Consider the mapping, w(z) = 1/z. In particular show that circles and straight lines map into circles or straight lines. To prove this, consider any circle, defined by
with 
and a constants.
Consider 
and 
separately.
Show that the former maps into a straight line, the later maps into
a circle. Find the center and radius.
Now consider
with and a constants and 
a real parameter.
Show that the image is a straight line through the origin
if 
and the image is a circle, otherwise.
Finally, consider the mapping
Describe the map in terms of a translation in the w-plane, a translation in the z-plane, and the mapping above.