Free Exponential Growth and Decay Worksheets with Answer Key - Free Printable

Let's solve each problem step by step using the general form of an exponential function:

$$
y = a \cdot (1 + r)^t
$$

Where:
- $ y $ is the value after time $ t $
- $ a $ is the initial value
- $ r $ is the rate of growth or decay (expressed as a decimal)
- If $ r > 0 $, it's exponential growth
- If $ r < 0 $, it's exponential decay

We can also write this as:
$$
y = a \cdot b^t
$$
where $ b = 1 + r $

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Problem 1: $ y = 1200 \cdot (1 + 0.3)^t $

A. Growth or Decay?
Since $ 1 + 0.3 = 1.3 > 1 $, this is exponential growth.

B. Initial Value?
The initial value is $ a = 1200 $.

C. Rate of Growth or Decay?
$ r = 0.3 = 30\% $ → Growth rate: 30%

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Problem 2: $ y = 55 \cdot (1 - 0.02)^t $

A. Growth or Decay?
$ 1 - 0.02 = 0.98 < 1 $ → Exponential decay

B. Initial Value?
$ a = 55 $

C. Rate of Decay?
$ r = -0.02 = -2\% $ → Decay rate: 2%

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Problem 3: $ y = 100 \cdot (1.25)^t $

A. Growth or Decay?
$ 1.25 > 1 $ → Exponential growth

B. Initial Value?
$ a = 100 $

C. Rate of Growth?
$ 1.25 = 1 + r $ → $ r = 0.25 = 25\% $ → Growth rate: 25%

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Problem 4: $ y = 5575 \cdot (0.65)^t $

A. Growth or Decay?
$ 0.65 < 1 $ → Exponential decay

B. Initial Value?
$ a = 5575 $

C. Rate of Decay?
$ 0.65 = 1 + r $ → $ r = -0.35 = -35\% $ → Decay rate: 35%

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Problem 5: $ y = 2000 \cdot (1.05)^t $

A. Growth or Decay?
$ 1.05 > 1 $ → Exponential growth

B. Initial Value?
$ a = 2000 $

C. Rate of Growth?
$ r = 0.05 = 5\% $ → Growth rate: 5%

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Problem 6: $ y = 14000 \cdot (0.92)^t $

A. Growth or Decay?
$ 0.92 < 1 $ → Exponential decay

B. Initial Value?
$ a = 14000 $

C. Rate of Decay?
$ r = 0.92 - 1 = -0.08 = -8\% $ → Decay rate: 8%

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Problem 7: $ y = 2250 \cdot (1 - 0.9)^t $

Simplify: $ 1 - 0.9 = 0.1 $, so $ y = 2250 \cdot (0.1)^t $

A. Growth or Decay?
$ 0.1 < 1 $ → Exponential decay

B. Initial Value?
$ a = 2250 $

C. Rate of Decay?
$ r = -0.9 = -90\% $ → Decay rate: 90%

> Note: This is a very rapid decay — only 10% remains each time period.

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Problem 8: $ y = 10 \cdot (1 + 0.04)^t $

A. Growth or Decay?
$ 1 + 0.04 = 1.04 > 1 $ → Exponential growth

B. Initial Value?
$ a = 10 $

C. Rate of Growth?
$ r = 0.04 = 4\% $ → Growth rate: 4%

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✔ Final Answers Summary:

| Problem | A. Type | B. Initial Value | C. Rate |
|--------|------------------|------------------|--------|
| 1 | Growth | 1200 | 30% |
| 2 | Decay | 55 | 2% |
| 3 | Growth | 100 | 25% |
| 4 | Decay | 5575 | 35% |
| 5 | Growth | 2000 | 5% |
| 6 | Decay | 14000 | 8% |
| 7 | Decay | 2250 | 90% |
| 8 | Growth | 10 | 4% |

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🔍 Explanation Recap:

- The initial value is always the coefficient in front of the exponential part.
- To determine growth or decay, check whether the base $ b $ is greater than or less than 1.
- The rate is found from $ r = b - 1 $. If $ r > 0 $, it’s growth; if $ r < 0 $, it’s decay.
- Rates are often expressed as percentages.

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