This is a cool demonstration showing what happens when I tried to find equations of all possible parabolas which passed through 3 given points. The slider for theta defines the angle by which the parabola is rotated around the origin.
The algebra required to get this was pretty intense and was supported by my CAS (Computer Algebra System). A complete explanation of how the equation was derived is explained here: http://casmusings.wordpress.com/2012/09/08/rotating-parabolas/ .