To determine the [tex]\(y\)[/tex]-value of [tex]\(P'\)[/tex] after the translation of the triangle [tex]\( PQR \)[/tex], we start with the given coordinates of point [tex]\( P \)[/tex], which are [tex]\( P(-2, 6) \)[/tex].
The translation rule provided is [tex]\((x, y) \rightarrow (x-2, y-16)\)[/tex], which indicates that for every point, you subtract 2 from its [tex]\(x\)[/tex]-coordinate and 16 from its [tex]\(y\)[/tex]-coordinate.
Let's apply this rule to point [tex]\(P\)[/tex]:
1. The original [tex]\(x\)[/tex]-coordinate of [tex]\(P\)[/tex] is [tex]\(-2\)[/tex].
2. According to the rule, the new [tex]\(x\)[/tex]-coordinate will be [tex]\(-2 - 2 = -4\)[/tex].
3. The original [tex]\(y\)[/tex]-coordinate of [tex]\(P\)[/tex] is [tex]\(6\)[/tex].
4. According to the rule, the new [tex]\(y\)[/tex]-coordinate will be [tex]\(6 - 16 = -10\)[/tex].
Hence, after the translation, the coordinates of [tex]\(P'\)[/tex] will be [tex]\((-4, -10)\)[/tex].
Therefore, the [tex]\(y\)[/tex]-value of [tex]\(P'\)[/tex] is [tex]\(-10\)[/tex].
The correct answer is [tex]\(-10\)[/tex].
