Parabola constructed graphically from 3 points using slopes
A special quality of the parabola is that the slope between any two points is the slope on the curve at half the horizontal distance between those two points. This property is used to geometrically make a graph of slope vs x and find the intersection of that line with a line parallel to the x-axis (where the slope would be zero). The vertical line through that intersection is the axis of symmetry for the parabola. With that axis, two of the 3 given points can be used to locate the focus (N), directrix (horizontal line through D) and vertex (T) of the parabola.
Read the discussion at http://www.geogebra.org/forum/viewtopic.php?f=2&t=2887 and compare the construction discussed by batmath (shown at http://www.batmath.it/interattive/ggb/parab_tre_pti/parab_tre_pti.htm) with what is given here. At http://mathpages.com/home/kmath546/kmath546.htm there are two methods described which tell how, given 4 points, one can determine the two possible directions of the y axis. Newton's method of constructing the parabola through 3 points and a given axis of symmetry is discussed in http://www2.washjeff.edu/users/mwoltermann/Dorrie/45.pdf.