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.