Let's analyze the effect of reflecting a point across different lines to determine which reflection will transform the point [tex]\((m, 0)\)[/tex] to the point [tex]\((0, -m)\)[/tex].
1. Reflection across the [tex]\(x\)[/tex]-axis:
When we reflect a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis, the new coordinates become [tex]\((x, -y)\)[/tex].
- For the point [tex]\((m, 0)\)[/tex], reflecting across the [tex]\(x\)[/tex]-axis results in [tex]\((m, 0)\)[/tex].
This reflection does not change the [tex]\(x\)[/tex]-coordinate and produces no change that moves [tex]\(m\)[/tex] to [tex]\(-m\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
When we reflect a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis, the new coordinates become [tex]\((-x, y)\)[/tex].
- For the point [tex]\((m, 0)\)[/tex], reflecting across the [tex]\(y\)[/tex]-axis results in [tex]\((-m, 0)\)[/tex].
This reflection changes the sign of the [tex]\(x\)[/tex]-coordinate but does not affect the [tex]\(y\)[/tex]-coordinate, so it does not place the point at [tex]\((0, -m)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
When we reflect a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex], the new coordinates become [tex]\((y, x)\)[/tex].
- For the point [tex]\((m, 0)\)[/tex], reflecting across the line [tex]\(y=x\)[/tex] would give [tex]\((0, m)\)[/tex].
This reflection swaps the coordinates, but the resulting point would be in the form [tex]\((0, m)\)[/tex], not [tex]\((0, -m)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
When we reflect a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex], the new coordinates become [tex]\((-y, -x)\)[/tex].
- For the point [tex]\((m, 0)\)[/tex], reflecting across the line [tex]\(y = -x\)[/tex] results in [tex]\((0, -m)\)[/tex].
This reflection swaps the coordinates and changes their signs.
Hence, reflecting the point [tex]\((m, 0)\)[/tex] across the line [tex]\(y = -x\)[/tex] produces the image located at [tex]\((0, -m)\)[/tex].
The correct answer is:
A reflection of the point across the line [tex]\(y = -x\)[/tex].
