Math worksheet on exponential growth and decay with questions and a grading scale.

It looks like you've uploaded a worksheet titled "Quiz: Exponent of Growth and Decay" with 10 questions related to exponential functions, specifically focusing on identifying whether a function represents growth or decay based on its base (the exponent), and calculating the percent change from the exponential expression.

Since I can't view images directly, I’ll solve and explain each question based on the text you provided, assuming this is what’s in the image. Here's the full solution:

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Quiz: Exponent of Growth and Decay

#### 1. Given: $ y = a \cdot b^x $, what is $ a $?
- Answer: $ a $ is the initial value or starting amount when $ x = 0 $. It's the y-intercept of the exponential function.
- Explanation: When $ x = 0 $, $ y = a \cdot b^0 = a \cdot 1 = a $. So $ a $ is the value at the beginning.

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#### 2. Is the graph of $ y = 1.8 \cdot 3^x $ an exponential growth or decay?
- Answer: Growth
- Explanation: The base $ b = 3 $, which is greater than 1. Since $ b > 1 $, the function increases as $ x $ increases → exponential growth.

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#### 3. Given: $ y = a \cdot b^x $, where $ b = 2 $
- Answer: This means the function doubles for every unit increase in $ x $.
- Explanation: If $ b = 2 $, then $ y $ multiplies by 2 each time $ x $ increases by 1. So it's doubling.

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#### 4. The equation $ y = a \cdot b^x $ is an exponential growth when $ b $ equals...
- Answer: $ b > 1 $
- Explanation: For exponential growth, the base $ b $ must be greater than 1. If $ b < 1 $, it's decay.

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#### 5. Is the graph of $ y = 5 \cdot 0.6^x $ an exponential growth or decay?
- Answer: Decay
- Explanation: The base $ b = 0.6 $, which is between 0 and 1. Since $ 0 < b < 1 $, the function decreases over time → exponential decay.

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#### 6. Given: $ y = a \cdot b^x $, where $ b = 0.5 $
- Answer: The function halves (decreases by half) for every unit increase in $ x $.
- Explanation: $ b = 0.5 $ means each step, the output is multiplied by 0.5 → it halves.

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#### 7. Is the graph of $ y = 2 \cdot 1.9^x $ an exponential growth or decay?
- Answer: Growth
- Explanation: Base $ b = 1.9 > 1 $, so the function grows exponentially.

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#### 8. The graph of $ y = 1.07 \cdot 1.05^x $ is which quadrant?
- Answer: Quadrant I
- Explanation:
- $ a = 1.07 > 0 $, so the initial value is positive.
- $ b = 1.05 > 1 $, so it's growing.
- Exponential functions with positive $ a $ and $ b > 1 $ start at $ (0, a) $ and increase to infinity.
- All values of $ y $ are positive, and $ x $ can be any real number.
- But since $ x $ typically starts at 0 and increases, and $ y > 0 $, the graph lies in Quadrant I (positive $ x $, positive $ y $).
- Note: For negative $ x $, it may extend into Quadrant II, but usually we consider $ x \geq 0 $ in such contexts.

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#### 9. Write an exponential equation using the percentage change for the function $ y = 2 \cdot 1.15^x $
- Answer: $ y = 2 \cdot (1 + 0.15)^x $
- Explanation:
- $ b = 1.15 $ → this means a 15% increase per unit of $ x $.
- So the percentage change is +15%.
- The general form is $ y = a(1 + r)^x $, where $ r $ is the rate of change.
- So $ r = 0.15 $, and $ y = 2(1 + 0.15)^x $

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#### 10. Given: $ y = 3 \cdot 0.8^x $, what is $ r $?
- Answer: $ r = -0.2 $ or $ -20\% $
- Explanation:
- $ b = 0.8 $, so $ b = 1 + r $
- Solve: $ 0.8 = 1 + r $ → $ r = -0.2 $
- So the rate of change is $ -20\% $, meaning it decreases by 20% each time $ x $ increases by 1.

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Bonus: Percentage Change Table (at bottom)

This table shows:
| Percent Change | -100% | -90% | -80% | -70% | -60% | -50% | -40% | -30% | -20% | -10% | 0% | 10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90% | 100% |
|----------------|-------|------|------|------|------|------|------|------|------|------|----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|------|
| Base $ b $ | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.6 | 1.7 | 1.8 | 1.9 | 2.0 | 2.0 |

- Note: $ b = 1 + r $, where $ r $ is the rate of change (in decimal).
- Example: 20% growth → $ r = 0.2 $ → $ b = 1.2 $
- 20% decay → $ r = -0.2 $ → $ b = 0.8 $

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✔ Summary of Key Concepts:

- Exponential Growth: $ b > 1 $ → increases
- Exponential Decay: $ 0 < b < 1 $ → decreases
- $ a $: Initial value (when $ x = 0 $)
- $ b = 1 + r $, where $ r $ is the rate of change (as a decimal)
- Use $ y = a(1 + r)^x $ to model real-world growth/decay

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