Exponential Modeling and Logarithms

Lesson 2

Math

Unit 5

11th Grade

Lesson 2 of 16

Objective


Analyze and construct exponential functions that model contexts.

Common Core Standards


Core Standards

  • F.IF.C.8.B — Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01 12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
  • F.IF.B.4 — For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
  • F.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).

Foundational Standards

  • F.LE.A.1.A
  • F.LE.A.1.B
  • F.LE.A.1.C

Criteria for Success


  1. Identify y-intercepts and rates of growth or decay from graphs, tables, and equations.
  2. Connect features of exponential functions to contextual situations.
  3. Write and analyze exponential functions that model contextual situations.

Tips for Teachers


This lesson requires some knowledge of simple function compositions and rational exponents. These skills should be reviewed before or during the lesson.

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Anchor Problems


Problem 1

The graph of a function of the form $${f(x)=ab^x}$$ is shown below. Find the values of $$a$$ and $$b$$.

Guiding Questions

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References

Illustrative Mathematics Two Points Determine an Exponential Function I

Two Points Determine an Exponential Function I, accessed on Feb. 22, 2018, 2:17 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem 2

A fisherman illegally introduces some fish into a lake, and they quickly propagate. The growth of the population of this new species (within a period of a few years) is modeled by $${P(x)=5b^x}$$, where $$x$$ is the time in weeks following the introduction and $$b$$ is a positive unknown base.

  1. Exactly how many fish did the fisherman release into the lake?
  2. Find $$b$$ is you know the lake contains $${33}$$ fish after eight weeks. Show step-by-step work.
  3. Instead, now suppose that $${P(x)=5b^x}$$and $$b=2$$ What is the weekly percent growth rate in this case? What does this mean in every-day language?

Guiding Questions

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References

Illustrative Mathematics Illegal Fish

Illegal Fish, accessed on Feb. 22, 2018, 2:17 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem 3

Do the following equations represent growth or decay?

$${y=2(3)^{x\over5}}$$                $${y=2\left ( 1\over3 \right )^x}$$                  $${y=2(3)^{-x}}$$                   $${y={1\over2}(3)^x}$$

Guiding Questions

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Target Task


According to Wikipedia, the International Basketball Federation (FIBA) requires that a basketball bounce to a height of $${1300}$$ mm when dropped from a height of $${1800}$$ mm.

  1. Suppose you drop a basketball and the ratio of each rebound height to the previous rebound height is $${1300}:{1800}$$. Let $$h$$ be the function that assigns to $$n$$ the rebound height of the ball (in mm) on the $$n^{th}$$ bounce. Complete the chart below, rounding to the nearest mm.
$$n$$ $$h(n)$$
0 {1800}
1  
2  
3  
  1. Write an expression for $$h(n)$$.

References

Illustrative Mathematics Basketball Rebounds

Basketball Rebounds, accessed on Feb. 22, 2018, 2:34 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include questions converting between graphs, tables, equations, and contextual situations.
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Lesson 1

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Lesson 3

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Modeling with and Interpreting Exponential Functions

Topic B: Definition and Meaning of Logarithms

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