Exponential Growth and Decay Worksheet featuring eight problems with equations and questions about growth or decay rates.

Let’s go through each problem one by one. We’ll look at the formula, figure out if it’s growth or decay, find the starting value (initial value), and then calculate the rate.

Remember:
- If the number inside the parentheses is greater than 1, it’s growth.
- If it’s less than 1, it’s decay.
- The initial value is the number multiplied in front — that’s your “starting point” when t = 0.
- For growth: rate = (base - 1) → convert to percent
- For decay: rate = (1 - base) → convert to percent

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Problem 1: y = 1200 · (1 + 0.3)^t

A. Base is 1 + 0.3 = 1.3 → greater than 1 → exponential growth

B. Initial value = 1200

C. Rate of growth = 0.3 → 30%

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Problem 2: y = 55 · (1 – 0.02)^t

A. Base is 1 – 0.02 = 0.98 → less than 1 → exponential decay

B. Initial value = 55

C. Rate of decay = 0.02 → 2%

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Problem 3: y = 100 · (1.25)^t

A. Base = 1.25 → greater than 1 → exponential growth

B. Initial value = 100

C. Rate of growth = 1.25 – 1 = 0.25 → 25%

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Problem 4: y = 5575 · (0.65)^t

A. Base = 0.65 → less than 1 → exponential decay

B. Initial value = 5575

C. Rate of decay = 1 – 0.65 = 0.35 → 35%

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Problem 5: y = 2000 · (1.05)^t

A. Base = 1.05 → greater than 1 → exponential growth

B. Initial value = 2000

C. Rate of growth = 1.05 – 1 = 0.05 → 5%

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Problem 6: y = 14000 · (0.92)^t

A. Base = 0.92 → less than 1 → exponential decay

B. Initial value = 14000

C. Rate of decay = 1 – 0.92 = 0.08 → 8%

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Problem 7: y = 2250 · (1 – 0.9)^t

First simplify inside: 1 – 0.9 = 0.1 → so base = 0.1

A. Base = 0.1 → less than 1 → exponential decay

B. Initial value = 2250

C. Rate of decay = 1 – 0.1 = 0.9 → 90%

*(Note: This is a very fast decay — losing 90% each time period!)*

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Problem 8: y = 10 · (1 + 0.04)^t

A. Base = 1 + 0.04 = 1.04 → greater than 1 → exponential growth

B. Initial value = 10

C. Rate of growth = 0.04 → 4%

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Final Answer:

1. A. Growth, B. 1200, C. 30%
2. A. Decay, B. 55, C. 2%
3. A. Growth, B. 100, C. 25%
4. A. Decay, B. 5575, C. 35%
5. A. Growth, B. 2000, C. 5%
6. A. Decay, B. 14000, C. 8%
7. A. Decay, B. 2250, C. 90%
8. A. Growth, B. 10, C. 4%