Alright, let's fill in the table by calculating the values of [tex]\( Y \)[/tex] for each given [tex]\( X \)[/tex] using the function [tex]\( Y = x^2 - 2x - 2 \)[/tex].
1. For [tex]\( X = 1 \)[/tex]:
[tex]\[
Y = 1^2 - 2 \cdot 1 - 2 \\
= 1 - 2 - 2 \\
= -3
\][/tex]
2. For [tex]\( X = 2 \)[/tex]:
[tex]\[
Y = 2^2 - 2 \cdot 2 - 2 \\
= 4 - 4 - 2 \\
= -2
\][/tex]
3. For [tex]\( X = 3 \)[/tex]:
[tex]\[
Y = 3^2 - 2 \cdot 3 - 2 \\
= 9 - 6 - 2 \\
= 1
\][/tex]
4. For [tex]\( X = 0 \)[/tex]:
[tex]\[
Y = 0^2 - 2 \cdot 0 - 2 \\
= 0 - 0 - 2 \\
= -2
\][/tex]
5. For [tex]\( X = -1 \)[/tex]:
[tex]\[
Y = (-1)^2 - 2 \cdot (-1) - 2 \\
= 1 + 2 - 2 \\
= 1
\][/tex]
6. For [tex]\( X = -2 \)[/tex]:
[tex]\[
Y = (-2)^2 - 2 \cdot (-2) - 2 \\
= 4 + 4 - 2 \\
= 6
\][/tex]
Now let's update the table with the calculated values of [tex]\( Y \)[/tex]:
[tex]\[
\begin{tabular}{l|l}
$X$ & $Y = x^2 - 2X - 2$ \\
\hline
1 & -3 \\
2 & -2 \\
3 & 1 \\
0 & -2 \\
-1 & 1 \\
-2 & 6 \\
\end{tabular}
\][/tex]
So, the completed table is as follows:
[tex]\[
\begin{array}{c|c}
X & Y = x^2 - 2X - 2 \\
\hline
1 & -3 \\
2 & -2 \\
3 & 1 \\
0 & -2 \\
-1 & 1 \\
-2 & 6 \\
\end{array}
\][/tex]
