Let's solve the problem by calculating the values of [tex]\( y \)[/tex] for specific values of [tex]\( x \)[/tex] and filling in the missing entries in the table.
Given the function:
[tex]\[ y = x^2 - 3x - 2 \][/tex]
We need to find the values for [tex]\( y \)[/tex] when [tex]\( x = -1 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( x = 3 \)[/tex], represented as [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] respectively.
1. Calculate [tex]\( y \)[/tex] when [tex]\( x = -1 \)[/tex] (this is [tex]\( A \)[/tex]):
[tex]\[ A = (-1)^2 - 3(-1) - 2 \][/tex]
[tex]\[ A = 1 + 3 - 2 \][/tex]
[tex]\[ A = 2 \][/tex]
2. Calculate [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex] (this is [tex]\( B \)[/tex]):
[tex]\[ B = (0)^2 - 3(0) - 2 \][/tex]
[tex]\[ B = 0 - 0 - 2 \][/tex]
[tex]\[ B = -2 \][/tex]
3. Calculate [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex] (this is [tex]\( C \)[/tex]):
[tex]\[ C = (3)^2 - 3(3) - 2 \][/tex]
[tex]\[ C = 9 - 9 - 2 \][/tex]
[tex]\[ C = -2 \][/tex]
So, the correct values to replace [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] in the table are:
[tex]\[ A = 2, \ B = -2, \ C = -2 \][/tex]
The completed table should be:
[tex]\[
\begin{tabular}{c||c|c|c|c|c}
$x$ & -1 & 0 & 1 & 2 & 3 \\
\hline
$y$ & 2 & -2 & -4 & -4 & -2 \\
\end{tabular}
\][/tex]
