 Another important feature of a graph is its asymptotes and this is a somewhat peculiar concept that takes a little bit of getting used to. So an asymptote of a graph is a line that the graph approaches more and more closely, either as x gets very large or as y gets very large. And typically we draw asymptotes using dashed line. For example, here's a graph and we have a point on the graph right around here. And so now what I'll do is I will see what happens if my x values get very large. And what that means for x values to get very large, positive, then I'm moving to the right on the graph. And so if I let my x values get very large and I move to the right, it looks like I'm approaching this horizontal line. Meanwhile, if I have a point on the graph and my x values get very large, negative, that means I'll be moving to the left on the graph. So here's a point on the graph and I'll move left and see what happens then. And it appears that I'm flattening out and I'm approaching this line y equals 1. So it appears that y equals 1, this horizontal dashed line, is going to be an asymptote of the graph. Now what happens is y gets very large. So if y gets very large and positive, what that means is I'm going to be going vertically upwards. So I'm going to be going following this line here or maybe I'll be coming up this way. We'll see what happens in either case there. So here I'm going to be at a point and I'm going to follow the graph upwards as far as I can. And it appears that I am approaching this vertical line and that looks like it's around x equals 3. And in fact, that same line, if y becomes large but negative, that means I'm going to go down, way, way, way down. Well, if I take this point, if I get a right on this point and I write the graph down, there I go, following that line as well. And so it appears that this line x equals 3 is also a vertical asymptote. Again, if y is getting very large, maybe I'm on this side of the graph and if y is getting very large, I'm moving up, up, up, up. And so here I'll follow that point up. And it appears that there's a vertical line around x equals negative 2 that the graph is approaching. And as y goes to negative, as y becomes negative very large, then it also appears that I might be following the graph down this way. So again, down, down, down, down. And following a point, seeing what happens, it does appear that this line x equals negative 2 is a vertical asymptote. So it appears that this graph has a couple of asymptotes. We have this horizontal asymptote, y equals 1. We have this vertical asymptote, x equals negative 2. We have a vertical asymptote, x equals positive 3. And as our x or y values get very large, positive or negative, as we get very far away to the right, very far away to the left, very far away above, and very far away below, then it appears that the points on the graph approach those lines. Well, the geometry of an asymptote, you look at it there it is, but to actually find them algebraically requires a little bit more effort. So to find the asymptotes for rational function, we'll go through the following steps. So one of the things we need to do is it's a rational function. It's a quotient of two polynomials, so we should do the division to end up with a proper rational function. We'll talk about that in a second. The horizontal asymptotes are going to correspond to the end behavior of the graph. The vertical asymptotes correspond to the behavior near the forbidden values. Well, wait a minute. This is a rational function, which means that there are certain values that we are not permitted to have. There are certain forbidden values, and we actually need to find those. And that really should be our first step. We should find the forbidden values first before we divide and then find the horizontal and vertical asymptotes. Okay, so let's take a look at an example. So we want to find the asymptotes of the graph y equals x minus 2 over x minus 5. And so the very, very, very, very, very first thing we should do any time we're confronted with the rational function is to find the forbidden values, those that are going to make the denominator equal to 0. And so our denominator x minus 5, so that tells me that x cannot be allowed to be 5. Now, the next thing we're going to do is we're going to divide. Now, since the quotient is x minus 5, it's x minus something we can use synthetic division. And you might wonder, well, why are we dividing? Don't we already have that division? And the answer is no, not really. What we don't have is what's called a proper rational function. And in this case, what that means is that the degree of the numerator has to be strictly less than the degree of the denominator. And this is the analog to proper fractions. A fraction like 3 7 is proper because the numerator 3 is less than the denominator 7. Here, we want to make sure we have a proper rational function. So we'll use synthetic division to perform the division. And I divide using the synthetic division algorithm. And remember, what that means is here's my quotient. Here's my remainder. So this rational expression is the same as this rational expression 1 plus remainder 3 over the original divisor. So as long as x is not equal to 5, my y graph, my y equals this, is the same thing. This is the same thing as 1 plus 3 over x minus 5. So as long as x is not equal to 5, I'm really looking at this graph. As x gets very large and positive, for x very large, very positive, this value is going to be close to 1. So that suggests that y equal to 1 is a horizontal asymptote. And I'll go ahead and write that down. As x goes to positive infinity, the graph approaches y equals 1. If x is positive, it's also worth performing the slight bit of additional analysis. If x is positive, 3 over x minus 5, that's positive over positive. This will be a small positive number. So not only does the graph approach y equals 1, it approaches y equals 1 from above. This is going to be slightly more than 1. As x gets large but negative, 3 over x minus 5 becomes a small negative number. And so that means that y equals 1 plus 3 over x minus 5, 1 plus a small negative number, that's going to be slightly less than 1. So as x goes to minus infinity, the graph approaches y equals 1 from below. Next we'll see what happens near our forbidden values, x equals 5. If x is slightly less than 5, then 3 over x minus 5, x is slightly less than 5. The denominator is a small negative number. So the quotient will become a large negative number. So as x approaches 5 but is slightly less, y becomes a large negative number. And so we might say that this x approaches 5 from below, slightly less than 5. y goes to negative infinity, negative large number. And if x is slightly more than 5, 3 over x minus 5, x minus 5 will be a small positive number. 3 over a small positive number is going to be a large positive number. So y equals 1 plus 3 over x minus 5, 1 plus a large positive number, this is going to become a large positive number. So as x approaches 5 from above, y goes to positive infinity. Now here's an important requirement. In order for a line to be an asymptote, the graph must approach it for either large values of x or large values of y. So as x goes to infinity, the graph approaches y equals 1. So y equals 1 is definitely a horizontal asymptote. This as x equals 5, as x gets close to 5, y gets close to plus or minus infinity. And so that says as y goes to infinity, x approaches 5. So that says that x equals 5 is going to be a vertical asymptote.