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.