To complete the table of values for the function [tex]\( y = x^2 + x \)[/tex], we will evaluate the expression for each given value of [tex]\( x \)[/tex] and fill in the corresponding [tex]\( y \)[/tex] values.
Let's go through each calculation:
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = (-2)^2 + (-2) = 4 - 2 = 2 \][/tex]
So, when [tex]\( x = -2 \)[/tex], [tex]\( y = 2 \)[/tex].
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = (-1)^2 + (-1) = 1 - 1 = 0 \][/tex]
So, when [tex]\( x = -1 \)[/tex], [tex]\( y = 0 \)[/tex]. This means [tex]\( A = 0 \)[/tex].
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 0^2 + 0 = 0 + 0 = 0 \][/tex]
So, when [tex]\( x = 0 \)[/tex], [tex]\( y = 0 \)[/tex]. This means [tex]\( B = 0 \)[/tex].
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 1^2 + 1 = 1 + 1 = 2 \][/tex]
So, when [tex]\( x = 1 \)[/tex], [tex]\( y = 2 \)[/tex].
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 2^2 + 2 = 4 + 2 = 6 \][/tex]
So, when [tex]\( x = 2 \)[/tex], [tex]\( y = 6 \)[/tex]. This means [tex]\( C = 6 \)[/tex].
Now, we can fill in the table:
[tex]\[
\begin{tabular}{c||c|c|c|c|c}
$x$ & -2 & -1 & 0 & 1 & 2 \\
\hline
$y$ & 2 & 0 & 0 & 2 & 6 \\
\end{tabular}
\][/tex]
Therefore, the numbers that replace [tex]\( A \)[/tex], [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are:
[tex]\[ A = 0, \quad B = 0, \quad C = 6. \][/tex]
