To determine which reflection of the point [tex]\((m, 0)\)[/tex] will produce the image located at [tex]\((0, -m)\)[/tex], we need to analyze the effect of each type of reflection on the coordinates [tex]\((m, 0)\)[/tex].
1. Reflection across the x-axis:
Reflecting a point [tex]\((x, y)\)[/tex] across the x-axis results in the point [tex]\((x, -y)\)[/tex].
[tex]\[
(m, 0) \rightarrow (m, -0) = (m, 0)
\][/tex]
The resulting point would be [tex]\((m, 0)\)[/tex], which is not [tex]\((0, -m)\)[/tex].
2. Reflection across the y-axis:
Reflecting a point [tex]\((x, y)\)[/tex] across the y-axis results in the point [tex]\((-x, y)\)[/tex].
[tex]\[
(m, 0) \rightarrow (-m, 0)
\][/tex]
The resulting point would be [tex]\((-m, 0)\)[/tex], which is not [tex]\((0, -m)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
Reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex] results in the point [tex]\((y, x)\)[/tex].
[tex]\[
(m, 0) \rightarrow (0, m)
\][/tex]
The resulting point would be [tex]\((0, m)\)[/tex], which is not [tex]\((0, -m)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
Reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in the point [tex]\((-y, -x)\)[/tex].
[tex]\[
(m, 0) \rightarrow (0, -m)
\][/tex]
The resulting point would be [tex]\((0, -m)\)[/tex], which matches the given point [tex]\((0, -m)\)[/tex].
Therefore, the reflection of the point [tex]\((m, 0)\)[/tex] that will produce the image located at [tex]\((0, -m)\)[/tex] is across the line [tex]\(y = -x\)[/tex].
Thus, the correct answer is:
- a reflection of the point across the line [tex]\(y = -x\)[/tex].
