

Standards Alignment

Building On

HSF-IF.B.4

Addressing

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSF-BF.A.1.a

HSF-IF.C.7.a

Building Toward

Standards Alignment

Building On

Addressing

HSF-IF.B.5

Building Toward