Exponential Growth and Decay Worksheet with questions on identifying growth or decay, initial values, and rates of change.

Let’s work through each problem one by one.

We’re given exponential functions in the form:
y = a · (1 ± r)^t

Where:
- a is the initial value (what you start with when t = 0)
- (1 + r) means growth — because you’re adding a percentage
- (1 – r) means decay — because you’re subtracting a percentage
- r is the rate (as a decimal) of growth or decay

So for each function:
A. If it’s (1 + something), it’s growth. If it’s (1 – something), it’s decay.
B. The number multiplying the parentheses is the initial value.
C. The “something” inside the parentheses (after the plus or minus) is the rate, written as a decimal. But sometimes they write it like (1 - 0.9) — that means the rate is 0.9, or 90%. We’ll just report it as shown unless asked to convert to percent.

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Problem 1: y = 1200·(1+0.3)^t
A. (1 + 0.3) → growth
B. Initial value = 1200
C. Rate = 0.3 (or 30%)

Problem 2: y = 55·(1–0.02)^t
A. (1 – 0.02) → decay
B. Initial value = 55
C. Rate = 0.02 (or 2%)

Problem 3: y = 100·(1.25)^t
Wait — this isn’t written as (1 + r). But 1.25 = 1 + 0.25 → so it’s still growth!
A. Growth (since base > 1)
B. Initial value = 100
C. Rate = 0.25 (because 1.25 - 1 = 0.25)

Problem 4: y = 5575·(0.65)^t
Again, not written as (1 - r), but 0.65 < 1 → decay
To find rate: 1 - 0.65 = 0.35 → so rate is 0.35
A. Decay
B. Initial value = 5575
C. Rate = 0.35 (or 35%)

Problem 5: y = 2000·(1.05)^t
1.05 = 1 + 0.05 → growth
A. Growth
B. Initial value = 2000
C. Rate = 0.05 (or 5%)

Problem 6: y = 14000·(0.92)^t
0.92 < 1 → decay
Rate = 1 - 0.92 = 0.08
A. Decay
B. Initial value = 14000
C. Rate = 0.08 (or 8%)

Problem 7: y = 2250·(1–0.9)^t
(1 - 0.9) → decay
A. Decay
B. Initial value = 2250
C. Rate = 0.9 (or 90%) ← yes, even though it’s big, mathematically correct

Problem 8: y = 10·(1+0.04)^t
(1 + 0.04) → growth
A. Growth
B. Initial value = 10
C. Rate = 0.04 (or 4%)

All done! Let me double-check Problem 7 — (1 - 0.9) = 0.1, so after one time unit, you have 10% left — meaning 90% decayed. So rate of decay is indeed 0.9. Correct.

Final Answer:
1. A. Growth, B. 1200, C. 0.3
2. A. Decay, B. 55, C. 0.02
3. A. Growth, B. 100, C. 0.25
4. A. Decay, B. 5575, C. 0.35
5. A. Growth, B. 2000, C. 0.05
6. A. Decay, B. 14000, C. 0.08
7. A. Decay, B. 2250, C. 0.9
8. A. Growth, B. 10, C. 0.04