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The function [tex]\( h(x) \)[/tex] is a continuous quadratic function with a domain of all real numbers. The table lists some of the values:

[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]

What are the vertex and range of [tex]\( h(x) \)[/tex]?

A. Vertex [tex]\((-4, 7)\)[/tex]; Range [tex]\(7 \leq y \leq \infty\)[/tex]

B. Vertex [tex]\((-4, 7)\)[/tex]; Range [tex]\(\infty \leq y \leq 7\)[/tex]

C. Vertex [tex]\((-3, 6)\)[/tex]; Range [tex]\(6 \leq y \leq \infty\)[/tex]

D. Vertex [tex]\((-3, 6)\)[/tex]; Range [tex]\(\infty \leq y \leq 6\)[/tex]


Sagot :

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]
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