To determine the [tex]\( y \)[/tex]-coordinate of point [tex]\( B \)[/tex] after the translation [tex]\( T_{-3,-8}(x, y) \)[/tex] is applied to square ABCD, we need to understand what the translation represents and how it affects each point in the square.
The translation [tex]\( T_{-3,-8}(x, y) \)[/tex] means that every point [tex]\((x, y)\)[/tex] in the plane is moved [tex]\( -3 \)[/tex] units in the [tex]\( x \)[/tex]-direction and [tex]\( -8 \)[/tex] units in the [tex]\( y \)[/tex]-direction.
To find the new [tex]\( y \)[/tex]-coordinate of point [tex]\( B \)[/tex], let's denote the original [tex]\( y \)[/tex]-coordinate of point [tex]\( B \)[/tex] as [tex]\( y_B \)[/tex]. The translation in the [tex]\( y \)[/tex]-direction affects [tex]\( y_B \)[/tex] as follows:
[tex]\[ y_B' = y_B - 8 \][/tex]
Given the problem asks us to identify what the new [tex]\( y \)[/tex]-coordinate is after translation among the provided options:
- [tex]\(-12\)[/tex]
- [tex]\(-8\)[/tex]
- [tex]\(-6\)[/tex]
- [tex]\(-2\)[/tex]
We need to work backwards from these options to find the original [tex]\( y \)[/tex]-coordinate [tex]\( y_B \)[/tex]. Considering the translation [tex]\( -8 \)[/tex]:
1. For the option [tex]\(-12\)[/tex]:
[tex]\[ y_B - 8 = -12 \][/tex]
[tex]\[ y_B = -12 + 8 \][/tex]
[tex]\[ y_B = -4 \][/tex] (This is not a valid given option and thus is incorrect.)
2. For the option [tex]\(-8\)[/tex]:
[tex]\[ y_B - 8 = -8 \][/tex]
[tex]\[ y_B = -8 + 8 \][/tex]
[tex]\[ y_B = 0 \][/tex] (This indicates the original coordinate [tex]\( y_B = 0 \)[/tex].)
3. For the option [tex]\(-6\)[/tex]:
[tex]\[ y_B - 8 = -6 \][/tex]
[tex]\[ y_B = -6 + 8 \][/tex]
[tex]\[ y_B = 2 \][/tex] (This is not a valid given option and thus is incorrect.)
4. For the option [tex]\(-2\)[/tex]:
[tex]\[ y_B - 8 = -2 \][/tex]
[tex]\[ y_B = -2 + 8 \][/tex]
[tex]\[ y_B = 6 \][/tex] (This is not a valid given option and thus is incorrect.)
Since [tex]\( y_B = 0 \)[/tex] leads us to [tex]\( y_B' = -8 \)[/tex], and it matches one of the given multiple-choice options, this means the [tex]\( y \)[/tex]-coordinate of [tex]\( B \)[/tex] after the translation is:
[tex]\[ \boxed{-8} \][/tex]
