To solve this problem, we need to determine how the point [tex]\( P \)[/tex] with coordinates [tex]\((-2, 6)\)[/tex] is affected by the given translation rule [tex]\((x, y) \rightarrow (x-2, y-16)\)[/tex].
Translation involves shifting a point or a shape from one position to another without rotating or changing its shape. Each point on the shape moves the same distance in the same direction. According to the translation rule, the [tex]\( x \)[/tex]-coordinate of each point decreases by 2 and the [tex]\( y \)[/tex]-coordinate decreases by 16.
Given the coordinates of point [tex]\( P \)[/tex]:
[tex]\[ P(-2, 6) \][/tex]
We will apply the translation rule to the y-coordinate of [tex]\( P \)[/tex]:
[tex]\[ y \rightarrow y - 16 \][/tex]
Here, the [tex]\( y \)[/tex]-coordinate of [tex]\( P \)[/tex] is 6. Applying the translation:
[tex]\[ y - 16 = 6 - 16 = -10 \][/tex]
Thus, 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] after the translation is [tex]\(\boxed{-10}\)[/tex].
