TITLE: How many parabolas can be formed from 3 points? (If we allow rotation)
QUESTION [25 upvotes]: If we allow the standard parabola to rotate, dilate and freely move across a Euclidean plane, then we can define at least one parabola from any 3 distinct points not aligned on a straight line.  
Is this parabola unique?  If not, how many different parabolas can be formed from any 3 distinct points? 

My intuition tells me 3, but I'm not seeing any obvious way to show this (if it's accurate, that is). 
A quick sketch of what I mean (sorry for the poor drawing accuracy)

Here we have 3 parabolas being formed by the same 3 distinct points.

REPLY [3 votes]: A parabola is a conic with a double point at infinity. For each point on the line at infinity other than the three points corresponding to the three lines determined by the given three points,  there is a unique parabola through the given three points not on a line. so there are a continuum of them.