Lesson 9

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.

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.

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)

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.

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.)

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.

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.)

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.

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.