Worksheet on exponential growth and decay with problems on calculating multipliers, identifying growth or decay, and solving real-world application questions.

Let’s go through each problem step by step. I’ll explain everything clearly so you can understand how to solve them on your own.

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Part 1: Given the growth/decay percentage, determine the multiplier.

The “multiplier” is what you multiply by each time period to get the new amount.

- For growth: Add the percent (as a decimal) to 1.
- For decay: Subtract the percent (as a decimal) from 1.

Let’s do each one:

1) 5% Growth
→ 5% = 0.05 → Multiplier = 1 + 0.05 = 1.05

2) 12% Decay
→ 12% = 0.12 → Multiplier = 1 - 0.12 = 0.88

3) 200% Growth
→ 200% = 2.00 → Multiplier = 1 + 2.00 = 3.00

4) 0.85% Decay
→ 0.85% = 0.0085 → Multiplier = 1 - 0.0085 = 0.9915

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Part 2: State whether each equation represents growth or decay.

Look at the base of the exponent (the number being raised to the power x).

- If base > 1 → Growth
- If base < 1 → Decay

5) f(x) = 3^x → Base = 3 → Growth

6) f(x) = 0.25^x → Base = 0.25 → Decay

7) f(x) = 1.01^x → Base = 1.01 → Growth

8) f(x) = 0.033^x → Base = 0.033 → Decay

9) f(x) = 6·5^x → Base = 5 → Growth (the 6 is just a starting value)

10) f(x) = 6·(1/2)^x → Base = 1/2 = 0.5 → Decay

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Part 3: Identify necessary information and solve.

We’re dealing with exponential models:
Final Amount = Initial Amount × (Multiplier)^time

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Problem 11: Movie Tickets

Current price = $9.75
Increasing 15% per year → This is growth
Time = 5 years

a. Growth or Decay? → Growth (price is increasing)

b. What is your multiplier?
→ 15% = 0.15 → Multiplier = 1 + 0.15 = 1.15

c. Is $9.75 your zero term or first term?
→ It’s the current price, before any years pass → That’s the zero term (at time x=0)

d. Write the explicit equation.
→ Let f(x) = cost after x years
→ f(x) = 9.75 × (1.15)^x

e. Solve for x = 5
→ f(5) = 9.75 × (1.15)^5
First calculate (1.15)^5:

1.15^2 = 1.3225
1.15^3 = 1.3225 × 1.15 ≈ 1.520875
1.15^4 ≈ 1.520875 × 1.15 ≈ 1.74900625
1.15^5 ≈ 1.74900625 × 1.15 ≈ 2.0113571875

Now multiply by 9.75:

9.75 × 2.0113571875 ≈ ?

Let me compute that carefully:

9.75 × 2 = 19.5
9.75 × 0.0113571875 ≈ let’s break it down:

First, 9.75 × 0.01 = 0.0975
9.75 × 0.001 = 0.00975
9.75 × 0.0003 = ~0.002925
9.75 × 0.0000571875 ≈ very small, maybe ignore for now? Better to use calculator-style multiplication.

Actually, better way:

9.75 × 2.0113571875
= (10 - 0.25) × 2.0113571875
= 10×2.0113571875 - 0.25×2.0113571875
= 20.113571875 - 0.502839296875
≈ 19.610732578125

So approximately $19.61

But let me double-check with direct multiplication:

2.0113571875 × 9.75

Break 9.75 into 9 + 0.75

2.0113571875 × 9 = 18.1022146875
2.0113571875 × 0.75 = 1.508517890625
Add them: 18.1022146875 + 1.508517890625 = 19.610732578125

Yes → Round to nearest cent: $19.61

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Problem 12: Computer Value

Purchased for $2000
Loses 20% of value each year → Decay
Time = 4 years

a. Growth or decay? → Decay (losing value)

b. What is your multiplier?
→ 20% = 0.20 → Multiplier = 1 - 0.20 = 0.80

c. Is $2000 your zero term or first term?
→ It’s the initial purchase price → Zero term (at time x=0)

d. Write the explicit equation.
→ Let f(x) = value after x years
→ f(x) = 2000 × (0.80)^x

e. Solve for x = 4
→ f(4) = 2000 × (0.80)^4

Calculate (0.80)^4:

0.80^2 = 0.64
0.80^4 = (0.64)^2 = 0.4096

Now multiply: 2000 × 0.4096 = ?

2000 × 0.4 = 800
2000 × 0.0096 = 19.2
Total = 800 + 19.2 = 819.2

So, $819.20

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Final Answer:

1) 1.05
2) 0.88
3) 3.00
4) 0.9915
5) Growth
6) Decay
7) Growth
8) Decay
9) Growth
10) Decay
11a) Growth
11b) 1.15
11c) Zero term
11d) f(x) = 9.75 × (1.15)^x
11e) $19.61
12a) Decay
12b) 0.80
12c) Zero term
12d) f(x) = 2000 × (0.80)^x
12e) $819.20