To determine the correct statement about the quadratic function [tex]\( g(x) \)[/tex] given by the table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & -5 & -4 & -3 & -2 & -1 & 0 \\
\hline
g(x) & -1 & 0 & -1 & -4 & -9 & -16 \\
\hline
\end{array}
\][/tex]
we need to analyze the function's behavior.
1. Identify the function type:
The function [tex]\( g(x) \)[/tex] is quadratic because it appears to follow a parabolic shape, typical of quadratic functions.
2. Determine the minimum (or maximum) value:
For a quadratic function, the vertex represents either the minimum or maximum value of the function. After examining the given values, we observe that [tex]\( g(x) \)[/tex] reaches its lowest value of [tex]\(-16\)[/tex] at [tex]\( x = 0 \)[/tex].
3. Verify the y-intercept:
The y-intercept of the function is the value of [tex]\( g(x) \)[/tex] when [tex]\( x = 0 \)[/tex]. From the table, we see [tex]\( g(0) = -16 \)[/tex]. Therefore, the minimum value occurs at the point where the function intersects the y-axis.
4. Conclusion:
Based on this analysis, the minimum value of [tex]\( g(x) \)[/tex] occurs at the function's y-intercept, specifically when [tex]\( x = 0 \)[/tex].
Therefore, the correct statement about the quadratic function [tex]\( g(x) \)[/tex] is:
[tex]\[
\boxed{D. \text{The minimum occurs at the function's y-intercept.}}
\][/tex]
