Let's analyze the given table systematically:
[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 are examining the function [tex]\( g \)[/tex] over the set of [tex]\( x \)[/tex]-values given: [tex]\( \{-5, -4, -3, -2, -1, 0\} \)[/tex].
### Step-by-Step Analysis:
1. Identifying the nature of the function:
- The function [tex]\( g \)[/tex] is quadratic, which means it could be concave up (bowl-shaped) or concave down (inverted bowl-shaped).
2. Behavior of [tex]\( g(x) \)[/tex]:
- As [tex]\( x \)[/tex] increases from [tex]\(-5\)[/tex] to [tex]\( 0 \)[/tex], we observe the following pattern in [tex]\( g(x) \)[/tex]:
[tex]\[
g(-5) = -1 \quad \text{(increasing to)} \quad g(-4) = 0 \quad \text{(then decreasing continuously)} \quad g(0) = -16
\][/tex]
- This indicates a downward trend in [tex]\( g(x) \)[/tex] after reaching [tex]\( x = -4 \)[/tex].
3. Finding the minimum value:
- From the values given, we identify the minimum value of [tex]\( g(x) \)[/tex] in this range:
[tex]\[
g(0) = -16
\][/tex]
- This is the lowest value of [tex]\( g(x) \)[/tex] over the given interval.
4. Location of the minimum value:
- The [tex]\( x \)[/tex]-value corresponding to this minimum value is [tex]\( x = 0 \)[/tex].
- In a function, when [tex]\( x = 0 \)[/tex], it also represents the [tex]\( y \)[/tex]-intercept, as it is the point where the graph intersects the [tex]\( y \)[/tex]-axis.
5. Selecting the correct statement:
- Comparing all given options, it is evident that the correct statement describes the minimum value's location related to the [tex]\( y \)[/tex]-intercept.
Therefore, the correct statement is:
D. The minimum occurs at the function's [tex]\( y \)[/tex]-intercept.
