To solve this problem, we need to apply the given translation rule to point [tex]\( P \)[/tex] in triangle [tex]\( PQR \)[/tex]. The coordinates of point [tex]\( P \)[/tex] are [tex]\((-2, 6)\)[/tex].
According to the translation rule [tex]\((x, y) \rightarrow (x-2, y-16)\)[/tex], we will translate the coordinates of point [tex]\( P \)[/tex] as follows:
1. x-coordinate:
[tex]\[
x_{\text{new}} = x_{\text{original}} - 2
\][/tex]
For [tex]\( P(-2,6) \)[/tex]:
[tex]\[
x' = -2 - 2 = -4
\][/tex]
2. y-coordinate:
[tex]\[
y_{\text{new}} = y_{\text{original}} - 16
\][/tex]
For [tex]\( P(-2,6) \)[/tex]:
[tex]\[
y' = 6 - 16 = -10
\][/tex]
Thus, the new coordinates of point [tex]\( P \)[/tex] after applying the translation are [tex]\((-4, -10)\)[/tex].
Therefore, the [tex]\( y \)[/tex]-value of [tex]\( P' \)[/tex] is:
[tex]\[
\boxed{-10}
\][/tex]
