First, we need to understand the transformation rule given in the problem: [tex]\((x, y) \rightarrow (x-2, y-16)\)[/tex]. This means each point of triangle [tex]\(PQR\)[/tex] will be translated as follows:
1. The [tex]\(x\)[/tex]-coordinate of each point is decreased by 2.
2. The [tex]\(y\)[/tex]-coordinate of each point is decreased by 16.
We are asked to find the new [tex]\(y\)[/tex]-value of point [tex]\(P\)[/tex] after the transformation. Start by identifying the original [tex]\((x, y)\)[/tex] coordinates of point [tex]\(P\)[/tex]. The coordinates of [tex]\(P\)[/tex] are given as [tex]\((-2, 6)\)[/tex].
According to the translation rule, the new [tex]\(y\)[/tex]-coordinate of point [tex]\(P\)[/tex], which we will call [tex]\(P'\)[/tex], is obtained by subtracting 16 from the original [tex]\(y\)[/tex]-coordinate of [tex]\(P\)[/tex].
The original [tex]\(y\)[/tex]-coordinate of [tex]\(P\)[/tex] is 6. Applying the translation transformation to the [tex]\(y\)[/tex]-coordinate, we perform the following calculation:
[tex]\[ 6 - 16 = -10 \][/tex]
Therefore, the new [tex]\(y\)[/tex]-coordinate of [tex]\(P'\)[/tex] is [tex]\(-10\)[/tex]. Thus, the [tex]\(y\)[/tex]-value of [tex]\(P'\)[/tex] (denoted as [tex]\(y_{P'}\)[/tex]) is [tex]\(-10\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{-10}
\][/tex]
