What is the image of $(5,-2)$ under the transformation $r_{y=x}$?
A. $(5,2)$
B. $(2,5)$
C. $(-2,5)$
D. $(-5,2)
Answer (1)
To reflect a point across the line y = x , the x and y coordinates are swapped.
Given the point ( 5 , − 2 ) , we swap the coordinates.
The new coordinates are ( − 2 , 5 ) .
The image of ( 5 , − 2 ) under the transformation r y = x is ( − 2 , 5 ) .
Explanation
Understanding the Transformation To find the image of a point under the transformation r y = x , we simply swap the x and y coordinates of the point.
Applying the Transformation Given the point ( 5 , − 2 ) , we swap the x and y coordinates to get ( − 2 , 5 ) .
Final Answer Therefore, the image of ( 5 , − 2 ) under the transformation r y = x is ( − 2 , 5 ) .
Examples
Reflecting points over the line y = x is useful in various applications, such as computer graphics, where you might need to mirror an object across a diagonal axis. For example, if you are designing a symmetrical user interface element and need to ensure that elements are mirrored correctly, this transformation helps to find the exact position of the mirrored element. Understanding transformations like this is also fundamental in more advanced mathematical concepts like linear algebra and matrix transformations.
