To determine which function has an average rate of change of -4 over the interval [tex]\([-2,2]\)[/tex], we will use the average rate of change formula for a function [tex]\(f(x)\)[/tex] over the interval [tex]\([a, b]\)[/tex]:
[tex]\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\][/tex]
Given the interval [tex]\([-2, 2]\)[/tex], we have [tex]\(a = -2\)[/tex] and [tex]\(b = 2\)[/tex].
Let's find the average rate of change for each function:
1. For [tex]\(n(x)\)[/tex]:
[tex]\[
\begin{aligned}
& n(-2) = -6, \\
& n(2) = 6, \\
& \text{Average rate of change} = \frac{n(2) - n(-2)}{2 - (-2)} = \frac{6 - (-6)}{4} = \frac{6 + 6}{4} = \frac{12}{4} = 3.
\end{aligned}
\][/tex]
2. For [tex]\(m(x)\)[/tex]:
[tex]\[
\begin{aligned}
& m(-2) = -12, \\
& m(2) = 4, \\
& \text{Average rate of change} = \frac{m(2) - m(-2)}{2 - (-2)} = \frac{4 - (-12)}{4} = \frac{4 + 12}{4} = \frac{16}{4} = 4.
\end{aligned}
\][/tex]
3. For [tex]\(q(x)\)[/tex]:
[tex]\[
\begin{aligned}
& q(-2) = -4, \\
& q(2) = -12, \\
& \text{Average rate of change} = \frac{q(2) - q(-2)}{2 - (-2)} = \frac{-12 - (-4)}{4} = \frac{-12 + 4}{4} = \frac{-8}{4} = -2.
\end{aligned}
\][/tex]
4. For [tex]\(p(x)\)[/tex]:
[tex]\[
\begin{aligned}
& p(-2) = 12, \\
& p(2) = -4, \\
& \text{Average rate of change} = \frac{p(2) - p(-2)}{2 - (-2)} = \frac{-4 - 12}{4} = \frac{-4 - 12}{4} = \frac{-16}{4} = -4.
\end{aligned}
\][/tex]
Given the above calculations, the function [tex]\(p(x)\)[/tex] has an average rate of change of -4 over the interval [tex]\([-2, 2]\)[/tex].
Therefore, the correct answer is:
D.
\begin{tabular}{|c|c|c|c|c|c|}
\hline[tex]$x$[/tex] & -2 & -1 & 0 & 1 & 2 \\
\hline[tex]$p(x)$[/tex] & 12 & 5 & 0 & -3 & -4 \\
\hline
\end{tabular}
