Answered

Get the answers you've been looking for with the help of IDNLearn.com's expert community. Discover the information you need from our experienced professionals who provide accurate and reliable answers to all your questions.

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

Sagot :

GinnyAnsweridn
To find the equation of [tex]\( h(x) \)[/tex] in vertex form using the given points, we have to test which equation fits all the data points in the table.

The possible equations are:
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]

We will verify which equation correctly calculates the values for all given [tex]\( x \)[/tex] in the table.

1. Test [tex]\( h(x) = (x - 2)^2 + 3 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]: [tex]\( h(-3) = (-3 - 2)^2 + 3 = (-5)^2 + 3 = 25 + 3 = 28 \)[/tex] (not matching)
- So, this equation is incorrect.

2. Test [tex]\( h(x) = (x - 1)^2 + 2 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]: [tex]\( h(-3) = (-3 - 1)^2 + 2 = (-4)^2 + 2 = 16 + 2 = 18 \)[/tex] (not matching)
- So, this equation is incorrect.

3. Test [tex]\( h(x) = (x + 1)^2 + 2 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]: [tex]\( h(-3) = (-3 + 1)^2 + 2 = (-2)^2 + 2 = 4 + 2 = 6 \)[/tex] (matching)
- For [tex]\( x = -2 \)[/tex]: [tex]\( h(-2) = (-2 + 1)^2 + 2 = (-1)^2 + 2 = 1 + 2 = 3 \)[/tex] (matching)
- For [tex]\( x = -1 \)[/tex]: [tex]\( h(-1) = (-1 + 1)^2 + 2 = (0)^2 + 2 = 0 + 2 = 2 \)[/tex] (matching)
- For [tex]\( x = 0 \)[/tex]: [tex]\( h(0) = (0 + 1)^2 + 2 = (1)^2 + 2 = 1 + 2 = 3 \)[/tex] (matching)
- For [tex]\( x = 1 \)[/tex]: [tex]\( h(1) = (1 + 1)^2 + 2 = (2)^2 + 2 = 4 + 2 = 6 \)[/tex] (matching)
- For [tex]\( x = 2 \)[/tex]: [tex]\( h(2) = (2 + 1)^2 + 2 = (3)^2 + 2 = 9 + 2 = 11 \)[/tex] (matching)
- For [tex]\( x = 3 \)[/tex]: [tex]\( h(3) = (3 + 1)^2 + 2 = (4)^2 + 2 = 16 + 2 = 18 \)[/tex] (matching)
- All points match, so this equation is correct.

4. Test [tex]\( h(x) = (x + 2)^2 + 3 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]: [tex]\( h(-3) = (-3 + 2)^2 + 3 = (-1)^2 + 3 = 1 + 3 = 4 \)[/tex] (not matching)
- So, this equation is incorrect.

Based on the evaluation, the correct equation of [tex]\( h(x) \)[/tex] in vertex form is:
[tex]\[ h(x) = (x + 1)^2 + 2 \][/tex]

So, the answer is [tex]\( h(x) = (x + 1)^2 + 2 \)[/tex], which corresponds to the third option, therefore:
[tex]\[ 3 \][/tex] is the correct choice.
We are delighted to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. Your questions find answers at IDNLearn.com. Thanks for visiting, and come back for more accurate and reliable solutions.