Lesson 10

Combining Functions

10.1: Notice and Wonder: Are Book Sales Improving? (5 minutes)

Warm-up

The purpose of this warm-up is to elicit the idea that we can combine functions to determine additional information about a situation. While students may notice and wonder many things about the table, the books sold per person is the important discussion point.

This warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).

Launch

Display the table for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Student Facing

What do you notice? What do you wonder?

$$t$$ (years since 2010) number of books sold
in the US (millions)
population of
the US (millions)
0 2,530 309.35
1 2,400 311.64
2 2,730 313.99
3 2,720 316.23
4 2,700 318.62
5 2,710 321.04
6 2,700 323.41

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the books sold per person does not come up during the conversation, ask students to discuss this idea.

10.2: How Many Books Can One Person Have? (10 minutes)

Activity

Continuing to work with the data from the warm-up, students now consider the two functions the data represent and the meaning of the new function created by their quotient. Expressions for each function are purposefully left out of this activity to help students focus on the meaning of the new function created in context.

In this activity, students are building skills that will help them in mathematical modeling (MP4). They don't decide which model to use, but students have an opportunity to reason about data in context and see how to combine known functions into new functions in order to view the story told by the data in a different way.

Monitor for students who are considering the scale of the numbers as they graph the data for $$B(t)$$. For example, while a change of almost 200 million books from 2010 to 2016 seems large, it is a change of less than 7% of the total. The decrease from 2,730 million to 2,700 million is a change of less than 2%.

Launch

Provide access to devices that can run Desmos or other graphing technology.

Writing, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their writing by providing them with multiple opportunities to clarify their explanations through conversation. Give students time to meet with 2–3 partners to share their response to the last question. Students should first check to see if they agree with each other about what the values of $$R(t)$$ tell them. Provide listeners with prompts for feedback that will help their partner add detail to strengthen and clarify their ideas. For example, students can ask their partner, “How did you calculate the new data points?” or “Where does _____ show…?” Next, provide students with 3–4 minutes to revise their initial draft based on feedback from their peers. This will help students produce written interpretations of combined functions that create new functions and what the new function can mean for a specific context.
Design Principle(s): Support sense-making; Optimize output (for explanation)
Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, students can plot the values of $$B$$ and $$R$$ using two different colors to highlight the difference of the functions based on context.
Supports accessibility for: Visual-spatial processing

Student Facing

The table shows the values of two functions, $$P$$ and $$B$$, where $$P(t)$$ is the population of the US, in millions, $$t$$ years after 2010, and $$B(t)$$ is the number of books sold per year, in millions, $$t$$ years after 2010.

$$t$$ (years since 2010) $$B(t)$$ (millions) $$P(t)$$ (millions) $$R(t)$$
0 2,530 309.35
1 2,400 311.64
2 2,730 313.99
3 2,720 316.23
4 2,700 318.62
5 2,710 321.04
6 2,700 323.41
1. Plot the values of $$B$$ as a function of $$t$$. What does the plot tell you about book sales?

2. How many books were sold per person in 2010 and 2016? What do these values tell you about book sales?
3. Define a new function $$R$$ by $$R(t) = \frac{B(t)}{P(t)}$$. Complete the table and then graph the values of $$R(t)$$. What do the values of $$R$$ tell you?

Anticipated Misconceptions

Students may be confused by how to set up and label the axes to graph the values of $$R(t)$$ in the table. Remind these students of the discussion from the warm-up about the number of books per person. They may find it helpful to write out the values for $$B(t)$$ and $$P(t)$$ in single units instead of millions. For example, the top row of the table would have 2,530,000,000 divided by 309,350,000 for $$R(0)$$.

Activity Synthesis

The purpose of this discussion is for students to understand that we can combine functions to make new functions and what that can mean for a specific context.

Invite students previously identified to share their explanations for the plot of $$B(t)$$, displaying their graphs for all to see to help students illustrate their ideas. Here are some questions for discussion:

• “What does $$R(0)=8.178$$ mean in this context?” (In 2010 the average number of books bought per person in the US was about 8 books.)
• “Compare $$R$$ with $$B$$. Describe how they are alike and how they are different.” (They both dip in 2011 and then go up, but then $$B(t)$$ doesn't change much while $$R(t)$$ appears to decrease. This is because the population is increasing while the number of books sold stays about the same, so the denominator is getting large while the numerator is about the same which means the quotient, $$R(t)$$, decreases.)
• “Let’s say $$C(t)$$ is the average cost of a book, in dollars, in the US. What would the function defined by $$C(t) \boldcdot R(t)$$ tell us?” (The average amount of money spent per person in the US on books in year $$t$$.)

Conclude the discussion by asking students which graph they think is most useful for tracking book sales. After a brief quiet think time, invite students to share their reasoning. Depending on perspective, such as a local bookseller versus a large publishing company, both graphs can be useful for planning for future sales.

Activity

This activity focuses on a specific aspect of the previous activity: combining two functions to make a new function. Here, students sketch the graph of the function defined as the sum of two original functions. To help students make generalizations about the process, in each question one of the starting functions is always the same while the other is either constant, linear, or quadratic (MP8).

Monitor for students who approach the activity in different ways based on how precise they make their sketch, such as:

• making a rough sketch by observation
• plotting a few points by summing together the $$y$$-values of the two functions and then connecting them with a curve
• measuring the distance of each function from the $$x$$-axis in a few locations and then combining those heights to determine the $$y$$-value of the new function at specific values of $$x$$
• making a table of values for $$E$$ and the other function, as in the previous activity, in order to find the sums before making a sketch of the new function

Also look out for possible incorrect approaches due to students not considering the numerical values of the functions and what it means to add, such as by:

• sketching a curve that is halfway between the two given graphs
• letting the horizontal asymptote dictate end behavior on the right regardless of the value of the other function

Ask students who tried either of these last two approaches if they would be willing to share during the discussion. If possible, identify students who first did a sketch incorrectly and then revised their work to share at the end of the discussion.

Launch

Arrange students in groups of 2–3. Tell students that in this activity they are going to sketch graphs of functions that are the sum of two other functions. Ask students to first complete the sketch of $$S$$ on their own and then check their work with their group before continuing on to the other graphs. Depending on time available, assign each group 3–5 of the graphs to complete.

Since this activity was designed to be completed without technology, ask students to put away any graphing devices.

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, students can use different color highlighters for each graph. Students can identify and label points on each graph. Remind students that they are sketching the sum of the two functions.
Supports accessibility for: Visual-spatial processing

Student Facing

1. Here are the graphs of two functions, $$E$$ and $$L$$. Define a new function $$S$$ by adding $$E$$ and $$L$$, so $$S(x) = E(x) + L(x)$$. On the same axes, sketch what you think the graph of $$S$$ looks like.

2. Sketch the graph of the sum of $$E$$ and each of the following functions.

Student Facing

Here are the graphs of two functions, $$U$$ and $$V$$. Define a new function $$W$$ by multiplying $$U$$ and $$V$$, so $$W(x) = U(x) V(x)$$. On the same axes, sketch what you think the graph of $$W$$ looks like.

Activity Synthesis

The purpose of this discussion is for students to share how they made their sketches, focusing on how the numerical value of the new function is defined by the sum of the numerical values of the two original functions. The discussion begins with students sharing techniques that worked for them and then focuses on techniques that did not work and why, calling back to the strategies that did work in order to make sense of what went wrong.

Invite previously identified students who graphed correctly to share their process in the order listed in the Activity Narrative. Display student graphs and any accompanying work for all to see during the explanation. Since these were only sketches, students did not have to make a detailed table, but using a table illustrates clearly that adding two functions together means adding together their output values.

Next, invite previously identified students who took an incorrect approach to share in the order listed in the Activity Narrative. Where possible, identify what function the student did graph. For example, if a student sketched the curve halfway between the two starting graphs, the function they graphed is the average of the two starting graphs instead of the sum. If a student’s sketch appears to have a horizontal asymptote at $$y=0$$ on the right side, it's possible they were thinking more of a function defined by the product of the two original functions.

If time allows, ask students to choose one of the questions and sketch the graph of a function defined by the difference between $$E$$ and the other function.

Lesson Synthesis

Lesson Synthesis

The purpose of this discussion is for students to consider past work from the perspective of combining functions. Ask students to think back on the work they have done with functions and to identify 2–3 contexts where it would make sense to add, subtract, multiply, or divide functions in order to make a new function. Encourage students to make a sketch of what the original functions and new function look like, including axis labels.

After some quiet think time, invite students to share their examples. Possible responses include:

• Adding together the functions for the total population of each of the states and territories since 1990 makes a function for the total population of the United States since 1990.
• Subtracting the function for total cost of producing $$x$$ units from the function for total income for producing $$x$$ units gives the function for total profit for $$x$$ units.
• Multiplying the price of a product by the demand as a function of the price to make a function of the total revenue at different prices.
• Dividing the total cost to make a certain number of items by the number of items made creates a new function that gives the average price per item.

It is important to note that when adding and subtracting, the units for the input and output of each function must match, but the same is not true for units when multiplying or dividing functions.

Student Lesson Summary

Student Facing

We can add, subtract, multiply, and divide functions to get new functions. For example, the cost in dollars of producing $$n$$ cups of lemonade at a lemonade stand is $$C(n) = 5 + 0.8n$$. The revenue (amount of money collected) from selling $$n$$ cups is $$R(n) = 2n$$ dollars. The profit $$P(n)$$ from selling $$n$$ cups is the revenue minus the cost, so

$$\displaystyle P(n) = R(n) - C(n) = 2n - (5 + 0.8n) = 1.2n - 5$$

Here are the graphs of $$C$$, $$R$$, and $$P$$. Can you see how each value on $$P$$ is the result of the difference between the corresponding points on $$R$$ and $$C$$?

The average profit per cup, $$A(n)$$, from selling $$n$$ cups, is the quotient of the profit and the number of cups, so

$$\displaystyle A(n) = \frac{P(n)}{n} = \frac{1.2n - 5}{n} = 1.2 - \frac5n$$

Here are the graphs of $$P$$ and $$A$$. Can you see how the value of $$A(n)$$ is the result of the quotient of $$P(n)$$ and $$n$$? Why does it make sense that both functions are negative when $$n<4\frac16$$ and positive when $$n>4\frac16$$?

Since $$n$$ can only be positive, $$P(n)$$ and $$A(n)$$ always have the same sign for a given $$n$$ value. Notice that for the average profit to be positive, the seller has to sell at least 5 cups (since $$4\frac16$$ is not in the domain, we must round up). It is also true that for a large number of cups, the average profit is close to \\$1.20 per cup.