To determine the \( y \)-value of the translated vertex \( P' \), we need to apply the given translation rule to the vertex \( P \).
The coordinates of vertex \( P \) are \( (-2, 6) \).
The translation rule given is:
[tex]\[ (x, y) \rightarrow (x-2, y-16) \][/tex]
We apply this translation rule to the coordinates of \( P \):
1. For the \( x \)-coordinate of \( P \):
[tex]\[ x_P = -2 \][/tex]
Applying the translation rule:
[tex]\[ x' = x_P - 2 = -2 - 2 = -4 \][/tex]
2. For the \( y \)-coordinate of \( P \):
[tex]\[ y_P = 6 \][/tex]
Applying the translation rule:
[tex]\[ y' = y_P - 16 = 6 - 16 = -10 \][/tex]
Thus, the \( y \)-value of the translated vertex \( P' \) is \( -10 \). Therefore, the correct answer is:
[tex]\[ \boxed{-10} \][/tex]
