# Constructing a Parabola

By

# 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= ax^{2} + 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.

RETURN