What is the pre-image of vertex [tex]$A^{\prime}$[/tex] if the rule that created the image is [tex]$r_{y \text {-axis }}(x, y) \rightarrow(-x, y)$[/tex]?
A. [tex]$A(-4,2)$[/tex]
B. [tex]$A(-2,-4)$[/tex]
C. [tex]$A(2,4)$[/tex]
D. [tex]$A(4,-2)$[/tex]
Answer (1)
The transformation is a reflection over the y-axis, defined as r y -axis ( x , y ) → ( − x , y ) .
To find the pre-image, apply the inverse transformation, which is the same reflection over the y-axis.
Apply the transformation to the image point A ′ ( − 4 , 2 ) : A ( x , y ) = ( − ( − 4 ) , 2 ) = ( 4 , 2 ) .
The pre-image of A ′ ( − 4 , 2 ) is ( 4 , 2 ) .
Explanation
Analyze the problem The problem states that the image of a vertex A is A ′ ( − 4 , 2 ) after a reflection over the y-axis. The rule for reflection over the y-axis is given by $r_{y "-axis
Determine the inverse transformation To find the pre-image of A ′ , we need to apply the inverse transformation. For a reflection over the y-axis, the inverse transformation is the same reflection over the y-axis. Therefore, we apply the rule $r_{y "-axis
Apply the inverse transformation Applying the transformation to A ′ ( − 4 , 2 ) , we get: A ( x , y ) = r y -axis ( − 4 , 2 ) = ( − ( − 4 ) , 2 ) = ( 4 , 2 ) Thus, the pre-image of A ′ ( − 4 , 2 ) is A ( 4 , 2 ) .
State the final answer The pre-image of vertex A ′ is ( 4 , 2 ) .
Examples
Reflections are commonly used in computer graphics to create symmetrical images or mirror effects. For instance, when designing a game, reflecting a character or object across the y-axis can quickly generate a mirrored version, saving development time and ensuring visual consistency. This principle is also applied in image editing software to create reflections in water or other reflective surfaces, enhancing the realism of the scene.
