I would do this a little bit differently.

First let

Then

This gives that

Now you can just parameterize each of the line segments.

The vertical line segment from (1,0) to (1,1) is given by

Under the transformation this gives

so the start and end points are (1,0) and (0,2) now if you elminate the paramter you get the curve

So this is a parabolic arc.

Now for the horizontal line segemtn from (1,1) to (0,1) we have

under the transformation this gives

If you eliminate the parameter you get

we get another parabolic region with start point (0,1) and end point (-1,0)

Finally the line can be paramterized by

under the transformation this gives

If we eliminate the parameter we get

another parabola with start point (-1,0) and end point (1,0)

Here is a plot of the transformation. The first curve is in green, the 2nd in red, and the last in blue.