Hey, everyone. Welcome back. So in the previous video, we talked about finding the asymptotes of a hyperbola. 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. That's the first step.
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 -3. And on our graph down here, well, 3 is between 2 and 4, so we're going to have a point right there, and then a point at -3, which is right about there.
Step 3 here tells us that we need to find our b points, and to find our b points, 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 because 8 times 8 is 64. 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. So if I go ahead and draw the b values, one's going to be up here at 8, and one's going to be down here at -8.
Now, our 4th step is going to be to find the asymptotes, and this is going to be a 2 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. This is what the box would look like. 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. 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 one 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. This is going to be what the hyperbola looks like.
So now that we have our 2 branches or curves, we have successfully graphed this hyperbola. 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 is equal to a2 plus 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 plus b2, which we said was equal to 64. It's going to be 9 + 64. 9 + 64 comes out to 73, which I'm going to write over here. So, we're going to have 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 - 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.