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Focus and Directrix of a Parabola - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

A parabola is the curve formed from all points that are equidistant from the directrix and the focus. The parabola focus is a point from which distances are measured in forming a conic and where these distances converge. A parabola directrix is a line from which distances are measured in forming a conic. The line perpendicular to the directrix and passing through the focus is called the axis of symmetry.

So there's a couple extra elements
that a parabola can present us.
So we're used to knowing how to graph them,
given a vertex and sort of how we
given the shape steep or narrow all that stuff.

A couple other things that come into play
are what are called the focus and
directex. What they are is basically
two different things.
One's a point, one's a line.

The focus is basically a point that is
directly above the vertex in an upward
facing parabola. Would be directly below
the vertex in a downward facing parabola.
What it is if you sort of think about
going to a football game or something
like that you see the guys on the sideline
with these big semi circle basically looking things.
Those are actually parabolas, and what
they are is it's when the sound comes
in there's actually a little
microphone at that focus.
So all the soundwaves come in and they
all get reflected to that single spot.
So no matter where anything comes down,
it reflects into this focus.

And then what the directex is, is not
quite at applicable because it's
not quite in this little parabolic
sound collecting thing but what it
is, is it's a line so that any point
on the parabola is going to be equi
distant from the focus and
that directex.

If you were to draw a line straight down
to the direct treks that line is going
to be exactly the same distance if you
drove it straight to the focus.
So what we have are these
two different elements.
My drawing as you can see is a little off.
This line should be exactly the same length as that.
But hopefully you get the idea what's going on.

So this parabola I have up here is just
a basic parabola, A, X squared where
A is positive because we have
an upward facing parabola.
What we're used to seeing is A dictating
the steepness of this.
If A is a number greater than
1, it probably becomes steeper.
A is less than 1, it becomes wider.
But actually A is a direct relationship
to this focus.

And really what we call is -- we're used
to saying parabolas are the equation
for quadratics are AX squared plus
B plus C, I want to point out that C
in the quadratic equation is different
than the C I'm going to talk about

The C that we're talking about here is really
the difference or, sorry, the distance
between your focus and your vertex.
And the absolute value of A is going
to be equal to 1 over 4C. That distance.
So that's actually one way we can figure
out exactly how steep or wide this
parabola is by knowing the distance
between the focus and the vertex.

The one other thing I want to talk about
is we know this distance C between
the focus and the vertex is C. We
also know that's going to be the
same distance from the point
to the directex. So by default right at that
vertex point that's going to be the same distance
C from the directex to the vertex.

So the main thing from this little talk is
the focus is a point where everything
you're creating lines that are coming
straight in are going to be deflected
into this focus.
And the relationship between the focus
and directex is any point on the
curve is going to be equidistant
between the two.

And lastly, this A, this coefficient on
X squared, can be defined by absolute
value of A is equal to 1 over 4C.
And the reason they have that absolute value
is this is just going to dictate the
magnitude of A. It doesn't
actually tell us the sign.

So upward facing parabola, A
is going to be positive.
Downward facing parabola we
know A has to be negative.
So we can throw those signs in afterward
but this is going to tell us the manitude
at least in numeric
value of that coefficient.

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