A parabola is the set of points equidistant from a line,
called the directrix, and a fixed point, called the focus.

Constructing
parabolas:

If you have Geometer's Sketchpad, try to construct your own
parabola by following these directions.

First
select the “show grid” option. This
will help you to place a focus and a directrix for your parabola. Place an arbitrary point on your graph,
which will be the focus, and an arbitrary line, which will be the
directrix. For my focus, I chose the
point (1,2) and for my directrix I chose the x-axis.

Now
you will need to construct a point on your directrix. To do this, highlight your directrix, and under “construct”
choose the “point on segment” option.
Now you have created a “moveable” point on the directrix. Highlight the point and the directrix and
construct a perpendicular line. This
step is important and you will soon see why!

Our
next step is to construct a segment between the focus and our “moveable”
point. Then construct the midpoint of
the new segment. This ensures that this
point on the directrix is
equidistant from the focus. So, when we move the point along the directrix, we
can see that the midpoint of this segment is always equidistant from our
“moveable” point and the focus.

Now, by highlighting the midpoint and the segment, construct a perpendicular line. This line and our other perpendicular line will intersect. The point where they intersect is the point where the focus and directrix are always equidistant! So when we plot a set of these points, we have a parabola!

So
you might be wondering how this is related to Calculus. You might realize from looking at the
picture, that the blue line is the tangent line to the parabola. This allows us to make even more
connections involving the first derivative of a function.

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