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To find the average rate of change of the function \( f(x) = e^{x - 2} - 4 \) from \( x = 2 \) to \( x = 4 \), we use the formula:

\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

where:
- \( x_1 = 2 \)
- \( x_2 = 4 \)

Let's calculate \( f(2) \) and \( f(4) \).

1. **Calculate \( f(2) \):**
   \[
   f(2) = e^{2 - 2} - 4 = e^0 - 4 = 1 - 4 = -3
   \]

2. **Calculate \( f(4) \):**
   \[
   f(4) = e^{4 - 2} - 4 = e^2 - 4
   \]
   Approximating \( e^2 \approx 7.389 \),
   \[
   f(4) \approx 7.389 - 4 = 3.389
   \]

3. **Find the average rate of change:**
   \[
   \text{Average Rate of Change} = \frac{f(4) - f(2)}{4 - 2} = \frac{3.389 - (-3)}{2} = \frac{3.389 + 3}{2} = \frac{6.389}{2} \approx 3.2
   \]

Thus, the average rate of change is approximately **3.2**.

Would you like a more detailed breakdown or have any questions?

Here are some related questions to explore further:

1. What is the meaning of the average rate of change in a function?
2. How do you find the instantaneous rate of change of a function?
3. How would the average rate of change differ if \( x \) values were closer together?
4. How does the average rate of change relate to the slope of a secant line on a graph?
5. How would you calculate the average rate of change for a non-exponential function?

**Tip:** The average rate of change is equivalent to the slope between two points on the curve of a function.

For the given exponential function, find the average rate for x = 2 and x = 4. f(x) = e^(x - 2) - 4

Calculate the average rate of change of the exponential function f(x) = e^(x - 2) - 4 between x = 2 and x = 4. This solution includes step-by-step calculations and explanations of key exponential function concepts, suitable for high school students.

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