To determine which reflection will produce an image with the given coordinates, we need to analyze how the coordinates change under various reflections.
1. Reflection across the [tex]\( x \)[/tex]-axis:
- For a reflection across the [tex]\( x \)[/tex]-axis, the coordinates [tex]\((x, y)\)[/tex] become [tex]\((x, -y)\)[/tex].
- Applying this to [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex]:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((-1, -4)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((4, -1)\)[/tex]
- The resulting points [tex]\((-1, -4)\)[/tex] and [tex]\((4, -1)\)[/tex] do not match the desired image points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].
2. Reflection across the [tex]\( y \)[/tex]-axis:
- For a reflection across the [tex]\( y \)[/tex]-axis, the coordinates [tex]\((x, y)\)[/tex] become [tex]\((-x, y)\)[/tex].
- Applying this to [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex]:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((1, 4)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((-4, 1)\)[/tex]
- The resulting points [tex]\((1, 4)\)[/tex] and [tex]\((-4, 1)\)[/tex] do not match the desired image points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].
3. Reflection across the line [tex]\( y = x \)[/tex]:
- For a reflection across the line [tex]\( y = x \)[/tex], the coordinates [tex]\((x, y)\)[/tex] become [tex]\((y, x)\)[/tex].
- Applying this to [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex]:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((4, -1)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((1, 4)\)[/tex]
- The resulting points [tex]\((4, -1)\)[/tex] and [tex]\((1, 4)\)[/tex] do not match the desired image points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].
4. Reflection across the line [tex]\( y = -x \)[/tex]:
- For a reflection across the line [tex]\( y = -x \)[/tex], the coordinates [tex]\((x, y)\)[/tex] become [tex]\((-y, -x)\)[/tex].
- Applying this to [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex]:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((-4, -(-1)) = (-4, 1)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes ( [tex]\(-1, -4\)[/tex] )
- The resulting points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex] match the desired image points exactly.
Hence, the correct reflection that produces the image with endpoints [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex] is:
- a reflection of the line segment across the line [tex]\( y = -x \)[/tex].
