Compass and Rule construction of a parabola
Given two points, and the slope at one of them

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Points A and P, and the Slope at A determine a Parabola.
The slope is adusted by moving point B.
Point E will always lie on the Parabola:

Steps in the construction of Point P:
Construct a vertical line through P... (black dashed line)
Locate Point on that line (C)
construct lines AP (Red dotted) and AC(black solid)
Construct line through C parallel to slope (AB) (blue dotted)
Construct D where line parallel to slope meets AP
construct vertical line through D (green dotted)
Where this vertical line (green dotted) crosses AC (point E) is a point on the parabola.

Pat Ballew, June 15, 2009, Created with GeoGebra