To solve the given problem, we need to determine which reflection will transform the endpoints [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex] of a line segment into [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex]:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- Reflecting [tex]\((-1, 4)\)[/tex] across the [tex]\(x\)[/tex]-axis results in [tex]\((-1, -4)\)[/tex].
- Reflecting [tex]\((4, 1)\)[/tex] across the [tex]\(x\)[/tex]-axis results in [tex]\((4, -1)\)[/tex].
- This does not match the desired endpoints [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
- Reflecting [tex]\((-1, 4)\)[/tex] across the [tex]\(y\)[/tex]-axis results in [tex]\((1, 4)\)[/tex].
- Reflecting [tex]\((4, 1)\)[/tex] across the [tex]\(y\)[/tex]-axis results in [tex]\((-4, 1)\)[/tex].
- This also does not match the desired endpoints [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- Reflecting [tex]\((-1, 4)\)[/tex] across the line [tex]\(y = x\)[/tex] results in [tex]\((4, -1)\)[/tex].
- Reflecting [tex]\((4, 1)\)[/tex] across the line [tex]\(y = x\)[/tex] results in [tex]\((1, 4)\)[/tex].
- This transformation does not match the desired endpoints [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- Reflecting [tex]\((-1, 4)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in [tex]\((-4, -1)\)[/tex].
- Reflecting [tex]\((4, 1)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in [tex]\((-1, -4)\)[/tex].
- This matches the desired endpoints [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].
Given that reflecting across the line [tex]\(y = -x\)[/tex] correctly transforms [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex] to [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex], the correct answer is the reflection across the line [tex]\(y = -x\)[/tex].
