A Parabola Passing Through Three Points
This applet demonstrates that it is usually possible to draw a parabola that passes through three given points A, B, and C.
In the applet, you may click and drag the points A, B, and C; the corresponding parabola through these points will be drawn. Notice that the curve becomes a straight line if you move one of the points so that it is collinear with the others. Also, if you move one of the points so that it has the same x-coordinate as one of the others, the curve will disappear, indicating that there is no parabola (with y a function of x) in this case.
Bradley and Stemkoski, Created with GeoGebra