Answer :
To determine which reflection of the point \((m, 0)\) will produce an image located at \((0, -m)\), let's consider the effects of reflecting a point across various lines:
1. Reflection across the \(x\)-axis:
When a point \((m, 0)\) is reflected across the \(x\)-axis, the new coordinates of the point will still be on the \(x\)-axis, specifically:
[tex]\[ (m, 0) \rightarrow (m, 0) \][/tex]
This reflection does not change the coordinates in a way that would give us \((0, -m)\).
2. Reflection across the \(y\)-axis:
When a point \((m, 0)\) is reflected across the \(y\)-axis, the new coordinates will be:
[tex]\[ (m, 0) \rightarrow (-m, 0) \][/tex]
This changes the sign of the \(x\)-coordinate, not the \(y\)-coordinate, and still places the point on the \(x\)-axis. Thus, this will not give us \((0, -m)\).
3. Reflection across the line \(y = x\):
When a point \((m, 0)\) is reflected across the line \(y = x\), the coordinates are swapped:
[tex]\[ (m, 0) \rightarrow (0, m) \][/tex]
This changes the coordinates but places the \(y\)-coordinate of the original point into the \(x\)-coordinate of the new point and vice versa. Thus, this will not result in \((0, -m)\).
4. Reflection across the line \(y = -x\):
When a point \((m, 0)\) is reflected across the line \(y = -x\), the coordinates are swapped and the signs are changed:
[tex]\[ (m, 0) \rightarrow (0, -m) \][/tex]
This reflection transforms the \(x\)-coordinate of the original point into the \(y\)-coordinate of the new point with the sign changed, and the \(y\)-coordinate of the original point into the \(x\)-coordinate of the new point.
Therefore, the correct reflection that produces an image located at \((0, -m)\) is a reflection of the point across the line \(y = -x\).
Hence, the answer is:
A reflection of the point across the line [tex]\(y=-x\)[/tex].
1. Reflection across the \(x\)-axis:
When a point \((m, 0)\) is reflected across the \(x\)-axis, the new coordinates of the point will still be on the \(x\)-axis, specifically:
[tex]\[ (m, 0) \rightarrow (m, 0) \][/tex]
This reflection does not change the coordinates in a way that would give us \((0, -m)\).
2. Reflection across the \(y\)-axis:
When a point \((m, 0)\) is reflected across the \(y\)-axis, the new coordinates will be:
[tex]\[ (m, 0) \rightarrow (-m, 0) \][/tex]
This changes the sign of the \(x\)-coordinate, not the \(y\)-coordinate, and still places the point on the \(x\)-axis. Thus, this will not give us \((0, -m)\).
3. Reflection across the line \(y = x\):
When a point \((m, 0)\) is reflected across the line \(y = x\), the coordinates are swapped:
[tex]\[ (m, 0) \rightarrow (0, m) \][/tex]
This changes the coordinates but places the \(y\)-coordinate of the original point into the \(x\)-coordinate of the new point and vice versa. Thus, this will not result in \((0, -m)\).
4. Reflection across the line \(y = -x\):
When a point \((m, 0)\) is reflected across the line \(y = -x\), the coordinates are swapped and the signs are changed:
[tex]\[ (m, 0) \rightarrow (0, -m) \][/tex]
This reflection transforms the \(x\)-coordinate of the original point into the \(y\)-coordinate of the new point with the sign changed, and the \(y\)-coordinate of the original point into the \(x\)-coordinate of the new point.
Therefore, the correct reflection that produces an image located at \((0, -m)\) is a reflection of the point across the line \(y = -x\).
Hence, the answer is:
A reflection of the point across the line [tex]\(y=-x\)[/tex].
