Let's analyze the given values of the function on the interval [tex]\([-2, 1]\)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & -12 \\
\hline
-1 & 0 \\
\hline
0 & 6 \\
\hline
1 & 7.5 \\
\hline
\end{array}
\][/tex]
We will look at how the [tex]\( y \)[/tex]-values change as [tex]\( x \)[/tex] increases from [tex]\(-2\)[/tex] to [tex]\(1\)[/tex]:
1. From [tex]\(-2\)[/tex] to [tex]\(-1\)[/tex]:
- When [tex]\( x = -2 \)[/tex], [tex]\( y = -12 \)[/tex]
- When [tex]\( x = -1 \)[/tex], [tex]\( y = 0 \)[/tex]
- The function value increases from [tex]\(-12\)[/tex] to [tex]\(0\)[/tex].
2. From [tex]\(-1\)[/tex] to [tex]\(0\)[/tex]:
- When [tex]\( x = -1 \)[/tex], [tex]\( y = 0 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 6 \)[/tex]
- The function value increases from [tex]\(0\)[/tex] to [tex]\(6\)[/tex].
3. From [tex]\(0\)[/tex] to [tex]\(1\)[/tex]:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 6 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 7.5 \)[/tex]
- The function value increases from [tex]\(6\)[/tex] to [tex]\(7.5\)[/tex].
In each of these intervals, the [tex]\( y \)[/tex]-value increases as the [tex]\( x \)[/tex]-value increases. Thus, the function is consistently increasing throughout the entire interval [tex]\([-2, 1]\)[/tex].
Therefore, the sentence that best describes the interval [tex]\([-2, 1]\)[/tex] is:
D. The function is increasing on the interval [tex]\([-2, 1]\)[/tex].
