 This video is going to talk about graphing rational functions. We have v of x equal to p of x divided by d of x. That gives us a rational function. And it's a rational function as long as d of x is not equal to zero. So the first thing we talk about when we're talking about graphs is the vertical asymptote. And in a vertical asymptote, it's just an imaginary line. We're going to draw it in in a second. That goes through the zeros of the denominator. And then we want to state what they are for this particular graph. Now if you look at your worksheet, you have a couple others. The first one you should find out that you have one over at that zero would be zero. So it should be the y-axis. And the other one you have one over x squared. Again, it's going to be zero. But you'll notice that one graphs on either side of the y-axis and one graphs in the first and third quadrants. Well, what makes it that happen? Well, let's look and see if we can figure that out. x minus two, it would be a zero of two. x minus two equal to zero, so we solve it x equal to two. So we have that vertical asymptote right here at x equal to two. And it's an imaginary line, so we draw it as a dotted line. And then we have x plus one. And if we set that equal to zero, then x is going to be equal to negative one. And you can see that we have that vertical asymptote right here. The graph gets real close to it, but it doesn't quite cross it. So if you notice at this x plus one zero, which is negative one, on this side of the asymptote, we are below the x-axis. And on this side of the asymptote, we've gone above the x-axis. And when we go to this vertical asymptote, in fact, I'm going to change colors on it. And this would be our x minus two quantity squared. You can see on this side of the asymptote, I'm above. And on this side of the asymptote, I stayed above. So we can say that even degree zero's don't change sign. And then we could say that the odd degree zero does change sign. So it goes across to x-axis. Now we want to talk about horizontal opposites and topes. And those we're going to be comparing the leading terms of each of our polynomials, of the numerator and the leading term of the denominator. Always follows this pattern. If the numerator degree is less than the denominator degree, then we know it's going to be a y equals zero asymptote. Always is. If n is equal to m, both those degrees are the same, then it will be y is equal to coefficients a over b. And if n is greater than m, there's not a horizontal asymptote. So you do need to know, though, that a graph can cross a horizontal asymptote. It will never cross a vertical asymptote, but it can cross a horizontal asymptote. We can just set the function equal to the asymptote and then solve. If you get a true statement, it crosses and it will tell you where. So let's look at this problem. Determine the asymptotes and then determine the graph across the horizontal asymptote or not. So we have 4x and we have x squared. This is m and this is m. So we would say that n is less than m, which is our first case up there. So we would say that the horizontal asymptote is y equal zero. Now the question is, does it cross it or not? Well, we're going to take our 4x plus 8, the whole thing, divided by the x squared plus 1, and we're going to set it equal to zero and solve. Well, I want to clear this fraction. So to clear the fraction, I'm going to multiply both sides by x squared plus 1. And when I multiply over here, they cancel out. That's why you multiply it by it. It's like I multiply by x squared over 1. Zero times anything is just zero. So now I'm left with 4x plus 8 equal to zero. And I'm running out of space, so let's move up here. So 4x would be equal to negative 8 and x is going to be equal to negative 2. And I came up with a statement that I can solve. x is equal to negative 2. So it will cross at x equal negative 2. It will cross at y equals zero, which is your x axis at x equal negative 2. And we could verify that if we wanted to, by just looking at a calculator. So let's pull over the calculator and see what we get. 4x plus 8 over x squared plus 1. So y equal, parentheses 4x plus 8. And I have to put it in parentheses because I want that whole numerator divided by, and I have to use parentheses again, the whole denominator. And I believe a standard window, which it works, so zoom 6. And if we see right here, it looks like negative 2 is where it crossed. And we can verify that by looking at our table. If we have nice numbers, let's go in here. Look at negative 2 and go over 1. We can verify that sure enough, at negative 2 it's at zero. It's on that x axis.