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### 1. Growth and Decay Factors:
- **If the growth rate is 4%, what is the growth factor?**
  - Growth factor $ b = 1 + r $, where $ r $ is the growth rate in decimal form.
  - Here, $ r = 0.04 $, so $ b = 1 + 0.04 = 1.04 $.

- **If the decay rate is 8%, what is the decay factor?**
  - Decay factor $ b = 1 - r $.
  - Here, $ r = 0.08 $, so $ b = 1 - 0.08 = 0.92 $.

- **If the growth factor is 1.012, what is the growth rate?**
  - Growth factor $ b = 1 + r $.
  - Given $ b = 1.012 $, the growth rate is $ r = 1.012 - 1 = 0.012 $, or 1.2%.

- **If the decay factor is 0.87, what is the decay rate?**
  - Decay factor $ b = 1 - r $.
  - Given $ b = 0.87 $, the decay rate is $ r = 1 - 0.87 = 0.13 $, or 13%.

### 5. Analyzing Exponential Growth:
- The table shows yearly profits, and by comparing year 1 to year 2, year 2 to year 3, and so on, you can see that the profits increase by a factor of approximately 1.1 each year. 
  - For example, $ \frac{2.42}{2.2} \approx 1.1 $ and $ \frac{2.662}{2.42} \approx 1.1 $, indicating a consistent growth.

- **What is the growth factor?**
  - The growth factor is 1.1.

### 7. Writing an Exponential Function:
- Let $ P(t) $ represent the yearly profit $ t $ years since 1997. The general form for an exponential growth model is $ P(t) = a \cdot b^t $, where:
  - $ a $ is the initial value (profit in year 0).
  - $ b $ is the growth factor.
  
  From the table, in year 0, the profit is 2 million dollars, so $ a = 2 $. The growth factor, as previously found, is $ b = 1.1 $. Therefore, the exponential function is:
  \[
  P(t) = 2(1.1)^t
  \]

### 8. Growth Percentage:
- **The yearly profits of this company are increasing by what percent?**
  - Since the growth factor is 1.1, the yearly growth rate $ r = 0.1 $, or 10%.

- **This percent is called** _the growth rate_.

### 9. Car Depreciation:
- The value of your car $ t $ years from now is given by $ V(t) = 18600(0.85)^t $.

  a. **What is the initial value of your car?**
     - The initial value is the value when $ t = 0 $. Plugging this into the equation:
     \[
     V(0) = 18600(0.85)^0 = 18600
     \]
     So, the initial value of the car is $18,600.

  b. **What is the decay rate? What is the practical meaning of this decay rate in this situation?**
     - The decay factor is 0.85, which means the car loses 15% of its value each year (since $ r = 1 - 0.85 = 0.15 $).
     - Practically, this means the car’s value decreases by 15% annually due to depreciation.

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Would you like further details on any of the steps, or do you have any specific questions?

### 5 Follow-up Questions:
1. How would the exponential model change if the growth factor were higher?
2. What if the decay rate for the car was 20% instead of 15%?
3. How can you use the decay factor to predict a car's value after 5 years?
4. How does changing the growth rate affect the exponential function's graph?
5. Can exponential decay be reversed, and if so, how?

### Tip:
When calculating growth or decay rates, always convert percentages to decimal form by dividing by 100. This ensures that you're working with the correct factors for exponential models.

The worksheet includes a variety of questions related to exponential growth and decay factors, profit modeling, and car depreciation.

This worksheet covers exponential growth and decay factors, modeling yearly profits of an airline company, and car depreciation calculations. Students learn to derive growth and decay rates, and model real-world situations using exponential functions. Suitable for students in Grades 9-11.

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