Exponential Functions: Understanding Growth and Decay with Graphs and Equations

Let's solve the problem step by step based on the image you provided. The task involves exponential functions of the form:

$$
f(x) = a(b)^x
$$

We are given:
- A general formula and explanation of exponential growth and decay.
- A table to complete for the function $ f(x) = a(b)^x $.
- Four specific functions to classify as growth or decay.

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🔹 Step 1: Complete the Table

The table shows how the function evolves with increasing $ x $. We are given:

| $ x $ | Process | $ y $ |
|--------|---------------------|---------------|
| 0 | | $ a $ |
| 1 | $ ab $ | $ ab $ |
| 2 | $ a(b)(b) $ | $ ab^2 $ |
| 3 | $ a(b)(b)(b) $ | $ ab^3 $ |
| 4 | $ a(b)(b)(b)(b) $ | $ ab^4 $ |
| 5 | | $ ab^5 $ |

So, we just need to fill in the "Process" column for $ x = 0 $ and $ x = 5 $.

- For $ x = 0 $: Since $ f(0) = a(b)^0 = a \cdot 1 = a $, the process is just $ a $ (no multiplication).
- For $ x = 5 $: The process is multiplying $ a $ by $ b $ five times: $ a(b)(b)(b)(b)(b) $

✔ Completed table:

| $ x $ | Process | $ y $ |
|--------|--------------------------|-------------|
| 0 | $ a $ | $ a $ |
| 1 | $ ab $ | $ ab $ |
| 2 | $ a(b)(b) $ | $ ab^2 $ |
| 3 | $ a(b)(b)(b) $ | $ ab^3 $ |
| 4 | $ a(b)(b)(b)(b) $ | $ ab^4 $ |
| 5 | $ a(b)(b)(b)(b)(b) $ | $ ab^5 $ |

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🔹 Step 2: Classify Each Function as Growth or Decay

Recall:
- If $ b > 1 $ → Growth
- If $ 0 < b < 1 $ → Decay

Now analyze each function:

#### 1. $ y = \frac{1}{2}(1.5)^x $

- Base $ b = 1.5 $
- Since $ 1.5 > 1 $ → Growth

✔ Answer: G

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#### 2. $ y = 2(0.5)^x $

- Base $ b = 0.5 $
- Since $ 0 < 0.5 < 1 $ → Decay

✔ Answer: D

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#### 3. $ y = 2^x $

- This is equivalent to $ y = 1 \cdot 2^x $, so $ a = 1 $, $ b = 2 $
- $ b = 2 > 1 $ → Growth

✔ Answer: G

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#### 4. $ y = 0.8^x $

- This is $ y = 1 \cdot (0.8)^x $, so $ b = 0.8 $
- $ 0 < 0.8 < 1 $ → Decay

✔ Answer: D

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✔ Final Answers:

Table (completed):

| $ x $ | Process | $ y $ |
|--------|-------------------------------|---------------|
| 0 | $ a $ | $ a $ |
| 1 | $ ab $ | $ ab $ |
| 2 | $ a(b)(b) $ | $ ab^2 $ |
| 3 | $ a(b)(b)(b) $ | $ ab^3 $ |
| 4 | $ a(b)(b)(b)(b) $ | $ ab^4 $ |
| 5 | $ a(b)(b)(b)(b)(b) $ | $ ab^5 $ |

Classifications:

1. $ y = \frac{1}{2}(1.5)^x $ → G
2. $ y = 2(0.5)^x $ → D
3. $ y = 2^x $ → G
4. $ y = 0.8^x $ → D

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Let me know if you'd like this explained visually or with graphs!