To determine the new coordinates of point [tex]\( P \)[/tex] after it has been translated according to the rule [tex]\((x, y) \rightarrow (x-2, y-16)\)[/tex], we will follow these steps:
1. Identify the original coordinates of point [tex]\(P\)[/tex]:
[tex]\( P(-2, 6) \)[/tex]
2. Understand the translation rule:
The rule given is [tex]\((x, y) \rightarrow (x-2, y-16)\)[/tex]. This means each [tex]\( x \)[/tex]-coordinate is decreased by 2, and each [tex]\( y \)[/tex]-coordinate is decreased by 16.
3. Apply the translation to the [tex]\(y\)[/tex]-coordinate of [tex]\(P\)[/tex]:
The original [tex]\( y \)[/tex]-coordinate of [tex]\( P \)[/tex] is [tex]\(6\)[/tex].
According to the translation rule, we need to subtract 16 from this [tex]\( y \)[/tex]-coordinate:
[tex]\[
y_{\text{new}} = 6 - 16
\][/tex]
4. Calculate the new [tex]\( y \)[/tex]-coordinate:
[tex]\[
y_{\text{new}} = 6 - 16 = -10
\][/tex]
So, the new [tex]\( y \)[/tex]-coordinate of point [tex]\( P' \)[/tex] after the translation is [tex]\(-10\)[/tex].
Therefore, the [tex]\( y \)[/tex]-value of [tex]\( P' \)[/tex] is:
[tex]\[
\boxed{-10}
\][/tex]
