To solve the problem, let's carefully analyze the given information and determine the vertex and range of the quadratic function [tex]\( h(x) \)[/tex].
1. Understanding the pattern:
The table lists values of [tex]\( h(x) \)[/tex] at specific points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & h(x) \\
\hline
-6 & 15 \\
\hline
-5 & 10 \\
\hline
-4 & 7 \\
\hline
-3 & 6 \\
\hline
-2 & 7 \\
\hline
-1 & 10 \\
\hline
\end{array}
\][/tex]
2. Identify the vertex:
For a quadratic function [tex]\( h(x) = ax^2 + bx + c \)[/tex], the vertex form is [tex]\( h(x) = a(x - h)^2 + k \)[/tex]. The vertex is the point where the function reaches its minimum or maximum value.
Observing the given values, [tex]\( h(x) \)[/tex] decreases until [tex]\( x = -3 \)[/tex], where it reaches the minimum value of 6, and then increases. Therefore, the vertex of the parabola is at:
[tex]\[
(-3, 6)
\][/tex]
3. Determine the range:
Since the quadratic function opens upwards (as indicated by the pattern of decreasing and then increasing values), the range is all [tex]\( y \)[/tex]-values starting from the minimum value 6 to positive infinity.
Thus, the range can be expressed as:
[tex]\[
6 \leq y \leq \infty
\][/tex]
4. Match with the given choices:
- Vertex [tex]\( (-3, 6) \)[/tex]; Range [tex]\( 6 \leq y \leq \infty \)[/tex]
The correct answer is:
Vertex [tex]\( (-3, 6) \)[/tex]; Range [tex]\( 6 \leq y \leq \infty \)[/tex]
