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

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

where $x_1 = 2$ and $x_2 = 4$.

Step 1: Calculate $f(2)$

$f(2) = e^{2 - 2} - 4 = e^0 - 4 = 1 - 4 = -3$

Step 2: Calculate $f(4)$

$f(4) = e^{4 - 2} - 4 = e^2 - 4$

Using an approximate value of $e^2 \approx 7.389$: $f(4) \approx 7.389 - 4 = 3.389$

Step 3: Substitute into the Average Rate of Change Formula

$\text{Average Rate of Change} = \frac{f(4) - f(2)}{4 - 2} = \frac{3.389 - (-3)}{2} = \frac{3.389 + 3}{2} = \frac{6.389}{2} = 3.1945$

Rounded to the nearest hundredth, the average rate of change is approximately:

$\boxed{3.19}$

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How does the average rate of change relate to the derivative at a point?
2. What is the significance of the exponential function in this context?
3. How would the average rate of change change if the interval was different?
4. Can this method be used for non-exponential functions?
5. How would you interpret the average rate of change for a real-world application?

Tip: The average rate of change over an interval for a function is similar to finding the slope of the secant line connecting two points on the function.