To determine the average rate of change for the given exponential function on the interval from [tex]\( x = 2 \)[/tex] to [tex]\( x = 4 \)[/tex], we follow these steps:
1. Identify the values of the function at [tex]\( x = 2 \)[/tex] and [tex]\( x = 4 \)[/tex]:
From the table, we have:
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 4 \)[/tex]
- When [tex]\( x = 4 \)[/tex], [tex]\( y = 16 \)[/tex]
2. Calculate the change in [tex]\( y \)[/tex] (Δy):
[tex]\[
Δy = y_{4} - y_{2} = 16 - 4 = 12
\][/tex]
3. Calculate the change in [tex]\( x \)[/tex] (Δx):
[tex]\[
Δx = x_{4} - x_{2} = 4 - 2 = 2
\][/tex]
4. Compute the average rate of change:
[tex]\[
\text{Average Rate of Change} = \frac{Δy}{Δx} = \frac{12}{2} = 6
\][/tex]
Therefore, the average rate of change for the function on the interval from [tex]\( x = 2 \)[/tex] to [tex]\( x = 4 \)[/tex] is [tex]\(\boxed{6}\)[/tex].
