The table describes the quadratic function [tex]$h(x)$[/tex].

\begin{tabular}{|c|l|}
\hline
[tex]$x$[/tex] & [tex]$h(x)$[/tex] \\
\hline
-3 & -2 \\
\hline
-2 & -3 \\
\hline
-1 & -2 \\
\hline
0 & 1 \\
\hline
1 & 6 \\
\hline
2 & 13 \\
\hline
3 & 22 \\
\hline
\end{tabular}

What is the equation of [tex]$h(x)$[/tex] in vertex form?

A. [tex]$h(x) = (x + 2)^2 - 3$[/tex]
B. [tex]$h(x) = (x + 1)^2 - 2$[/tex]
C. [tex]$h(x) = (x - 1)^2 + 2$[/tex]
D. [tex]$h(x) = (x - 2)^2 + 3$[/tex]



Answer :

To solve this problem of determining the equation of \( h(x) \) in vertex form given the table of values, we'll go through a systematic approach.

First, let's recall the forms of quadratic functions:
- Standard form: \( h(x) = ax^2 + bx + c \)
- Vertex form: \( h(x) = a(x - h_v)^2 + k \), where \((h_v, k)\) is the vertex of the parabola.

Given the data points in the table:
[tex]\[ \begin{array}{|c|c|} \hline x & h(x) \\ \hline -3 & -2 \\ \hline -2 & -3 \\ \hline -1 & -2 \\ \hline 0 & 1 \\ \hline 1 & 6 \\ \hline 2 & 13 \\ \hline 3 & 22 \\ \hline \end{array} \][/tex]

1. Fit a Quadratic Function:
We fit a quadratic function \( h(x) = ax^2 + bx + c \) using the given points.

2. Determine the Coefficients:
The quadratic function that fits these points can be determined, and we find the coefficients to be \( a = 1 \), \( b = 4 \), and \( c = 1 \). Thus, the quadratic function in standard form is:
[tex]\[ h(x) = x^2 + 4x + 1 \][/tex]

3. Convert to Vertex Form:
To convert to vertex form, we use the relationship \( x = -\frac{b}{2a} \) to find the x-coordinate of the vertex. Here \( a = 1 \) and \( b = 4 \):
[tex]\[ h_v = -\frac{4}{2 \cdot 1} = -2 \][/tex]
We then find the corresponding \( y \)-value (k) by substituting \( h_v = -2 \) back into the quadratic equation:
[tex]\[ k = h(-2) = (1)(-2)^2 + 4(-2) + 1 = 4 - 8 + 1 = -3 \][/tex]

Therefore, the vertex form of the quadratic equation is:
[tex]\[ h(x) = 1(x - (-2))^2 + (-3) \][/tex]
Simplifying, we get:
[tex]\[ h(x) = (x + 2)^2 - 3 \][/tex]

4. Match with Given Options:
We compare our derived vertex form with the given options:
- \( (x+2)^2 - 3 \)
- \( (x+1)^2 - 2 \)
- \( (x-1)^2 + 2 \)
- \( (x-2)^2 + 3 \)

We see that the vertex form \( h(x) = (x + 2)^2 - 3 \) corresponds perfectly with the first option.

Thus, the correct equation in vertex form is:
[tex]\[ h(x) = (x + 2)^2 - 3 \][/tex]

Therefore, the correct choice is:
[tex]\[ \boxed{h(x) = (x + 2)^2 - 3} \][/tex]