A point has the coordinates $(m, 0)$ and $m \neq 0$. Which reflection of the point will produce an image located at $(0,-m)$?
A. a reflection of the point across the $x$-axis
B. a reflection of the point across the $y$-axis
C. a reflection of the point across the line $y=x$
D. a reflection of the point across the line $y=-x
Answer (2)
Reflection across the x-axis: ( m , 0 ) → ( m , 0 ) .
Reflection across the y-axis: ( m , 0 ) → ( − m , 0 ) .
Reflection across the line y = x : ( m , 0 ) → ( 0 , m ) .
Reflection across the line y = − x : ( m , 0 ) → ( 0 , − m ) .
Thus, the reflection across the line y = − x produces the image ( 0 , − m ) .
The answer is: a reflection of the point across the line y = − x
Explanation
Analyze possible reflections. Let's analyze the possible reflections to determine which one maps the point ( m , 0 ) to ( 0 , − m ) .
Reflection across the x-axis: When reflecting a point across the x-axis, the x-coordinate remains the same, and the y-coordinate changes sign. So, the image of ( m , 0 ) would be ( m , − 0 ) = ( m , 0 ) . This is not ( 0 , − m ) .
Reflection across the y-axis: When reflecting a point across the y-axis, the y-coordinate remains the same, and the x-coordinate changes sign. So, the image of ( m , 0 ) would be ( − m , 0 ) . This is not ( 0 , − m ) .
Reflection across the line y = x : When reflecting a point across the line y = x , the x and y coordinates are swapped. So, the image of ( m , 0 ) would be ( 0 , m ) . This is not ( 0 , − m ) .
Reflection across the line y = − x : When reflecting a point across the line y = − x , the x and y coordinates are swapped and their signs are changed. So, the image of ( m , 0 ) would be ( 0 , − m ) . This matches the given image point.
Conclusion. Therefore, the reflection across the line y = − x maps the point ( m , 0 ) to ( 0 , − m ) .
Examples
Reflections are used in various fields, such as physics (optics), computer graphics, and even art. For example, when designing a symmetrical building or object, reflections across lines or planes ensure that the two halves are mirror images of each other. In computer graphics, reflections are used to create realistic images of objects in mirrors or water.
The correct reflection that produces the image ( 0 , − m ) from the point ( m , 0 ) is across the line y = − x . Therefore, the answer is: a reflection of the point across the line y = − x .
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