Verify that the mapping z --> z^2, (2 € c) will map the line y = -3 to a parabola. find its cartesian equation and sketch it. determine the direction of the movement of the image point as the pre-image point moves from left to right on the line y = -3.

1. We have the complex map which transform the plane to the plane . This is

.

Now, the complex numbers on the line are of the form . if we substitute those values into the map we get

.

So,

.

This means that we can consider the above equation as the parametric equation of the image of the line by the map .

It is not difficult to notice that this is the parametric equation of a parabola. In order to obtain its Cartesian equation in the plane we only need to "eliminate" the parameter . This can be done by the following relation:

that gives the Cartesian equation

. In the usual coordinates we have the equation

.

In the attached figure we have its graph.

2. From the parametric equation we can see that if moves from to the image point moves from the branch of the parabola situated on the first quadrant crossing the horizontal axis when and then going through the branch on the fourth quadrant to infinity.