Constructing a Parabola




Audrey V. Simmons


Parabolas are usually graphed by using the quadratic equation given for the particular situation.  We know that if the equation is written in vertex form

y = a(x-h)2 + k   that the vertex is (h,k).  Values of  a and k tell us other information about the shape of the graph. We can plot additional values and draw the graph.


 Sometimes the quadratic equation is written in standard form   y= ax2 + bx + c .  In this case, we know that the value for h (x value in the vertex) is equal to     .  We can use h to find the value of k (our y coordinate).  By substituting other values for x and y we can draw the graph.


This time we would like to construct a parabola.  By definition, a parabola is a set of points (locus) equidistant from a line and a fixed point.  The line is called the directrix.  The fixed point is called the focus. We will pick our focus so that it is not on the directrix.


Step 1

Draw a line.  Pick an arbitrary point F that is not on the line.




Step 2


Pick another arbitrary point called P in this illustration. This time the point should be on the directrix.  Draw a perpendicular line from point P.  We will be using this line to find a point that is equidistant from both the directrix and the focus.



Step 3


Draw a segment connecting our focus (F) with the arbitrary point (P) on the directrix.  Find the midpoint of this segment and draw a perpendicular line through that point. Remember all points on this line will be equidistant from both the focus and point P.



Step 4


Since all points on the perpendicular bisector are the same distance from F and P we can draw an isosceles triangle with points F, P, X. The point where the perpendicular bisector and the perpendicular to the directrix intersect is a point on the parabola.


Step 5


To see the parabola or locus of points equidistant form both the directrix and the focus click HERE.


Step 6


To see a trace of the tangent line constructed at the constructed point click HERE.


Step 7


Click HERE to see the locus command.