Hey, everyone. Welcome back. So, in the previous video, we talked about finding the asymptotes of a hyperbola. And in this video, we're going to see if we can take all this knowledge that we've learned about hyperbolas and combine them into graphing hyperbolas from just the equation. Now, this process has a lot of steps to it, but if you've watched the previous videos up to this point, hopefully, all these steps will make a lot of sense. And don't worry because I'm going to be explaining them all in this video. So, without further ado, let's get right into this.

Now here we have an example where we're asked to graph the hyperbola and identify the foci given this equation. Our first step should be to determine whether the hyperbola is horizontal or vertical, and I can do that by just looking at the equation that we have. Recall that the first term that you see in the denominator is the a² term, and this a² term is underneath x². Since we see an x that shows up first, that means this hyperbola will be oriented here on the x-axis. So because of this, we could say that we are dealing with a horizontal hyperbola.

Now, our next step should be to identify the vertices. And because we have a horizontal hyperbola, we only are going to care about the x-coordinate for the vertices here. So we're going to use this version of the coordinates. Now to find out what a is, well, a² is equal to 9. So if a² is equal to 9, that means a is going to be the square root of 9, and the square root of 9 is equal to 3. So our vertices are going to be at 3 and at negative 3.

Now, step 3 here tells us that we need to find our b values, and to find our b values, we need to look at this other value that we have in the denominator. This corresponds with b, and we see that b² is equal to 64, which means b is going to be the square root of 64. The square root of 64 turns out to be 8. So that means that our b values are going to now follow this pattern, and by the way, it's this pattern because since we had our vertices on the horizontal axis, our b values are going to be on the vertical axis. So we're going to have 0, 8, and 0, -8.

Now, our fourth step is going to be to find the asymptotes, and this is going to be a two-step process. Step a is going to be to draw a box through the vertices and b points. So basically, with all the points that we found so far, I can connect these all with a rectangular box. And step b for drawing the asymptotes is to draw two lines through the corners of the box. So if I draw lines through the corners, one line is going to look like this, and another line is going to look like that. So these are the two lines that we need to draw, which are also the asymptotes for our hyperbola.

And our last step, which is step 5, is going to be to draw branches at the vertices approaching the asymptotes, which are these lines. So if I start at the vertices here, I can draw a curve that looks like this, and that looks like that. Notice how it approaches these lines that we see. And another curve is going to be drawn like this and like that as well. So this is going to be what the hyperbola looks like. So now that we have our two branches or curves, we have successfully graphed this hyperbola.

And as a last step, we're asked to also find the foci of this hyperbola. Well, to do this, recall that we need to use this equation, which is c² ≡ a² + b². Our c² is going to equal a², and we said up here that a² is equal to 9, so it's going to be 9 + b², which we said was equal to 64. It's gonna be 9 + 64. 9 + 64 comes out to 73, which I'm going to write over here. So we're going to have the c² is equal to 73, and then we're going to take the square root on both sides of this equation, giving us that c is equal to the square root of 73, and the square root of 73 is approximately equal to 8.54. This is just the approximate value for the square root of 73, but to put in this value here, we can say that our foci are going to be on the x-axis because it's a horizontal hyperbola. So we're going to have the square root of 73 and 0, and then we're going to have negative square root of 73 and 0 as our foci. And since we said that the c value is approximately equal to 8.54, and it's on the x-axis, that means one of our foci is going to be right about there, and another foci is going to be right about here. So these are the foci for the hyperbola, and that is how you can graph hyperbolas if you're only given the equation. So I hope you found this video helpful. Thanks for watching, and let's move on.