THE PARABOLA
General Definition of Parabola
A parabola is a conic section formed by the intersection of a cone and a plane parallel to the
side of the cone. 1
Mathematical Definition of a Parabola
Given a point in a plane, F, and a line D that does not contain F, a parabola is the set of all points
equidistant from F and D.
F is called the focus
D is called the directrix
V is the vertex (see sketch)
F
V
D
Equations of a Parabola
Standard parabolas can be vertical or horizontal. Vertical parabolas open up or down.
Horizontal parabolas open left or right. The following equation is for a vertical parabola.
(x – a)2 = 4c(y - b)
(a,b) locates the vertex
|c| is the focal length.
The focal length is the distance from the vertex to the focus
If c > 0 the parabola opens up
If c < 0 the parabola opens down
Notes
Another form of the equation for the vertical parabola is ax2 + bx + cy + d
a and c are non-zero. a, b, c, and d are integers
For a horizontal parabola, the equation is (y – b)2 = 4c(x – a)
Another form of the equation for the horizontal parabola is ay2 + by + cx + d
a and c are non-zero. a, b, c, and d are integers
The latus rectum, also called the focal diameter, is parallel to the directrix, passes through the
focus and ends at two points on opposite sides of the parabola
The focal diameter = |4p|
References
1. http://dictionary.reference.com/browse/parabola
2. http://www.stitzzeager.com/Precalculus/Stitz_Zeager_Open_Source_Precalculus_files/SPreCalc07152011.pd
f
3. http://www.college-cram.com/study/precalculus/conic-sections/study-sheet-of-conicsections/