What is the image of point $A(4,2)$ after the composition of transformations defined by $R_{90^{\circ}} \circ r_{y=x}$?
1) $(-4,2)$
2) $(4,-2)$
3) $(-4,-2)$
4) $(2,-4)$
Answer (1)
Reflect point A ( 4 , 2 ) across the line y = x to get A ′ ( 2 , 4 ) .
Rotate A ′ ( 2 , 4 ) by 9 0 ∘ counterclockwise to get A ′′ ( − 4 , 2 ) .
The final image of point A is ( − 4 , 2 ) .
Explanation
Analyze the problem Let's analyze the problem. We are given a point A ( 4 , 2 ) and a composition of two transformations: a reflection across the line y = x , denoted by r y = x , followed by a rotation of 9 0 ∘ counterclockwise, denoted by R 9 0 ∘ . We need to find the image of point A after applying both transformations in the specified order.
Apply the reflection First, we apply the reflection r y = x to point A ( 4 , 2 ) . The rule for reflection across the line y = x is ( x , y ) → ( y , x ) . Therefore, the image of A ( 4 , 2 ) after the reflection is A ′ ( 2 , 4 ) .
Apply the rotation Next, we apply the rotation R 9 0 ∘ to the image A ′ ( 2 , 4 ) . The rule for a 90 -degree counterclockwise rotation is ( x , y ) → ( − y , x ) . Therefore, the image of A ′ ( 2 , 4 ) after the rotation is A ′′ ( − 4 , 2 ) .
Find the final image The image of point A ( 4 , 2 ) after the composition of transformations R 9 0 ∘ ∘ r y = x is A ′′ ( − 4 , 2 ) . Comparing this result with the given options, we see that option 1) is the correct one.
Examples
Understanding transformations is crucial in computer graphics for tasks like rotating, reflecting, and positioning objects on the screen. For example, when designing a game, you might use these transformations to mirror an enemy's movement or rotate a character to face a different direction. By applying a series of transformations, you can create complex animations and visual effects.
