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!

Share

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

now.

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.