To find the [tex]\( y \)[/tex]-value of the translated point [tex]\( P' \)[/tex] of vertex [tex]\( P \)[/tex] after applying the translation rule, follow these steps:
1. Identify the coordinates of point [tex]\( P \)[/tex]:
The coordinates of point [tex]\( P \)[/tex] are given as [tex]\( P(-2, 6) \)[/tex].
2. Understand the translation rule:
The rule for translation is given as [tex]\( (x, y) \rightarrow (x - 2, y - 16) \)[/tex].
3. Apply the translation rule to point [tex]\( P \)[/tex]:
- For the [tex]\( x \)[/tex]-coordinate of [tex]\( P \)[/tex], which is [tex]\(-2\)[/tex]:
[tex]\[
x_{\text{new}} = -2 - 2 = -4
\][/tex]
- For the [tex]\( y \)[/tex]-coordinate of [tex]\( P \)[/tex], which is [tex]\( 6 \)[/tex]:
[tex]\[
y_{\text{new}} = 6 - 16 = -10
\][/tex]
4. Determine the translated coordinates of [tex]\( P' \)[/tex]:
After applying the translation rule, the coordinates of [tex]\( P \)[/tex] change to [tex]\( P'(-4, -10) \)[/tex].
5. Extract the [tex]\( y \)[/tex]-value of [tex]\( P' \)[/tex]:
The [tex]\( y \)[/tex]-value of the translated point [tex]\( P' \)[/tex] is [tex]\(-10\)[/tex].
Therefore, the [tex]\( y \)[/tex]-value of [tex]\( P' \)[/tex] is [tex]\(\boxed{-10}\)[/tex].
