To begin this exploration, I must first construct a parabola. Some might ask how you would go about that. Well, lets first define what a parabola is!!! A parabola is the set of points equidistant from a line, called the directrix, and a fixed point, called the focus. Assume the focus is not on the line. First, let's construct the focus and directrix.
Notice, the focus is not on the given line ( the directrix).
The next step in the construction of a parabola is to construct a point
on the directrix and draw a segment from that point to the focus.
The next step of construction is to make the perpendicular bisector of the segment connecting the focus to the directrix as follows:
Construct a line perpendicular to the directrix and going through the point on the directrix. The point where the two perependicular line constructions meet is the set of locus you will trace to make a parabola.
Now, I will show you what the trace looks like after running through a animation.
The trace is the set of points that are equidistant from the directrix and the focus point.
How would the graph looked if I traced the line tangent to the parabola?
The parabola shape will change depending on the location of the focus, with respect to the directrix. Let's examine a few different shapes.
Easily some conclusions can be drawn. The further the focus is
from the directrix the wider the parabola shape will be and the closer the
focus is to the directrix the narrower the parabola shape.
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