To find the correct reflection that transforms the endpoints of the line segment from [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex] to [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex], let’s analyze each given option for reflection:
1. Reflection across the x-axis:
- When reflecting a point across the x-axis, the y-coordinate changes its sign.
- Applying this reflection:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((-1, -4)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((4, -1)\)[/tex]
- The new reflected endpoints are [tex]\((-1, -4)\)[/tex] and [tex]\((4, -1)\)[/tex], which do not match what we need.
2. Reflection across the y-axis:
- When reflecting a point across the y-axis, the x-coordinate changes its sign.
- Applying this reflection:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((1, 4)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((-4, 1)\)[/tex]
- The new reflected endpoints are [tex]\((1, 4)\)[/tex] and [tex]\((-4, 1)\)[/tex], which again do not match what we need.
3. Reflection across the line [tex]\(y = x\)[/tex]:
- When reflecting a point across the line [tex]\(y = x\)[/tex], the coordinates of the point are swapped.
- Applying this reflection:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((4, -1)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((1, 4)\)[/tex]
- The new reflected endpoints are [tex]\((4, -1)\)[/tex] and [tex]\((1, 4)\)[/tex], which still do not match what we need.
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- When reflecting a point across the line [tex]\(y = -x\)[/tex], the coordinates of the point swap places and change signs.
- Applying this reflection:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((-4, 1)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((-1, -4)\)[/tex]
- The new reflected endpoints are [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex], which match exactly what we need.
Thus, the transformation that will produce the correct image of endpoints is:
A reflection of the line segment across the line [tex]\(y = -x\)[/tex].
