 This video is going to talk about rational functions. So rational function is this, v of x is equal to p of x over d of x, where p of x and d of x are polynomials, and we just have to remember that d of x can't be equal to zero, because remember we can't divide by zero or we have something undefined. So the domain of v of x then is going to be all reals except for those values that make d of x zero or the zeros of d of x. So we want to look at a graph and see how this applies and what behaviors happen in these graphs. So the graph of, this is just a graph of f of x equal 1 over x, most general rational function you can find. And we want to see what their in behaviors are. So we have, we're going to look at when x is heading toward negative infinity, so that's going all the way to the left here on the x axis and continuing on forever. And it looks like this graph is getting real close to that x axis, but it's not quite touching here and in fact it never will touch. So we would say that y approaches zero, it just never quite skits there. And then as we go to the other direction and all the way to the right and to infinity on the x axis, we see that again y is tending towards zero, but not crossing it. So let's see what happens when we have x approaching the vertical asymptote, but we looked at before it was the horizontal asymptote. Now we're looking at the vertical asymptote, which in this case happens to be, we'll find out the y axis. Now we're looking at x and it's approaching that vertical asymptote of the y axis from the left. That's what x approaches zero on the negative side. So on the negative side of zero, on the left hand side of that asymptote, we can see that as we get closer and closer to zero, this graph is going lower and lower or heading toward negative infinity. And when we come to the other side, approach it from the right or the positive side, then as we get closer and closer to zero, this graph is going to increase, which would mean that we're going toward positive infinity. So those are the behaviors that these graphs have. So let's take a look at a couple of examples. This is the graph of g of x. And if you look at this, this looks very much like a transformation. And in fact, it is. This would be our k. And remember, k moves you up and down. And this would be our h over here. And remember, that moves you left and right. And this minus two means that we're actually going to go down two. And when it says x minus one, remember that means you go the opposite direction. So we're actually going to go to the right one. So think of the graph that we just saw. It had a vertical asymptote of the y-axis. And now if you look, it looks like we've gone over one. And this is now our vertical asymptote. So there's our h. And then the horizontal asymptote was the x-axis in the original one over x, the parent function. And now you can see that it has gone down two. And in fact, that is exactly what it's done from the x-axis. It has gone down one and then two. And because of the program I used to make this, it looks like it's crossing that negative two, but it really isn't. It's just tending toward it. So we want to use mathematical notation to describe the n behavior, just like we did before. So we're saying, okay, as x goes toward negative infinity, what happens to y? Well, as x goes to negative infinity, y is getting close to this negative two. So y approaches negative two. And as x approaches positive infinity in the other direction, we see over here that it's also heading toward this negative two down here. So x approaches positive infinity, then y approaches the negative two. So that's the n's in behavior. But how it happens here in the middle at our vertical asymptote. Well, then we can see that our vertical asymptote here is actually this one. So x is going to approach one on the positive side of one and on the negative side of one. So I'll just put both of these in here. I'm going to do the positive one first, because that's what I wrote first. So as x comes here and approaches negative one on the positive side. So on the right hand side of the vertical asymptote, we can see this thing is going to positive infinity. It's going up forever. So y is approaching positive infinity. And as it approaches from the negative side, this time instead of being negative infinity, it's again going to positive infinity. So if we want to state the horizontal asymptote then, that would be our horizontal line. So it's a y as a negative two. So now let's see if we can take all this that we know. We want to sketch the transformation. So let's look at these transformations. We've got a negative two, which means it's going to go down two. And we've got just a plain old x, so it's going to stay. It's not going to shift left or right. And then we've also got this negative one. And remember that means that we're going to change the y values. So if I graph my little graph here, I'm going to put in black here, the one over x. And that was the first and third quadrants. So it looks something like this. Now we need to reflect it. Remember the last thing you always do is the up and down. So we need to reflect that first. So reflecting it is going to make it go, all our y values are going to change. So we're going to be now in the second and fourth quadrants. And then finally, and I'll do that in red since that's the final thing we're going to do, we have to move everything down too. So instead of tending toward the x-axis, we can come down here and call this negative two. And that's going to be our horizontal asymptote. So that's going to be like our x-axis now. So we were above that in the, oops, a little bit over too far, in the third quadrant. And below that in the fourth quadrant. And that's a horrible sketch. My pen is not helping me very much, but that would be our graph. It would tend toward that y-axis just not quite cross it. So let's answer these rest of these questions then. Label the horizontal and vertical asymptotes, okay? Well, we said that this one was going to be y equal negative two. And the vertical one is in our black. And the vertical asymptote is going to be x equal zero. That's the y-axis. And then we want to know the x and y intercepts. Well, we have an x-intercept right here. And since our y-axis is our asymptote, the y-intercept, we can say it does not exist. But we do have an x-intercept. But remember, if you're trying to find an intercept, we would say that zero is our y. And that's equal to negative one over x minus two. And if we want to clear our fraction, we multiply everything by x. So this zero times x would be zero. And negative one times x will cancel the x on the bottom. So we just have negative one. And then we have negative two x. And if we bring the negative two x to the other side, we have two x is equal to negative one. And then solving for x, we would say that x is equal to negative one half. And if we look at our graph, that's sure enough, that looks like negative one half right there. So let's look at an application in this case then. A large city has initiated a new recycling effort and want to distribute recycling bins for use in separating the materials. And they want to approximate the costs by this function, negative 22,000 over P minus 100 minus 220, where C of P is the cost in thousands of dollars and distribute the bin to P percent of the population. So we want to find the cost to distribute bins to 25% of the population. Well, that means P is going to be 25. And in this case, it's just going to be a plug-and-check. So we have negative 22,000 over 25 minus 100. And then that fraction minus 220. And I'm just going to pull up my calculator and go to my home screen. And in here, I'm going to put negative 22,000 divided by, and then in parentheses, 25 minus 100 minus the 220. And we find out that the cost is going to be about $73,000. And we want to do the same situation, but now we want to know what happens if the city tries to give the recycling bins to 100% of the population. And it says, hint, take a look. And I actually took the time to make some screenshots of those. So here's our graph. And we can see our asymptotes here are at the x-axis. And then this one, we would have to figure out what that was. But we can figure out what that is by looking at our table here. Because you see the error here, that's going to be my vertical asymptote. At 100, we have an error. So what happens is we try to get to 100%. We can get to 99%, but we just can't quite get to 100%. Never reaches or never can reach 100%. And again, that's because of this right here.