To find the average rate of change of the exponential function $f(x) = e^{x-2} - 6$ over the interval $x = 2$ and $x = 6$, we use the formula for the average rate of change:

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

Step 1: Evaluate $f(2)$

Substitute $x = 2$ into $f(x)$: $f(2) = e^{2-2} - 6 = e^0 - 6 = 1 - 6 = -5$

Step 2: Evaluate $f(6)$

Substitute $x = 6$ into $f(x)$: $f(6) = e^{6-2} - 6 = e^4 - 6$

Step 3: Compute the Average Rate of Change

Now substitute $f(2) = -5$ and $f(6) = e^4 - 6$ into the formula: $\text{Average Rate of Change} = \frac{f(6) - f(2)}{6 - 2} = \frac{(e^4 - 6) - (-5)}{4}$

Simplify the numerator: $\text{Average Rate of Change} = \frac{e^4 - 6 + 5}{4} = \frac{e^4 - 1}{4}$

Final Answer:

The average rate of change is: $\boxed{\frac{e^4 - 1}{4}}$

Let me know if you'd like further clarification or details!

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Here are 5 related questions you might explore:

1. How do you find the instantaneous rate of change for $f(x)$ at a given point?
2. What does the average rate of change represent graphically for this function?
3. How would the average rate of change differ for a different interval, say $x = 1$ to $x = 5$?
4. Can you compute the exact value of the average rate of change using a calculator for $e^4$?
5. How does the average rate of change of an exponential function compare to that of a linear function?

Tip: The average rate of change is essentially the slope of the secant line connecting two points on the curve!