SM2 14.4-1 Writing the Equation of a Hyperbola Given the Graph

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[00:00:00]Hello and welcome. So for the graph that we  see here for a hyperbola, we're going to be writing the equation. Now there's two kinds  of equations. I've already written them off to the side here. We'll see one where you have  the x minus h over a squared, um, before the negative. The one where it'll essentially switch  the other term and be out for the negative. Now essentially this determines whether or not you  have a horizontal or a vertical orientation for your hyperbola. If your x is positive that means  it's going to be horizontal. If your y is positive it'll be vertical. So where we see a vertical  hyperbola right where it's going up and down, we're going to be using this equation. All  right. So if you have horizontal you'll want to use the other equation there. Okay so we  are looking at vertical so let's use that one.

[00:00:49]Okay so first off for any hyperbola you got to  know a few things. You got to know the center, you have to know your a and your b. That's the  bare minimum you need for at least a graph here or at least even write the equation in uh with  just information. You got to know that much. So the center is always h comma k. That's why we got  to know it because that'll be your h and your k inside the equation here. Well the center we can  see because that's going to be the intersection of the oblique asymptotes which is that point right  there, meaning that your center is at (5, 2).

[00:01:25]So that means that your h value is 5 and  your k is 2. So right then and there we know that for our final equation where we're going to  be ending up at is we will have y minus k, which we just said was 2 squared over whatever b squared  is, minus our x minus our h which we just said was 5 squared over whatever we find our a squared to be  and then equals 1. Okay so now we've got to find our a and b. Now this one's not too bad either.  All right so remember a is always the horizontal distance to either a vertex or a co-vertex. Now  in this case it's going to be a co-vertex because your hyperbola is vertical. All right and then b  is going to be the vertical distance. All right so from center to a vertex. Right, vertex is going  to be the point, the turning point on your curves for the hyperbola. So that's a distance of two, so  that means your b value equals two. If we were to square both of those that means b squared is four.  So that means we have a 4 under our y minus 2 squared down there in our equation. Now we're just  going to find the a. Now this one takes a little bit of work. So a co-vertex is essentially  if you were to graph the little box here right. So go from your, let me not use that, let  me use highlighter instead it's a little easier to see. So if you go from your vertex, right, and then  you essentially go until you hit the line on the right, go until they hit the line on the left.  Right, same thing for the bottom curve. Do the go to the right, go to the left. And now  you'll notice that if you just connect these two points together it forms a little  box and that is how you can easily find your co-vertex or your covertices I should  say. Right because they're on the same y value when it's a vertical hyperbola, would  be the same x value for horizontal hyperbola.

[00:03:26]Okay and then essentially it's just the same as  the center and then middle of the box, middle of the lines for the box. Okay so that means that we  have a distance of three. That's your a value so a equals three. So if you have a equals three,  square both sides that means a squared is nine.

[00:03:47]Tadaa. That's your equation. Okay so big thing is just  finding the center, a value, b value. Done.

[00:03:56]Okay all right, that's it, thanks for watching.