To determine which reflection of the point [tex]\((m, 0)\)[/tex] will produce an image located at [tex]\((0, -m)\)[/tex], we need to consider the effect of reflecting a point across different lines.
1. Reflection across the [tex]\(x\)[/tex]-axis:
- Reflecting a point [tex]\((a, b)\)[/tex] across the [tex]\(x\)[/tex]-axis changes its coordinates to [tex]\((a, -b)\)[/tex].
- For the point [tex]\((m, 0)\)[/tex], the reflected coordinates would be [tex]\((m, -0) = (m, 0)\)[/tex].
- Clearly, this reflection does not change the point, so it does not produce the desired image [tex]\((0, -m)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
- Reflecting a point [tex]\((a, b)\)[/tex] across the [tex]\(y\)[/tex]-axis changes its coordinates to [tex]\((-a, b)\)[/tex].
- For the point [tex]\((m, 0)\)[/tex], the reflected coordinates would be [tex]\((-m, 0)\)[/tex].
- This does not match the desired image coordinates of [tex]\((0, -m)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- Reflecting a point [tex]\((a, b)\)[/tex] across the line [tex]\(y = x\)[/tex] swaps the coordinates to [tex]\((b, a)\)[/tex].
- For the point [tex]\((m, 0)\)[/tex], the reflected coordinates would be [tex]\((0, m)\)[/tex].
- Again, this does not match the desired image coordinates of [tex]\((0, -m)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- Reflecting a point [tex]\((a, b)\)[/tex] across the line [tex]\(y = -x\)[/tex] swaps and negates the coordinates to [tex]\((-b, -a)\)[/tex].
- For the point [tex]\((m, 0)\)[/tex], the reflected coordinates would be [tex]\((0, -m)\)[/tex].
- This matches the desired image coordinates of [tex]\((0, -m)\)[/tex].
Hence, the correct reflection that produces the image at [tex]\((0, -m)\)[/tex] is a reflection of the point across the line [tex]\(y = -x\)[/tex]. Therefore, the answer is:
a reflection of the point across the line [tex]\(y = -x\)[/tex]
