Hey, everyone. Welcome back. So one thing that you may have learned in math classes up until now is how to draw the graphs of equations and functions and look at their behaviors. One very common behavior that you'll see in a graph is where a graph gets really, really close to a value. But it never actually quite touches it. Look at this purple graph and as it's all the way out to the right, you'll see that it gets close to a value of two, but it never actually quite gets there. This is a perfect example of what we call an Asymptote. I'm gonna go ahead and review asymptotes in this video because you'll need to be able to identify them and also draw them and write their equations. Let me do some examples together. Let's get started here. An Asymptote is really just like kind of like an imaginary line that a graph gets really close to, but it never actually touches or crosses. So this purple graph over here gets really close as we go to the right to a value of two, but it never actually touches it or crosses that barrier. Notice how if we keep extending this purple line, it'll get really, really, really close to zero, like 0.00001 but it never actually gets there. So this is an example of a horizontal Asymptote, we draw this with a sort of dashed line like this just to kind of separate it from the graph. What you'll notice here is that on the opposite side of the graph, the same thing I happens if you look at the bottom. Now this graph is gonna approach this value from the left side or so as it goes off to the left, it gets really flat, never quite gets to two, but it gets really, really, really close. So we draw this as a horizontal Asymptote. Now, what's the equation for it? Well, these things are usually just gonna be vertical or horizontal lines. So we just write them as horizontal or vertical line equations. Horizontal lines are written as Y equals some number and this value in this case is just equal to two. So that's the horizontal Asymptote. All right. Now, if you actually look at this graph here, there's actually another example of this behavior happening. Look at the tops and the bottom of this graph notice how as we come in from the left, this purple graph gets really, really, really steep because really vertical, it never actually quite gets to a value of the X equals one, but it gets really, really close. And then once we cross that barrier over here, on the other side, we'll see that. Now the graph is approaching sort of it's coming in from the top all the way down like this. So what happens usually at these asymptotes is that it kind of sets up like a barrier in which the graph can't cross it. It'll kind of go off to infinity each direction. But you'll see the graph coming in from opposing sides like this very common behavior for an Asymptote. And in this particular case, the vertical Asymptote is X equals one. That's really all there is to it. We just draw them as dash lines and then write their equations. All right. Let's take a look at the next example over here. We're gonna have this other purple graph and we're gonna identify and we're gonna sketch all the asymptotes of the function below. Notice how it actually doesn't matter what this function actually is. All we really just need to do is just figure out and identify where these asymptotes are happening. So let's take a look at for any horizon until asymptotes here, this purple graph you'll see uh goes off to the right and the left kind of like our graph above did. And usually that's a good place to look for asymptotes. Notice how on the right side, this thing gets really flat and it approaches a value of zero, but never actually quite gets there on the left side. The same thing happens, it approaches a value of zero but never quite gets there. So there is a horizontal Asymptote here at Y equals zero. It's perfectly fine for asymptotes to be on the X or Y axis. Um That'll actually very commonly happen. All right, there's no other horizontal asymptotes here because notice how the graph doesn't get close to a horizontal or, or AAA value as it goes off to either end. Um And we're crossing all of the other values over here, right? So let's take a look for any vertical asymptotes. Now, now, what we see here is that the uh graphs get really steep at two certain values. Um And you'll notice that as the graph comes in from the left, it goes really vertical and never actually quite gets to a value of negative two. And um the graph would just sort of go off to infinity there. Then what happens is on the other side, let take a look at the sort of middle upside down U over here as we go off to the left over here, you'll see that this graph also approaches a value of negative two. So this is gonna be X equals negative two. That's that vertical Asymptote. There's actually one more over here because this graph ends up being really symmetrical. You'll see that the same exact behavior happens over here. On the right side. This graph gets really close to a value of two. Never gets there. And on the other side of this U gets close to a value of two but never gets there. So this is a ho sorry vertical Asymptote at X equals two. My bad, this is actually supposed to be negative two. All right. So those are the three asymptotes that guide the behavior of this graph. That's really all you need to know. Thanks for watching and let me know if you have any questions.