# Parabola constructed graphically from 3 points using slopes

- Author:
- smichr_1

- Topic:
- Parabola

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.