Let's start by understanding the translation transformation [tex]\( T_{-3,-8}(x, y) \)[/tex]. This notation indicates a translation where every point [tex]\((x, y)\)[/tex] in the plane is moved by [tex]\(-3\)[/tex] units horizontally (along the x-axis) and [tex]\(-8\)[/tex] units vertically (along the y-axis).
When a translation is applied to any point [tex]\((x, y)\)[/tex], the new coordinates after the translation [tex]\( T_{-3,-8} \)[/tex] will be:
[tex]\[
(x', y') = (x - 3, y - 8)
\][/tex]
To find the [tex]\( y \)[/tex]-coordinate of point [tex]\( B \)[/tex] after the translation, we only need to focus on the vertical change in the [tex]\( y \)[/tex]-coordinate. Suppose the original [tex]\( y \)[/tex]-coordinate of [tex]\( B \)[/tex] is [tex]\( y \)[/tex]. After translating by [tex]\(-8\)[/tex] units vertically, the new [tex]\( y \)[/tex]-coordinate of [tex]\( B \)[/tex] will be:
[tex]\[
y' = y - 8
\][/tex]
The answer to the question specifies that the [tex]\( y \)[/tex]-coordinate of [tex]\( B \)[/tex] after translation is [tex]\(-8\)[/tex]. This means we are given the result directly. Therefore, the [tex]\( y \)[/tex]-coordinate of point [tex]\( B \)[/tex] post-translation is:
[tex]\[
y' = -8
\][/tex]
Given the options:
- [tex]\(-12\)[/tex]
- [tex]\(-8\)[/tex]
- [tex]\(-6\)[/tex]
- [tex]\(-2\)[/tex]
The correct [tex]\( y \)[/tex]-coordinate of point [tex]\( B \)[/tex] after applying the translation [tex]\( T_{-3,-8} \)[/tex] is:
[tex]\[
-8
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{-8}
\][/tex]
