To determine the constant of variation for the given table, we need to use the fact that for a linear relationship [tex]\( f(x) = kx \)[/tex], where [tex]\( k \)[/tex] is the constant of variation.
Let's start by examining two pairs of points from the table to find [tex]\( k \)[/tex]:
1. The points [tex]\((-8, -2)\)[/tex]
2. The points [tex]\((-4, -1)\)[/tex]
Using the general form [tex]\( f(x) = kx \)[/tex] (or [tex]\( y = kx \)[/tex]), we can find [tex]\( k \)[/tex] by dividing [tex]\( f(x) \)[/tex] by [tex]\( x \)[/tex]:
For the point [tex]\((-8, -2)\)[/tex]:
[tex]\[
k = \frac{f(x)}{x} = \frac{-2}{-8} = \frac{1}{4}
\][/tex]
For the point [tex]\((-4, -1)\)[/tex]:
[tex]\[
k = \frac{f(x)}{x} = \frac{-1}{-4} = \frac{1}{4}
\][/tex]
Both calculations give us the same constant of variation [tex]\( k \)[/tex].
Therefore, the constant of variation for the given table is
[tex]\[
\boxed{\frac{1}{4}}
\][/tex]
