To solve this problem, we need to determine both the vertex and the range of the quadratic function [tex]\( h(x) \)[/tex].
1. Inspection of Data Points for Vertex:
[tex]\( h(x) \)[/tex] is a quadratic function, and by definition, its graph is a parabola. The vertex is the turning point of the parabola. From the given table, we can extract the following information:
\begin{equation}
\begin{array}{|c|c|}
\hline
x & h(x) \\
\hline
-6 & 12 \\
\hline
-5 & 7 \\
\hline
-4 & 4 \\
\hline
-3 & 3 \\
\hline
-2 & 4 \\
\hline
-1 & 7 \\
\hline
\end{array}
\end{equation}
We notice that [tex]\( h(x) \)[/tex] reaches its minimum value when [tex]\( x = -3 \)[/tex]. The value of [tex]\( h(-3) \)[/tex] is 3. Therefore, the vertex of the parabola, where the function has its minimum value, is at [tex]\( (-3, 3) \)[/tex].
2. Determining the Range:
Since the vertex [tex]\((-3, 3)\)[/tex] represents the minimum point for the quadratic function (as the values of [tex]\( h(x) \)[/tex] increase on both sides of [tex]\( x = -3 \)[/tex]), the range of [tex]\( h(x) \)[/tex] includes all real numbers greater than or equal to the minimum value.
Therefore, the range of [tex]\( h(x) \)[/tex] is:
[tex]\[ [3, \infty) \][/tex]
Thus, the solution details the vertex and the range of the quadratic function [tex]\( h(x) \)[/tex].
Answer:
- Vertex: [tex]\((-3, 3)\)[/tex]
- Range: [tex]\(3 \leq y < \infty\)[/tex]
Among the choices listed, the correct one is:
Vertex [tex]\((-3, 3)\)[/tex], Range [tex]\(3 \leq y \leq \infty\)[/tex]
