To solve this problem, let's reflect each point of triangle [tex]\(PQR\)[/tex] across the [tex]\(y\)[/tex]-axis and determine the new coordinates of the reflected triangle [tex]\(P'Q'R'\)[/tex].
Reflecting a point across the [tex]\(y\)[/tex]-axis means changing the sign of the [tex]\(x\)[/tex]-coordinate while keeping the [tex]\(y\)[/tex]-coordinate the same. We apply this transformation to each vertex of the triangle:
1. For point [tex]\(P(-2, -4)\)[/tex]:
- The [tex]\(x\)[/tex]-coordinate is [tex]\(-2\)[/tex].
- Reflecting across the [tex]\(y\)[/tex]-axis changes the [tex]\(x\)[/tex]-coordinate to [tex]\(2\)[/tex].
- The [tex]\(y\)[/tex]-coordinate remains [tex]\(-4\)[/tex].
- So, the new coordinates of [tex]\(P'\)[/tex] are [tex]\((2, -4)\)[/tex].
2. For point [tex]\(Q(-3, -1)\)[/tex]:
- The [tex]\(x\)[/tex]-coordinate is [tex]\(-3\)[/tex].
- Reflecting across the [tex]\(y\)[/tex]-axis changes the [tex]\(x\)[/tex]-coordinate to [tex]\(3\)[/tex].
- The [tex]\(y\)[/tex]-coordinate remains [tex]\(-1\)[/tex].
- So, the new coordinates of [tex]\(Q'\)[/tex] are [tex]\((3, -1)\)[/tex].
3. For point [tex]\(R(-4, -4)\)[/tex]:
- The [tex]\(x\)[/tex]-coordinate is [tex]\(-4\)[/tex].
- Reflecting across the [tex]\(y\)[/tex]-axis changes the [tex]\(x\)[/tex]-coordinate to [tex]\(4\)[/tex].
- The [tex]\(y\)[/tex]-coordinate remains [tex]\(-4\)[/tex].
- So, the new coordinates of [tex]\(R'\)[/tex] are [tex]\((4, -4)\)[/tex].
After reflecting all three points, the coordinates of the reflected triangle [tex]\(P'Q'R'\)[/tex] are:
- [tex]\(P' = (2, -4)\)[/tex]
- [tex]\(Q' = (3, -1)\)[/tex]
- [tex]\(R' = (4, -4)\)[/tex]
Therefore, the correct answer is:
D. [tex]\(P' (2,-4), Q' (3,-1), R' (4,-4)\)[/tex]
