Question Video: Sketching the Image of a Transformation After a Given Coordinate Transformation Mathematics

Video Transcript

The given figure shows a triangle on the coordinate plane. Sketch the image of the triangle after the geometric transformation 𝑥, 𝑦 is mapped to negative 𝑦, 𝑥.

We are told that the geometric transformation we need to apply is at a general point with coordinates 𝑥, 𝑦 is mapped to the point with coordinates negative 𝑦, 𝑥. This means that the 𝑥- and 𝑦-coordinates swap around, and then we also change the sign of the new 𝑥-coordinate. We can apply this mapping to the coordinates of each vertex of triangle 𝐴𝐵𝐶 individually.

The coordinates of vertex 𝐴 are two, four. Applying the given mapping, so swapping the coordinates around and then changing the sign of the new 𝑥-coordinate, gives the point 𝐴 prime with coordinates negative four, two. We can plot this point on the coordinate grid to show the image of point 𝐴. The coordinates of point 𝐵 are three, one. Applying the given mapping gives the coordinates of 𝐵 prime as negative one, three, and then we can also plot this point. Finally, the coordinates of point 𝐶 are one, one, which under the given transformation is mapped to negative one, one. Plotting point 𝐶 prime and then connecting the three points together gives the image of triangle 𝐴𝐵𝐶 after the given transformation.

Although it isn’t required, we can also observe that the type of transformation that has been applied is a rotation, because the orientation of the triangle has changed: it’s now on its side compared to its original orientation.