Question Video: Identifying the Type of Geometric Transformation Given the Coordinate Tansformation Mathematics

Video Transcript

The vertices of a square are transformed by the transformation 𝑥, 𝑦 is mapped to negative 𝑥, 𝑦. Which of the following geometric transformations is performed? (a) Translation, (b) rotation, or (c) reflection.

We’re told that the vertices of this square are transformed by the transformation 𝑥, 𝑦 is mapped to negative 𝑥, 𝑦. So, for each vertex, the 𝑥-coordinate changes sign, and the 𝑦-coordinate stays the same. We’re not given the coordinates of the vertices of this square. So we may find it helpful to choose some arbitrary coordinates ourselves to help with visualizing the transformation. Let’s choose the points one, one; one, two; two, two; and two, one to be the coordinates of 𝐴, 𝐵, 𝐶, and 𝐷, respectively.

Under the given transformation, point 𝐴 will be mapped to the point 𝐴 prime with coordinates negative one, one. And we can add the image of point 𝐴 to the coordinate grid. Point 𝐵 with coordinates one, two will be mapped to negative one, two. Point 𝐶 will be mapped to negative two, two. And point 𝐷 will be mapped to negative two, one.

Now we need to compare the two shapes to determine the type of transformation that has occurred. We can see that the square has been flipped over. For example, the vertices 𝐴 and 𝐵 are originally on the left of the square, and in its image, they are on the right. The type of transformation is therefore a reflection. Whilst it’s not required, we can also identify that the position of the mirror line is along the 𝑦-axis. So the answer is option (c), a reflection.