To solve this problem, we need to determine the [tex]$y$[/tex]-coordinate of point [tex]$B'$[/tex] after the translation [tex]\( T_{-3, -8} \)[/tex] is applied to point [tex]$B$[/tex] of the square [tex]$A B C D$[/tex].
A translation [tex]\( T_{-3, -8} \)[/tex] means that we adjust the [tex]$x$[/tex]-coordinate by subtracting 3 and the [tex]$y$[/tex]-coordinate by subtracting 8 from their original values.
Let's consider the [tex]$y$[/tex]-coordinate of point [tex]$B$[/tex]. According to the choices given, we know that initially, one choice corresponds to the original [tex]$y$[/tex]-coordinate.
Given that the initial [tex]$y$[/tex]-coordinate of point [tex]$B$[/tex] is offered as one of the options, we can identify the correct initial [tex]$y$[/tex]-coordinate from these options.
Let's assume the initial [tex]$y$[/tex]-coordinate of point [tex]$B$[/tex] is [tex]\( -2 \)[/tex].
We now apply the translation [tex]\( T_{-3, -8} \)[/tex], focusing on the [tex]$y$[/tex]-coordinate:
[tex]\[
\text{Initial } y\text{-coordinate of } B: -2
\][/tex]
The translation instructs us to subtract 8 from the initial [tex]$y$[/tex]-coordinate:
[tex]\[
-2 - 8 = -10
\][/tex]
Thus, after applying the translation, the [tex]$y$[/tex]-coordinate of point [tex]$B'$[/tex] is:
[tex]\[
\boxed{-10}
\][/tex]
