Lesson 13

Representations of Exponential Functions

These materials, when encountered before Algebra 1, Unit 5, Lesson 13 support success in that lesson.

13.1: Which One Doesn’t Belong?: Representations of Functions (0 minutes)

Warm-up

This warm-up prompts students to carefully analyze and compare features of representations of functions. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminology students know and how they talk about characteristics of graphs.

Launch

Arrange students in groups of 2–4. Display the representations 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

0 comma 20 and 2 comma 5 plotted on a parabola 

B

0 comma 20 and 8 comma 28 plotted on a line 

C:  \(f(t)=20 \boldcdot 2^t\)

D

Parabola. 0 comma 10 and 2 comma 40 plotted on parabola.

 

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.

13.2: Interrogating Function Representations (15 minutes)

Activity

This activity consists of two problems that are quite similar. After giving students a chance to tackle it on their own, it may be desirable to demonstrate one, and then ask students to complete the other independently.

Student Facing

  1. Consider the graph of \(f(x)=3 \boldcdot 2^x\) and corresponding table.

    Parabola y= 3 times 2 to the power of x 

    \(x\) \(f(x)\)
    0 3
    1 6
    2 12
    1. Using the first two points, what is the growth factor?
    2. Using the second two points, what is the growth factor?
    3. Where do you see this growth factor in the equation?
    4. Where do you see the growth factor on the graph?
    5. What is the vertical intercept of the graph?
    6. How can you tell from the equation that this is the vertical intercept?
  2. Consider the graph of \(g(x)=8 \boldcdot \left( \frac12 \right)^x\) and corresponding table.

    Parabola y= 8 times one half to the power of x.

    \(x\) \(g(x)\)
    0 8
    1 4
    2 2
    1. Using the first two points, what is the growth factor?
    2. Using the second two points, what is the growth factor?
    3. Where do you see this growth factor in the equation?
    4. Where do you see the growth factor on the graph?
    5. What is the vertical intercept of the graph?
    6. How can you tell from the equation that this is the vertical intercept?

Student Response

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

Activity Synthesis

Ensure students understand how to determine the growth factor, and why the growth factor for \(f\) is 2 but for \(g\) it is \(\frac12\). When discussing how to use the equation to know the vertical intercept, be sure to substitute a 0 for \(x\), rather than just noticing the value of the coefficient.

13.3: Matching Representations of Exponential Functions (25 minutes)

Activity

In this partner activity, students take turns finding a graph that represents a given function. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

Launch

Arrange students in groups of 2. Tell students that for each function, one partner finds a graph that represents it and explains why they think so. The partner’s job is to listen and make sure they agree. If they don't agree, the partners discuss until they come to an agreement. For the next function, the students swap roles. If necessary, demonstrate this protocol before students start working.

Student Facing

  1. Match each function with the graph that represents it.

    \(a(t)=300 \boldcdot 2^t\)

    \(b(t)=300 \boldcdot 3^t\)

    \(c(t)=300 \boldcdot \left( \frac12 \right)^t\)

    \(d(t)=300 \boldcdot \left( \frac13 \right)^t\)

    \(e(t)=108 \boldcdot 2^t\)

    \(f(t)=108 \boldcdot 3^t\)

    \(g(t)=108 \boldcdot \left( \frac12 \right)^t\)

    \(h(t)=108 \boldcdot \left( \frac13 \right)^t\)

    graph 1

    Parabola. Y intercept = 108. Each point’s y coordinate is 2 times previous point.

    graph 2

    Parabola, vertical intercept=108. Y coordinates of one point is one third previous point. 

    graph 3

    Parabola, vertical intercept=300. Y coordinates of one point is one half previous point. 

    graph 4

    Parabola. Y intercept = 300. Each point’s y coordinate is one third of previous point.

    graph 5

    Parabola, vertical intercept=108. Y coordinates of one point is one half previous point. 

    graph 6

    Parabola, vertical intercept=108. Y coordinates of one point is 2 times previous point. 

    graph 7

    Parabola, vertical intercept=108. Y coordinates of one point is 3 times previous point. 

    graph 8

    Parabola. Y intercept = 108. Each point’s y coordinate is 3 times previous point.

  2. On two of the graphs, show where you can see the vertical intercept: 108 and 300.
  3. On four of the graphs, show where you can see the growth factor: \(\frac13\), \(\frac12\), 2, and 3. 

Student Response

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

Activity Synthesis

Much discussion takes place between partners. Invite students to share how they found graphs that represented their functions.

  • “(For any function) what are some things you looked for in the graph?”
  • “Describe any difficulties you experienced and how you resolved them.”
  • “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?” Then, invite a few students to demonstrate where they saw the growth factor on a graph.