Select the correct answer.

The table represents the quadratic function [tex]g[/tex]. Which statement is true about the function?

\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & -5 & -4 & -3 & -2 & -1 & 0 \\
\hline
[tex]$g(x)$[/tex] & -1 & 0 & -1 & -4 & -9 & -16 \\
\hline
\end{tabular}

A. The minimum occurs at the function's [tex]$x$[/tex]-intercept.

B. The maximum occurs at the function's [tex]$x$[/tex]-intercept.

C. The maximum occurs at the function's [tex]$y$[/tex]-intercept.

D. The minimum occurs at the function's [tex]$y$[/tex]-intercept.



Answer :

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.