To determine the new [tex]\(y\)[/tex]-coordinate when a translation of [tex]\(T_{-3,-8}(x, y)\)[/tex] is applied to point [tex]\(B\)[/tex] of square [tex]\(ABCD\)[/tex], we need to understand how the translation affects the [tex]\(y\)[/tex]-coordinate.
Given the translation [tex]\(T_{-3, -8}(x, y)\)[/tex], this means every point [tex]\((x, y)\)[/tex] on the square is moved 3 units to the left (affecting [tex]\(x\)[/tex]) and 8 units down (affecting [tex]\(y\)[/tex]). The translation of [tex]\((x, y)\)[/tex] [tex]\(\rightarrow\)[/tex] [tex]\((x - 3, y - 8)\)[/tex].
Our specific task is to find out which of the given initial [tex]\(y\)[/tex]-coordinates of point [tex]\(B\)[/tex] correctly results in a certain translated [tex]\(y\)[/tex]-coordinate after applying the translation.
Let's look at the given options for the initial [tex]\(y\)[/tex]-coordinate of [tex]\(B\)[/tex]:
1. [tex]\(-12\)[/tex]
2. [tex]\(10\)[/tex]
3. [tex]\(-6\)[/tex]
4. [tex]\(-2\)[/tex]
We apply the translation step-by-step calculation for each potential initial [tex]\(y\)[/tex]-coordinate to see what the new [tex]\(y\)[/tex]-coordinate of [tex]\(B\)[/tex] would be after translation:
1. Initial [tex]\(y\)[/tex]-coordinate: [tex]\(-12\)[/tex]
[tex]\[ y_B = -12 - 8 = -20 \][/tex]
2. Initial [tex]\(y\)[/tex]-coordinate: [tex]\(10\)[/tex]
[tex]\[ y_B = 10 - 8 = 2 \][/tex]
3. Initial [tex]\(y\)[/tex]-coordinate: [tex]\(-6\)[/tex]
[tex]\[ y_B = -6 - 8 = -14 \][/tex]
4. Initial [tex]\(y\)[/tex]-coordinate: [tex]\(-2\)[/tex]
[tex]\[ y_B = -2 - 8 = -10 \][/tex]
After applying the translation to each of these given initial [tex]\(y\)[/tex]-coordinates, we get the results:
- [tex]\(-12\)[/tex] translates to [tex]\(-20\)[/tex]
- [tex]\(10\)[/tex] translates to [tex]\(2\)[/tex]
- [tex]\(-6\)[/tex] translates to [tex]\(-14\)[/tex]
- [tex]\(-2\)[/tex] translates to [tex]\(-10\)[/tex]
With this detailed step-by-step solution, we see that the new [tex]\(y\)[/tex]-coordinate would clearly depend on which initial value is being translated. This confirms that we have correctly applied the translation [tex]\(T_{-3,-8}(x, y)\)[/tex]. Given the options, we can analyze if any specific condition needs to be met for a typical [tex]\(y\)[/tex]-coordinate in the problem context. However, if considering the output for just the resultant position due to translation, these translations illustrate the mathematical application of translations to each option.
