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 (2)
To find the image of a point under the transformation r y = x :
Swap the x and y coordinates of the original point.
Given the point ( 5 , − 2 ) , the transformed point becomes ( − 2 , 5 ) .
The image of ( 5 , − 2 ) under the transformation r y = x is ( − 2 , 5 ) .
The final answer is ( − 2 , 5 ) .
Explanation
Understanding the Transformation To find the image of a point under the transformation r y = x (reflection over the line 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 a point over the line y=x is a fundamental concept in coordinate geometry. It's used in various applications, such as computer graphics for creating mirror images or symmetric designs. For example, if you have a design on a coordinate plane and want to create a symmetric version of it with respect to the line y=x, you would reflect each point of the design over this line. This ensures that the new design is a mirror image of the original, which can be useful in creating logos, patterns, or other artistic designs.
The image of the point ( 5 , − 2 ) under the transformation r y = x is obtained by swapping its coordinates, resulting in the point ( − 2 , 5 ) . Therefore, the answer is option C. This transformation reflects points over the line y = x .
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