To find the [tex]$y$[/tex]-value of [tex]$P'$[/tex], which is the new position of vertex [tex]$P$[/tex] after the translation, we'll need to follow the given translation rule [tex]\((x, y) \rightarrow (x-2, y-16)\)[/tex].
Let's begin by applying this rule to the coordinates of point [tex]$P$[/tex] [tex]\((-2, 6)\)[/tex].
1. Identify the original coordinates of point [tex]\(P\)[/tex]:
[tex]\[
P = (-2, 6)
\][/tex]
2. Apply the translation rule to the [tex]\(x\)[/tex]-coordinate:
[tex]\[
x' = x - 2 \Rightarrow x' = -2 - 2 = -4
\][/tex]
3. Apply the translation rule to the [tex]\(y\)[/tex]-coordinate:
[tex]\[
y' = y - 16 \Rightarrow y' = 6 - 16 = -10
\][/tex]
Therefore, the translated coordinates of point [tex]\(P\)[/tex] are [tex]\((-4, -10)\)[/tex].
The [tex]\(y\)[/tex]-value of [tex]\(P'\)[/tex] is [tex]\(-10\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{-10}
\][/tex]
