A line segment has endpoints at $(-4,-6)$ and $(-6,4)$. Which reflection will produce an image with endpoints at $(4,-6)$ and $(6,4)?

A. a reflection of the line segment across the $x$-axis
B. a reflection of the line segment across the $y$-axis
C. a reflection of the line segment across the line $y=x$
D. a reflection of the line segment across the line $y=-x

Check the transformation of the point (-4, -6) and (-6, 4) after reflection across the x-axis.
Check the transformation of the point (-4, -6) and (-6, 4) after reflection across the y-axis.
Check the transformation of the point (-4, -6) and (-6, 4) after reflection across the line y = x .
Check the transformation of the point (-4, -6) and (-6, 4) after reflection across the line y = − x .
The reflection across the y-axis produces the image with endpoints at ( 4 , − 6 ) and ( 6 , 4 ) , so the answer is \boxed{a reflection of the line segment across the y -axis} .

Explanation

Analyze the problem Let's analyze the effect of each reflection on the given endpoints of the line segment. The original endpoints are ( − 4 , − 6 ) and ( − 6 , 4 ) . We want to find the reflection that transforms these points to ( 4 , − 6 ) and ( 6 , 4 ) .

Reflection across the x-axis

Reflection across the x-axis: The transformation is ( x , y ) → ( x , − y ) . Applying this to the original endpoints:

( − 4 , − 6 ) → ( − 4 , 6 )
( − 6 , 4 ) → ( − 6 , − 4 ) This does not produce the desired image.


Reflection across the y-axis

Reflection across the y-axis: The transformation is ( x , y ) → ( − x , y ) . Applying this to the original endpoints:

( − 4 , − 6 ) → ( 4 , − 6 )
( − 6 , 4 ) → ( 6 , 4 ) This matches the desired image.


Reflection across the line y=x

Reflection across the line y=x: The transformation is ( x , y ) → ( y , x ) . Applying this to the original endpoints:

( − 4 , − 6 ) → ( − 6 , − 4 )
( − 6 , 4 ) → ( 4 , − 6 ) This does not produce the desired image.


Reflection across the line y=-x

Reflection across the line y=-x: The transformation is ( x , y ) → ( − y , − x ) . Applying this to the original endpoints:

( − 4 , − 6 ) → ( 6 , 4 )
( − 6 , 4 ) → ( − 4 , 6 ) This does not produce the desired image.


Conclusion The reflection across the y-axis produces the image with endpoints at ( 4 , − 6 ) and ( 6 , 4 ) .


Examples
Reflections are used in computer graphics to create symmetrical images or mirror effects. For example, in game development, reflecting a character or object across a central axis can create a mirrored version of it, saving development time and resources. Understanding reflections helps in creating realistic and visually appealing graphics.

Answered by GinnyAnswer | 2025-07-08

The reflection that transforms the line segment with endpoints at (-4, -6) and (-6, 4) to an image with endpoints at (4, -6) and (6, 4) is the reflection across the y-axis. Thus, the answer is B. a reflection of the line segment across the y-axis.
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Answered by Anonymous | 2025-08-04