To solve the problem of finding the [tex]\( y \)[/tex]-value of [tex]\( P' \)[/tex] after applying the given translation rule to the vertex [tex]\( P(-2,6) \)[/tex] of triangle [tex]\( PQR \)[/tex]:
1. Identify the Translation Rule:
The translation rule given is [tex]\((x, y) \rightarrow (x-2, y-16)\)[/tex].
2. Apply the Translation to Point [tex]\( P \)[/tex]:
- The original coordinates of [tex]\( P \)[/tex] are [tex]\((x, y) = (-2,6)\)[/tex].
- Applying the translation rule to these coordinates:
- For the [tex]\( x \)[/tex]-coordinate: [tex]\( x' = -2 - 2 \)[/tex]
- For the [tex]\( y \)[/tex]-coordinate: [tex]\( y' = 6 - 16 \)[/tex]
3. Calculate the Translated Coordinates:
- Calculate the new [tex]\( x \)[/tex]-coordinate: [tex]\( x' = -2 - 2 = -4 \)[/tex]
- Calculate the new [tex]\( y \)[/tex]-coordinate: [tex]\( y' = 6 - 16 = -10 \)[/tex]
4. Find the Translated [tex]\( y \)[/tex]-coordinate:
- The [tex]\( y \)[/tex]-value of [tex]\( P' \)[/tex] after translation is [tex]\( -10 \)[/tex].
Therefore, the [tex]\( y \)[/tex]-value of [tex]\( P' \)[/tex] is [tex]\( -10 \)[/tex].
So the correct answer is [tex]\(\boxed{-10}\)[/tex].
