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.
Draw a line. Pick an arbitrary point F that is not on the line.
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.
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.
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.
To see the parabola or locus of points equidistant form both the directrix and the focus click HERE.
To see a trace of the tangent line constructed at the constructed point click HERE.
Click HERE to see the locus command.