# Exponents and Exponential Functions

## Objective

Solve exponential growth and exponential decay application problems.

## Common Core Standards

### Core Standards

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• 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.LE.A.1 — Distinguish between situations that can be modeled with linear functions and with exponential functions.

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• 8.F.B.4

## Criteria for Success

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1. Identify when a situation has a constant rate of change, an increasing rate of change, or a decreasing rate of change.
2. Write exponential growth and decay functions for real-world situations.
3. Interpret exponential functions in context of their situations.

## Tips for Teachers

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• In regard to pacing, this lesson may be extended over two days, especially if the Desmos modeling activity “iPhone 6s Opening Weekend Sales,” included in the Problem Set Guidance, is used.
• This lesson only includes one Anchor Problem in order to allow for adequate time for independent practice or other targeted instruction based on the needs of your classroom.
• The Anchor Problem and the Target Task both address exponential growth; ensure that students also encounter problems that target exponential decay.

## Anchor Problems

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Leo leaves his house one morning and notices a small plant growing in the yard that he has never noticed before. Out of curiosity, he grabs a ruler and measures it; the plant is 3 cm tall. Exactly one week later, Leo notices the plant has grown quite a bit, so he measures it again; now it is 9 cm tall. And one week after that, it measures 27 cm.

Leo knows that it is unlikely that the plant will continue to triple in height each week indefinitely, but he starts to wonder about the height of the plant before he started to measure it and how he could model its growth mathematically. Suppose that the plant follows a rule, “triples in height each week.”

1. Read the information contained in the table to understand what Leo has written so far, and then complete the table. Write any heights that are less than 1 cm as fractions.
 Week ${-4}$ ${-3}$ ${-2}$ ${-1 }$ $0$ $1$ $2$ $3$ $4$ $5$ $w$ Height (cm) $3$ $9$ $27$ Height Expression $3^2$ $3^3$
1. Express the height of the plant, $h$, as a function of the week it was measured, $w$.
2. Explain in words the meaning of $h(0)$
3. Use your function to find the height of the plant on Week {-4}. Write this value as a fraction. Does the result of the function agree with what you wrote in the table?

#### Guiding Questions

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#### References

Illustrative Mathematics Predicting the PastPart A

Predicting the Past, accessed on May 16, 2018, 4:51 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 Set

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

## Target Task

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Mr. Miller starts working for a technology company this year. His salary the first year is $40,000. According to the company’s employee handbook, each following year Mr. Miller works at the company, he is eligible for a raise equal to 2–5% of his previous year’s salary. Mr. Miller calculates the range of his raise on his first year’s salary. He adds that amount as his raise for each following year. Mr. Miller thinks that: • in his second year working at the company, he would be earning a salary between$40,800 and $42,000, and • in his third year, he would be earning a salary between$41,600 and \$44,000.
1. Based on this reasoning, what salary range would Mr. Miller expect to earn in his tenth year at the company?
2. Mr. Miller’s reasoning is incorrect. Show with diagrams, equations, expressions, or words why his reasoning is incorrect.