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

[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $h(x)$ \\
\hline
-3 & 6 \\
\hline
-2 & 3 \\
\hline
-1 & 2 \\
\hline
0 & 3 \\
\hline
1 & 6 \\
\hline
2 & 11 \\
\hline
3 & 18 \\
\hline
\end{tabular}
\][/tex]

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]

Sagot :

GinnyAnswer
Let's determine the equation of [tex]\( h(x) \)[/tex] in vertex form using the given points in the table. The vertex form of a quadratic function is given by:

[tex]\[ h(x) = a(x - h)^2 + k \][/tex]

where [tex]\((h, k)\)[/tex] is the vertex of the parabola.

We are provided with four possible equations:
1. [tex]\( h(x) = (x - 2)^2 + 3 \)[/tex]
2. [tex]\( h(x) = (x - 1)^2 + 2 \)[/tex]
3. [tex]\( h(x) = (x + 1)^2 + 2 \)[/tex]
4. [tex]\( h(x) = (x + 2)^2 + 3 \)[/tex]

To find the correct equation, we need to check which of these equations fits all the given points.

Let's analyze the points one by one for each equation.

First, check [tex]\( h(x) \)[/tex] for [tex]\( h(x) = (x - 2)^2 + 3 \)[/tex]:

1. For [tex]\( x = -3 \)[/tex], [tex]\( h(-3) = ((-3) - 2)^2 + 3 = 25 + 3 = 28 \)[/tex] (does not match 6)
2. This fails, so option 1 is incorrect.

Second, check [tex]\( h(x) \)[/tex] for [tex]\( h(x) = (x - 1)^2 + 2 \)[/tex]:

1. For [tex]\( x = -3 \)[/tex], [tex]\( h(-3) = ((-3) - 1)^2 + 2 = 16 + 2 = 18 \)[/tex] (does not match 6)
2. This fails, so option 2 is incorrect.

Third, check [tex]\( h(x) \)[/tex] for [tex]\( h(x) = (x + 1)^2 + 2 \)[/tex]:

1. For [tex]\( x = -3 \)[/tex], [tex]\( h(-3) = ((-3) + 1)^2 + 2 = 4 + 2 = 6 \)[/tex] (matches 6)
2. For [tex]\( x = -2 \)[/tex], [tex]\( h(-2) = ((-2) + 1)^2 + 2 = 1 + 2 = 3 \)[/tex] (matches 3)
3. For [tex]\( x = -1 \)[/tex], [tex]\( h(-1) = ((-1) + 1)^2 + 2 = 0 + 2 = 2 \)[/tex] (matches 2)
4. For [tex]\( x = 0 \)[/tex], [tex]\( h(0) = (0 + 1)^2 + 2 = 1 + 2 = 3 \)[/tex] (matches 3)
5. For [tex]\( x = 1 \)[/tex], [tex]\( h(1) = (1 + 1)^2 + 2 = 4 + 2 = 6 \)[/tex] (matches 6)
6. For [tex]\( x = 2 \)[/tex], [tex]\( h(2) = (2 + 1)^2 + 2 = 9 + 2 = 11 \)[/tex] (matches 11)
7. For [tex]\( x = 3 \)[/tex], [tex]\( h(3) = (3 + 1)^2 + 2 = 16 + 2 = 18 \)[/tex] (matches 18)

Thus, all points match for [tex]\( h(x) = (x + 1)^2 + 2 \)[/tex].

Fourth, check [tex]\( h(x) \)[/tex] for [tex]\( h(x) = (x + 2)^2 + 3 \)[/tex]:

1. For [tex]\( x = -3 \)[/tex], [tex]\( h(-3) = ((-3) + 2)^2 + 3 = 1 + 3 = 4 \)[/tex] (does not match 6)
2. This fails, so option 4 is incorrect.

Conclusively, the correct equation of [tex]\( h(x) \)[/tex] in vertex form is:

[tex]\[ h(x) = (x + 1)^2 + 2 \][/tex]

Which corresponds to the third option.
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