Solved) - Exponential Growth and Decay Worksheet 1. y = 1200-(1+ ... - Free Printable

Based on the image you provided, this is an “Exponential Growth and Decay Worksheet” where you are asked to analyze several exponential functions of the form `y = a * b^x`.

The circled problems are:

1. y = 0.32(0.98)^x
2. y = 0.775(1.04)^x
3. y = 14,000(0.92)^x

For each of these, you need to answer three questions:
- a. Does the function represent exponential growth or exponential decay?
- b. What is your initial value?
- c. What is the rate of growth or rate of decay?

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🔍 General Rules for Exponential Functions: y = a * b^x

- Initial Value (a): This is the value of `y` when `x = 0`. So, `a` is always the initial value.
- Growth vs. Decay:
- If `b > 1`, the function represents exponential growth.
- If `0 < b < 1`, the function represents exponential decay.
- Rate of Growth/Decay:
- For growth: Rate = `(b - 1) * 100%`
- For decay: Rate = `(1 - b) * 100%`

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## ✔ Problem 1: y = 0.32(0.98)^x

- a. Growth or Decay?
- Base `b = 0.98`
- Since `0 < 0.98 < 1`, this is exponential decay.

- b. Initial Value?
- `a = 0.32`
- So, initial value = 0.32

- c. Rate of Decay?
- Rate = `(1 - b) * 100% = (1 - 0.98) * 100% = 0.02 * 100% = 2%`
- So, rate of decay = 2%

> 📌 Answer for #1:
> a. Exponential decay
> b. 0.32
> c. 2%

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## ✔ Problem 2: y = 0.775(1.04)^x

- a. Growth or Decay?
- Base `b = 1.04`
- Since `1.04 > 1`, this is exponential growth.

- b. Initial Value?
- `a = 0.775`
- So, initial value = 0.775

- c. Rate of Growth?
- Rate = `(b - 1) * 100% = (1.04 - 1) * 100% = 0.04 * 100% = 4%`
- So, rate of growth = 4%

> 📌 Answer for #2:
> a. Exponential growth
> b. 0.775
> c. 4%

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## ✔ Problem 3: y = 14,000(0.92)^x

- a. Growth or Decay?
- Base `b = 0.92`
- Since `0 < 0.92 < 1`, this is exponential decay.

- b. Initial Value?
- `a = 14,000`
- So, initial value = 14,000

- c. Rate of Decay?
- Rate = `(1 - b) * 100% = (1 - 0.92) * 100% = 0.08 * 100% = 8%`
- So, rate of decay = 8%

> 📌 Answer for #3:
> a. Exponential decay
> b. 14,000
> c. 8%

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## ✔ Final Answers Summary:

| Problem | Function | a. Type | b. Initial Value | c. Rate |
|---------|--------------------|-----------------|------------------|-------------|
| 1 | y = 0.32(0.98)^x | Decay | 0.32 | 2% |
| 2 | y = 0.775(1.04)^x | Growth | 0.775 | 4% |
| 3 | y = 14,000(0.92)^x | Decay | 14,000 | 8% |

You can now fill in these answers on your worksheet! Let me know if you’d like help with the other problems too.