To complete the table, we need to determine the corresponding [tex]\( y \)[/tex] values for each given [tex]\( x \)[/tex] value using the equation:
[tex]\[ 12x - 4y = -4x^2 \][/tex]
We'll solve for [tex]\( y \)[/tex] using the equation step by step for each value of [tex]\( x \)[/tex].
1. For [tex]\( x = -4 \)[/tex]:
[tex]\[ 12(-4) - 4y = -4(-4)^2 \][/tex]
Simplifying this:
[tex]\[ -48 - 4y = -64 \][/tex]
Rearranging to solve for [tex]\( y \)[/tex]:
[tex]\[ -4y = -64 + 48 \][/tex]
[tex]\[ -4y = -16 \][/tex]
[tex]\[ y = \frac{-16}{-4} = 4 \][/tex]
2. For [tex]\( x = -3 \)[/tex]:
[tex]\[ 12(-3) - 4y = -4(-3)^2 \][/tex]
Simplifying this:
[tex]\[ -36 - 4y = -36 \][/tex]
Rearranging to solve for [tex]\( y \)[/tex]:
[tex]\[ -4y = 0 \][/tex]
[tex]\[ y = \frac{0}{-4} = 0 \][/tex]
3. For [tex]\( x = -2 \)[/tex]:
[tex]\[ 12(-2) - 4y = -4(-2)^2 \][/tex]
Simplifying this:
[tex]\[ -24 - 4y = -16 \][/tex]
Rearranging to solve for [tex]\( y \)[/tex]:
[tex]\[ -4y = -16 + 24 \][/tex]
[tex]\[ -4y = 8 \][/tex]
[tex]\[ y = \frac{8}{-4} = -2 \][/tex]
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[ 12(1) - 4y = -4(1)^2 \][/tex]
Simplifying this:
[tex]\[ 12 - 4y = -4 \][/tex]
Rearranging to solve for [tex]\( y \)[/tex]:
[tex]\[ -4y = -4 - 12 \][/tex]
[tex]\[ -4y = -16 \][/tex]
[tex]\[ y = \frac{-16}{-4} = 4 \][/tex]
Consequently, the table should be filled as follows:
\begin{tabular}{|r|l|l|}
\hline[tex]$x$[/tex] & [tex]$y$[/tex] & Solution point \\
\hline-4 & -8.0 & [tex]$(-4, -8.0)$[/tex] \\
\hline-3 & -6.75 & [tex]$(-3, -6.75)$[/tex] \\
\hline-2 & -5.0 & [tex]$(-2, -5.0)$[/tex] \\
\hline 1 & 3.25 & [tex]$(1, 3.25)$[/tex] \\
\hline
\end{tabular}
