# Construct a Parabola

This is an applet to construct a parabola from its definition. If needed, Free graph paper is available.

__Definition:__ A parabola is the set of all points M in a plane such that the distance from M to a fixed point F, the focus, is equal to the distance from M to a fixed line called the directrix.

The axis of the parabola is the line through F and perpendicular to the directrix. Point V on the axis and halfway between the focus F and the directrix is called the vertex.
It can be shown that the equation of a parabola with a horizontal axis and opening towards increasing x values is given by:
y^{2} = 4ax
where a is a constant. A tutorial to understand of the equation of a parabola is included in this site.
Follow the steps in the tutorial below to construct a parabola using the above definition. The equation is used to verify the construction of the parabola. Examples of applications of the parabolic shape as Parabolic Reflectors and Antennas are included.
__TUTORIAL__
1 - click on the button above "click here to start" to start the applet and MAXIMIZE the window obtained. The slider in the top left panel can be used to change the value of a, do not use it now, a=1.
2 - Before you start the construction, note the following.
The directrix: vertical line at x = -1.
The vertex: point V at (0,0).
The focus: point F at (1,0).
A marker: point M at (1,2), this point can be moved around freely.
Point D on the directrix and has the same y-coordinate as point M.
The distances d(F,M) and d(D,M) are both equal to 2. (their values and the coordinates of points M and D are displayed on the top left of the main panel).
3 - Construction:
a - Start by pressing the button "Plot Points", this will plot point M (in blue) as an element of the set of points whose distances from the directrix and the focus are equal.
b - Drag point M horizontally to a new position. Note the distance d(D,M).
c - Now drag point M vertically untill the distances d(D,M) and d(F,M) are equal or close in value.
d - Now press the button "Plot Points" to plot this point.
4 - Drag point M to a new position and repeat step 3 to plot another point.
5 - Repeat step 4 as many times as you can to plot points whose distances from the vertex and the focus are equal.
6 - You may also want to plot points in quadrant IV.
7 - Once you have enough points plotted, press the button "Plot/Delete Parabola" to plot the whole parabola in order to verify that all points whose distances from the vertex and the focus are equal, can be described by one single equation given above with a = 1. You may also want to drag point M along the parabola and see that the distance from M to F is equal (or very close) to the distance from M to the directrix.
8 - Use the slider to change the value of a. How does the coordinates of the focus F change? How does the directrix (vertical line) change? Find the coordinates of the focus F and the equation of the directrix (vertical line) in terms of a.
9 - Use the slider to change a (a = 2 for example), delete the points and the parabola and repeat the steps to plots points and then the parabola.
9- Exercise: Sketch a parabola on paper. Draw a line through the focus F that is perpendicular to the axis of the parabola. This line intersect the parabola at two points M and N. Show that distance d(M,N) is equal to twice the distance from F to the directrix.
More references and links to topics related to the parabola
Use of parabolic shapes as Parabolic Reflectors and Antannas.
Interactive tutorial on how to find the equation of a parabola.
Interactive tutorial on Equation of a Parabola.
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Updated: February 2015
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