Lesson 9

Comparing Graphs

9.1: Population Growth (5 minutes)


In this warm-up, students compare functions by analyzing graphs and statements in function notation. The work here prepares students to make more sophisticated comparisons later in the lesson.

Student Facing

This graph shows the populations of Baltimore and Cleveland in the 20th century. \(B(t)\) is the population of Baltimore in year \(t\). \(C(t)\) is the population of Cleveland in year \(t\).

Two functions. year and population.
  1. Estimate \(B(1930)\) and explain what it means in this situation.
  2. Here are pairs of statements about the two populations. In each pair, which statement is true? Be prepared to explain how you know.

    1. \(B(2000) > C(2000)\) or \(B(2000) < C(2000)\)
    2. \(B(1900) = C(1900)\) or \(B(1900) > C(1900)\)
  3. Were the two cities’ populations ever the same? If so, when?

Student Response

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

Activity Synthesis

Invite students to share their response to the first question. After students give a reasonable estimate of the population of Baltimore (about 800,000), display the statement \(B(1930) = 800,\!000\) for all to see. Make sure students can interpret it to mean: “In 1930, the population of Baltimore was about 800,000 people.”

Next, ask students to explain how they knew which statement in each pair of inequalities is true, and how they knew that there were two points in time when Baltimore and Cleveland had the same population.

Ask students how we could use function notation to express that the populations of Baltimore and Cleveland were equal in 1910. If no students mention \(B(1910)=C(1910)\) or \(B(t)=C(t)\) for \(t=1910\), bring these up and display these statements for all to see.

9.2: Wired or Wireless? (20 minutes)


In this activity, students continue to compare two functions by studying graphs and statements in function notation, as well as interpreting them in terms of a situation. They also revisit the meaning of a solution to an equation such as \(C(t) = 20\), both abstractly and in context, and apply what they know about average rate of change to compare the trend shown by each graph.

Additionally, the activity draws students’ attention to the point where the two graphs intersect, its meaning in context, and its corresponding representation in function notation.


Ask students how many of them have a landline phone at home and how many only have cell phones. If students are unfamiliar with landline phones, explain as needed.

Display the graphs for all to see. Discuss questions such as:

  • “How would you describe the trends in phone ownership over the years?” (Cell phones are increasing in use and landlines are decreasing.)
  • “How would you describe the shape of the graph of each function?” (Both are roughly linear. The value of \(H\) is decreasing over time, so the graph slants downward. The value of \(C\) is increasing over time, so the graph slants upward.)
  • “In what year did about 60% of homes have a landline?” (2012)
Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to help students compare and interpret two functions. Display the student task statement and graph, leaving out the questions. Ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving the overall shape of the graphs, trends over specific time intervals, and the intersection point of the two graphs. The process of creating mathematical questions, without the pressure of producing answers or solutions, prompts students to make sense of the given information and to activate the language of mathematical questions. This work helps to prepare students to process the actual questions. 
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation, such as: “One thing that is the same is . . .”, “One thing that is different is . . .”, “_____ represents _____.”
Supports accessibility for: Language; Social-emotional skills

Student Facing

\(H(t)\) is the percentage of homes in the United States that have a landline phone in year \(t\). \(C(t)\) is the percentage of homes with only a cell phone. Here are the graphs of \(H\) and \(C\).

Graph of 2 functions. Blue solid line represents percentage of homes in US with a landline phone in year t. Red dotted line represents the percentage of homes with only a cell phone.
  1. Estimate \(H(2006)\) and \(C(2006)\). Explain what each value tells us about the phones.
  2. What is the approximate solution to \(C(t)=20\)? Explain what the solution means in this situation.
  3. Determine if each equation is true. Be prepared to explain how you know.

    1. \(C(2011) = H(2011)\)
    2. \(C(2015) = H(2015)\)
  4. Between 2004 and 2015, did the percentage of homes with landlines decrease at the same rate at which the percentage of cell-phones-only homes increased? Explain or show 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?

  1. Explain why the statement \(C(t) + H(t) \leq 100\) is true in this situation.
  2. What value does \(C(t) + H(t)\) appear to take between 2004 and 2017? How much does this value vary in that interval?

Student Response

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

Anticipated Misconceptions

Students encounter percentages as the output of a function for the first time in this activity. Some students might think that the output of the functions here must be number of homes, and that they cannot estimate any output values because only percentages are known. Clarify that percent is the unit used in this case, as we are studying how the proportion of the two groups (rather than the actual number of homes in each group) changed over time.

Activity Synthesis

Focus the discussion on the meaning of equations such as \(C(t) = 20\) and \(C(2015)=H(2015)\), and on the meaning of the average rate of change of each function.

Select students to share their responses. Highlight the following points, if not already mentioned in students’ explanations:

  • \(C(t) = 20\) means “in year \(t\), 20% of homes relied only on cell phones” and, based on the graph of \(C\), the value of \(t\) that makes that statement true is 2008.
  • \(C(2015) = H(2015)\) means “in year 2015, the percentage of homes with only cell phones and the percentage of homes with landlines are equal.” We know this is true because at \(t=2015\) the two graphs intersect, which also means they share the same output value.
  • The average rate change for \(C\) is positive because, in the measured interval, the value of \(C\) increased overall. The average rate of \(H\) is negative because the value of \(H\) decreased overall.
  • An average rate of change of 3.8% per year for \(C\) means that the percentage of homes relying only on cell phones grew by about 3.8% each year.
  • An average rate of change of -4% per year means the percentage of homes that used only landlines fell by 4% each year.

If time permits, discuss with students:

  • “The average rate of change for \(C\) is 4% per year, while the average rate of change for \(H\) is -3.8% per year, which is very close to -4%. Why might the rate at which one increased be so close to the rate at which the other decreased? Could it be a coincidence?” (One possible explanation is that, as people relied more on cell phones, they relied less on landlines, and they discontinued using landlines around the same time they acquired new cell phones. These people essentially went from one group to the other.)

9.3: Audience of TV Shows (15 minutes)

Optional activity

Previously, students compared functions by analyzing their graphs on the same coordinate plane. Each graph was a continuous graph.

In this optional activity, students compare functions represented in separate graphs. Each graph is a discrete graph, showing the viewership of three TV shows as functions of the episode number. Students interpret features of the graphs and relate them to descriptions about the shows and to statements in function notation. They use their analyses to draw conclusions about the popularity of the shows and to sketch a possible graph for a fourth TV show.

The work here requires students to make sense of quantities and their relationships while attending to their representations (MP2). In sketching a graph that matches a description, students need to be careful about showing correspondence to the quantities in the situation (MP6), including by using appropriate scale and marks.


Arrange students in groups of 2. Give students a few minutes of quiet work time, then time to discuss their thinking with their partner. Follow with a whole-class discussion.

Student Facing

The number of people who watched a TV episode is a function of that show’s episode number. Here are three graphs of three functions—\(A, B\), and \(C\)—representing three different TV shows.

Show A

10 data points on coordinate plane.

Show B

A graph. 

Show C

10 data points on coordinate plane.
  1. Match each description with a graph that could represent the situation described. One of the descriptions has no corresponding graph.

    1. This show has a good core audience. They had a guest star in the fifth episode that brought in some new viewers, but most of them stopped watching after that.
    2. This show is one of the most popular shows, and its audience keeps increasing.
    3. This show has a small audience, but it’s improving, so more people are noticing.
    4. This show started out huge. Even though it feels like it crashed, it still has more viewers than another show.
  2. Which is greatest, \(A(7)\), \(B(7)\), or \(C(7)\)? Explain what the answer tells us about the shows.
  3. Sketch a graph of the viewership of the fourth TV show that did not have a matching graph.

    A blank graph, with origin O. The horizontal axis, episode number, scale from 0 to 10 by 2s. The vertical axis, audience in millions, scale is 6 units.

Student Response

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

Activity Synthesis

Focus the discussion on how students made their matches. Ask students to explain how parts of the descriptions and features of the graphs led them to believe that a pair of representations belong together.

Next, invite students to share their graph of the fourth TV show. Display the graphs for all to see and discuss how the graphs are alike and how they are different. Because each graph is created based on the same description, they should share some common features. If they look drastically different, solicit possible reasons. (Possible explanations include differences in interpretation of the description or in the choice of scale for the vertical axis, errors in reading the description, and plotting errors.)

If time permits, ask students,

  • “How are the graphs or the functions in this activity different than those in earlier activities?” (They show points rather than lines. The input is episode number, while in other cases, the input was time. Each function is shown on a different coordinate plane.)
  • “How is the work of comparing functions here like comparing functions in earlier activities?” (They all involve comparing the output values of points on a graph, interpreting the points, making sense of verbal descriptions, and some estimating.)
  • “How is the work of comparing functions here different than in earlier activities?” (Here we are comparing function values across three separate graphs, which is trickier than when the graphs are all on the same coordinate plane. Figuring out which function has a greater or lesser value was also easier when the functions were represented with lines or curves. It is harder to do with a set of points, especially when they are not in the same image.)
Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion as students explain how they matched the description to the graph. Display the following sentence frames for all to see: “_____ matches _____ because . . .” and “I noticed _____, so . . . .” Encourage students to challenge each other when they disagree. This routine will help students explain how they used parts of the descriptions and features of the graphs to find matches.
Design Principle(s): Support sense-making

9.4: Functions $f$ and $g$ (10 minutes)


In this activity, students compare graphs and statements that represent functions without a context. Because no concrete information is given, students need to rely on their understanding of function notation and points on a graph to make comparisons and to interpret intersections of the graphs.

Previously, students saw equations such as \(f(8) = g(8)\) and interpreted them in terms of a situation. Here, they begin to reason more abstractly about statements of the form \(f(x) = g(x)\) and relate it to one or more points where the graphs of \(f\) and \(g\) intersect. They see that a value of \(x\) that makes this equation true, or a solution to the equation, is the input value of such an intersection.

Student Facing

  1. Here are graphs that represent two functions, \(f\) and \(g\).

    Decide which function value is greater for each given input. Be prepared to explain your reasoning.

    1. \(f(2)\) or \(g(2)\)
    2. \(f(4)\) or \(g(4)\)
    3. \(f(6)\) or \(g(6)\)
    4. \(f(8)\) or \(g(8)\)
    A graph. 


  2. Is there a value of \(x\) at which the equation \(f(x)=g(x)\) is true? Explain your reasoning.
  3. Identify at least two values of \(x\) at which the inequality \(f(x) < g(x)\) is true.

Student Response

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

Activity Synthesis

Display the graphs of \(f\) and \(g\) for all to see. Invite students to share their responses to the first set of questions. As they point out the greater function value in each pair, mark the point on the graph and write a corresponding statement in function notation: \(f(2) < g(2)\), \(f(4) = g(4)\), \(f(6) > g(6)\), and \(f(8)>g(8)\). Emphasize that the function value that is greater in each pair has a higher vertical value on the coordinate plane for the same input value.

Next, discuss if there are values of \(x\) that make \(f(x)=g(x)\) true and how we can tell. Point out that earlier we wrote \(f(4)=g(4)\) because the values of \(f\) and \(g\) are equal when the input is 4. This means that \((4,f(4))\) and \((4,g(4))\) are coordinates of the same point.

So we can interpret an equation such as \(f(x)=g(x)\) to mean that the values of \(f\) and \(g\) are equal when the input is \(x\), and that \(x\) must be the horizontal value of the intersection of both graphs, which is a point that they share.

Conversing: MLR2 Collect and Display. During the synthesis, listen for and collect the language students use to share their interpretations of equations written as \(f(x)=g(x)\). Write students’ words and phrases on a visual display. Be sure to emphasize words and phrases such as, “the values are equal,” “intersect,” “horizontal value,” “input value,” and “common point.” Remind students to borrow language from the display as needed. This will help students begin to understand statements of the form \(f(x)=g(x)\) and make connections to the graph.
Design Principle(s): Maximize meta-awareness; Support sense-making

Lesson Synthesis

Lesson Synthesis

Refer back to the population graphs for Baltimore and Cleveland from 1900 to 2010, which students saw in the warm-up. Display the graphs for all to see.

Two functions. year and population.

Present students with the following statements, one at a time, about the populations of the two cities. Tell students that their job is to explain how they could tell from the graphs that each statement is true, and to translate each verbal description into a statement with the same meaning but written in function notation.

Consider using a three-column graphic organizer (as shown here and in the Lesson Summary) to organize students’ responses.

What can we tell about the
    How can we tell?     How can we convey this with function notation?

In 2010, Baltimore had more people than Cleveland.


Baltimore and Cleveland had the same population twice in the past century, in 1910 and around 1944.


After the mid-1940s, Cleveland has had a smaller population than Baltimore.


In the first half of the 20th century, the population of Cleveland grew at a faster rate than that of Baltimore.


Since 1950, the population of Cleveland has dropped at a faster rate than that of Baltimore.


9.5: Cool-down - A Toy Rocket and a Drone Again (5 minutes)


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

Student Lesson Summary

Student Facing

Graphs are very useful for comparing two or more functions. Here are graphs of functions \(C\) and \(T\), which give the populations (in millions) of California and Texas in year \(x\).

Graph of the populations of two states over time on a coordinate plane. 
What can we tell about the
How can we tell? How can we convey this
with function notation?

In the early 1900s, California had a smaller population than Texas.

The graph of \(C\) is below the graph of \(T\) when \(x\) is 1900.

\(C(1900) < T(1900)\)

Around 1935, the two states had the same population of about 5 million people.

The graphs intersect at about \((1935, 5)\).

\(C(1935) = 5\) and \(T(1935)=5\), and \(C(1935)=T(1935)\)

After 1935, California has had more people than Texas.

When \(x\) is greater than 1935, the graph of \(C(x)\) is above that of \(T(x)\) .

\(C(x) > T(x)\) for \(x>1935\)

Both populations have increased over time, with no periods of decline.

Both graphs slant upward from left to right.


From 1900 to 2010, the population of California has risen faster than that of Texas. California had a greater average rate of change.

If we draw a line to connect the points for 1900 and 2010 on each graph, the line for \(C\) has a greater slope than that for \(T\).