Let's analyze the given table step-by-step to determine the true statement about the quadratic function [tex]\( g(x) \)[/tex]:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$x$ & -5 & -4 & -3 & -2 & -1 & 0 \\
\hline
$g(x)$ & -1 & 0 & -1 & -4 & -9 & -16 \\
\hline
\end{tabular}
\][/tex]
1. We need to determine where the minimum value of the function [tex]\( g(x) \)[/tex] occurs. The minimum value is the smallest value of [tex]\( g(x) \)[/tex] present in the table.
2. We observe the values of [tex]\( g(x) \)[/tex]:
- When [tex]\( x = -5 \)[/tex], [tex]\( g(x) = -1 \)[/tex]
- When [tex]\( x = -4 \)[/tex], [tex]\( g(x) = 0 \)[/tex]
- When [tex]\( x = -3 \)[/tex], [tex]\( g(x) = -1 \)[/tex]
- When [tex]\( x = -2 \)[/tex], [tex]\( g(x) = -4 \)[/tex]
- When [tex]\( x = -1 \)[/tex], [tex]\( g(x) = -9 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( g(x) = -16 \)[/tex]
3. From these values, we see:
- The maximum value of [tex]\( g(x) \)[/tex] is [tex]\( 0 \)[/tex] at [tex]\( x = -4 \)[/tex].
- The minimum value of [tex]\( g(x) \)[/tex] is [tex]\( -16 \)[/tex] at [tex]\( x = 0 \)[/tex].
4. Since the minimum value of the function [tex]\( g(x) \)[/tex] occurs at the point where [tex]\( x = 0 \)[/tex], this point is the y-intercept of the function.
Therefore, the correct statement is:
D. The minimum occurs at the function's y-intercept.
