Figure $WXYZ$ is transformed using the rule ${T}_{-4,2}(x, y)$. Point $W$ of the pre-image is at $(1,6)$. What are the coordinates of point $W'$ on the final image?
A. $(-5,8)$
B. $(-3,-8)$
C. $(5,-8)$
D. $(3,8)$
Answer (1)
Reflect the point W ( 1 , 6 ) across the x-axis, resulting in W ′ ( 1 , − 6 ) .
Translate W ′ ( 1 , − 6 ) by the vector ( − 4 , 2 ) .
Calculate the new coordinates: x ′′ = 1 + ( − 4 ) = − 3 and y ′′ = − 6 + 2 = − 4 .
The final coordinates of point W ′′ are ( − 3 , − 4 ) .
Explanation
Analyze the problem The problem describes a transformation of a point W ( 1 , 6 ) in the coordinate plane. The transformation consists of two steps: a reflection across the x-axis, followed by a translation using the vector T ( − 4 , 2 ) . We need to find the coordinates of the final image point W ′′ .
Reflect across x-axis First, we reflect the point W ( 1 , 6 ) across the x-axis. When a point is reflected across the x-axis, its x-coordinate remains the same, and the y-coordinate changes sign. Therefore, the image of W after the reflection, which we'll call W ′ , has coordinates ( 1 , − 6 ) .
Translate the point Next, we translate the point W ′ ( 1 , − 6 ) using the translation vector T ( − 4 , 2 ) . This means we add − 4 to the x-coordinate and 2 to the y-coordinate of W ′ . So, the coordinates of the final image point W ′′ are:
x ′′ = 1 + ( − 4 ) = − 3 y ′′ = − 6 + 2 = − 4
Therefore, W ′′ = ( − 3 , − 4 ) .
State the final answer The final coordinates of point W ′′ after the reflection and translation are ( − 3 , − 4 ) .
Examples
Understanding transformations like reflections and translations is crucial in computer graphics and animation. For example, when creating a mirrored image of an object or moving an object across the screen, these transformations are applied to each point of the object. This ensures that the object is displayed correctly after the transformation. In architecture, transformations help in visualizing how a building design changes when rotated or moved to a different location.
