If the domain of a coordinate transformation that is reflected over the $y$-axis is $(-6,2),(4,5),(2,-1)$, what is the range?
A. $(6,2),(-4,5),(-2,-1)$
B. $(6,-2),(-4,-5),(-2,1)$
C. $(-6,-2),(4,-5),(2,1)$
D. $(-6,-2),(-4,-5),(-2,-1)$
Answer (1)
Reflect each point over the y-axis by changing the sign of the x-coordinate.
Apply the transformation ( x , y ) r i g h t a rro w ( − x , y ) to each point in the domain.
Transform ( − 6 , 2 ) to ( 6 , 2 ) .
Transform ( 4 , 5 ) to ( − 4 , 5 ) and ( 2 , − 1 ) to ( − 2 , − 1 ) .
The range is ( 6 , 2 ) , ( − 4 , 5 ) , ( − 2 , − 1 ) .
Explanation
Analyze the problem and the given data. The problem asks us to find the range of a coordinate transformation that reflects points over the y-axis. The domain of the transformation is given as the set of points ( − 6 , 2 ) , ( 4 , 5 ) , and ( 2 , − 1 ) . A reflection over the y-axis changes the sign of the x-coordinate while leaving the y-coordinate unchanged. Therefore, the transformation rule is ( x , y ) r i g h t a rro w ( − x , y ) .
Apply the transformation to each point in the domain. To find the range, we apply the transformation rule to each point in the domain:
( − 6 , 2 ) r i g h t a rro w ( − ( − 6 ) , 2 ) = ( 6 , 2 )
( 4 , 5 ) r i g h t a rro w ( − 4 , 5 )
( 2 , − 1 ) r i g h t a rro w ( − 2 , − 1 )
Determine the range. The range is the set of transformed points, which is ( 6 , 2 ) , ( − 4 , 5 ) , and ( − 2 , − 1 ) .
Examples
Coordinate transformations, like reflections, are used in computer graphics to manipulate objects in a virtual space. For instance, reflecting a building design over the y-axis can help architects visualize mirrored layouts or create symmetrical structures. Understanding these transformations allows designers to efficiently create and modify complex models, ensuring that designs are both aesthetically pleasing and structurally sound.
