To determine which table corresponds to the function [tex]\( f(x) = -x^2 + 4 \)[/tex], we need to evaluate this function at the given [tex]\( x \)[/tex]-values in each table and compare the results.
First, we calculate the function values:
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = -(-2)^2 + 4 = -4 + 4 = 0
\][/tex]
2. For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = -(0)^2 + 4 = 0 + 4 = 4
\][/tex]
3. For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = -(2)^2 + 4 = -4 + 4 = 0
\][/tex]
Thus, the function [tex]\( f(x) = -x^2 + 4 \)[/tex] at [tex]\( x = -2, 0, 2 \)[/tex] yields the table of values:
[tex]\[
[(-2, 0), (0, 4), (2, 0)]
\][/tex]
Now, we compare this calculated table with the provided options:
1. First table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
-2 & 0 \\
\hline
0 & 4 \\
\hline
2 & 8 \\
\hline
\end{tabular}
\][/tex]
This table is [tex]\((-2, 0), (0, 4), (2, 8)\)[/tex], which does not match the calculated table.
2. Second table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
-2 & 8 \\
\hline
0 & 4 \\
\hline
2 & 6 \\
\hline
\end{tabular}
\][/tex]
This table is [tex]\((-2, 8), (0, 4), (2, 6)\)[/tex], which does not match the calculated table.
3. Third table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
-2 & 0 \\
\hline
0 & 4 \\
\hline
2 & 0 \\
\hline
\end{tabular}
\][/tex]
This table is [tex]\((-2, 0), (0, 4), (2, 0)\)[/tex], which exactly matches the calculated table.
Therefore, the table that corresponds with the function [tex]\( f(x) = -x^2 + 4 \)[/tex] is the third table.
