To find the average rate of change of [tex]\( f(x) \)[/tex] over the interval [tex]\([-3, 2]\)[/tex], we will use the values of [tex]\( f(x) \)[/tex] at [tex]\( x = -3 \)[/tex] and [tex]\( x = 2 \)[/tex].
From the table, we have:
[tex]\[
f(-3) = -36
\][/tex]
[tex]\[
f(2) = 4
\][/tex]
The formula for the average rate of change of a function [tex]\( f(x) \)[/tex] over an interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\][/tex]
Here, [tex]\( a = -3 \)[/tex] and [tex]\( b = 2 \)[/tex].
Substituting the values in the formula:
[tex]\[
\text{Average rate of change} = \frac{f(2) - f(-3)}{2 - (-3)}
\][/tex]
Now, plug in the given values:
[tex]\[
\text{Average rate of change} = \frac{4 - (-36)}{2 - (-3)}
\][/tex]
Simplify the expression by adding inside the numerator and subtracting in the denominator:
[tex]\[
\text{Average rate of change} = \frac{4 + 36}{2 + 3}
\][/tex]
[tex]\[
\text{Average rate of change} = \frac{40}{5}
\][/tex]
[tex]\[
\text{Average rate of change} = 8
\][/tex]
Therefore, the average rate of change of [tex]\( f(x) \)[/tex] over the interval [tex]\([-3, 2]\)[/tex] is:
[tex]\[
\boxed{8}
\][/tex]
So, the correct answer is [tex]\( \text{C. 8} \)[/tex].
