To solve the problem of finding the values for [tex]\( y = x^2 + 3x - 2 \)[/tex] and filling in the table for the given [tex]\( x \)[/tex] values, we will compute [tex]\( y \)[/tex] for each [tex]\( x \)[/tex] value step-by-step. This will help us find the values for [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex].
### Step-by-Step Solution
1. Calculate [tex]\( y \)[/tex] when [tex]\( x = -3 \)[/tex]:
[tex]\[
y = (-3)^2 + 3(-3) - 2
\][/tex]
[tex]\[
y = 9 - 9 - 2
\][/tex]
[tex]\[
y = -2
\][/tex]
Thus, [tex]\( A = -2 \)[/tex].
2. Calculate [tex]\( y \)[/tex] when [tex]\( x = -2 \)[/tex]:
[tex]\[
y = (-2)^2 + 3(-2) - 2
\][/tex]
[tex]\[
y = 4 - 6 - 2
\][/tex]
[tex]\[
y = -4
\][/tex]
This value is already provided in the table.
3. Calculate [tex]\( y \)[/tex] when [tex]\( x = -1 \)[/tex]:
[tex]\[
y = (-1)^2 + 3(-1) - 2
\][/tex]
[tex]\[
y = 1 - 3 - 2
\][/tex]
[tex]\[
y = -4
\][/tex]
This value is also already provided in the table.
4. Calculate [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 0^2 + 3(0) - 2
\][/tex]
[tex]\[
y = 0 + 0 - 2
\][/tex]
[tex]\[
y = -2
\][/tex]
Thus, [tex]\( B = -2 \)[/tex].
5. Calculate [tex]\( y \)[/tex] when [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 1^2 + 3(1) - 2
\][/tex]
[tex]\[
y = 1 + 3 - 2
\][/tex]
[tex]\[
y = 2
\][/tex]
Thus, [tex]\( C = 2 \)[/tex].
Now we can complete the table with the calculated values:
[tex]\[
\begin{tabular}{c||c|c|c|c|c}
$x$ & -3 & -2 & -1 & 0 & 1 \\
\hline
$y$ & -2 & -4 & -4 & -2 & 2 \\
\end{tabular}
\][/tex]
So, the numbers that replace [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] are [tex]\( A = -2 \)[/tex], [tex]\( B = -2 \)[/tex], and [tex]\( C = 2 \)[/tex].
