Reasoning about Exponential Graphs (Part 2)

5 min

Narrative

This Warm-up prompts students to compare four functions. 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 functions for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three functions that go together and can explain why. Next, tell students to share their response with their group and then together find as many sets of three as they can.

Student Task

Which three go together? Why do they go together?

A(n)=8⋅3nA(n) = 8 \boldcdot 3^n

B(n)=2⋅8nB(n) = 2 \boldcdot 8^n

C(n)=8+2nC(n) = 8+2n

D(n)=8⋅(12)nD(n) = 8 \boldcdot \left( \frac{1}{2} \right)^n

Sample Response

Sample responses:

  • A, B, and C go together because they are increasing.
  • A, B, and D go together because they are exponential functions.
  • A, C, and D go together because the vertical intercept is (0,8)(0,8).
  • B, C, and D go together because they use a 2 in the function.

Synthesis

Invite each group to share one reason why a particular set of three go together. 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 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 “exponential function,” “growth factor,” “initial amount,” and “linear.” 

Standards
Building Toward
  • F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
  • F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
  • F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
  • F-LE.5·Interpret the parameters in a linear or exponential function in terms of a context.
  • F-LE.5·Interpret the parameters in a linear or exponential function in terms of a context.
  • F-LE.5·Interpret the parameters in a linear or exponential function in terms of a context.
  • F-LE.5·Interpret the parameters in a linear or exponential function in terms of a context.
  • HSF-LE.A.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
  • HSF-LE.B.5·Interpret the parameters in a linear or exponential function in terms of a context.

15 min

15 min