Exercise 2 Page 691

We are told that the coordinate grid below shows the final image of a point that was rotated $90^{\circ }$ clockwise about the origin, dilated by a scale factor of $2,$ and shifted $7$ units right.

To determine the original coordinates of the point, let's reverse each of these steps starting with the last one. The point was shifted $7$ units right, so we need to shift it horizontally $7$ units left. Let's do this!

Before that, the point was dilated by a scale factor of $2.$ Let's recall that if a point $(x,y)$ is dilated about the origin, both $x\text{-}$ and $y\text{-}$coordinate of the point are multiplied by the scale factor $k$ of the dilation. In our case the scale of the dilation is $2,$ so the original coordinates were doubled. From the diagram, we know that the coordinates of the image are $(\text{-}6,2).$ Dividing each coordinate by $2,$ we can find the coordinates of $P$ before dilation. Let's plot it on the coordinate plane!

Before dilation, the point was rotated $90^{\circ }$ clockwise around the origin. Hence, to find the initial position of the point, let's rotate it $90^{\circ }$ counterclockwise.

We conclude that the original coordinates of the point $P$ are $(\text{-}1,\text{-}3).$