 So it's our goal to be able to graph rational functions without the use of technology whatsoever. And a lot of the techniques involved here are gonna be similar to those we use to graph polynomial functions. We need to know the in behavior. We need to know how does the function behave near an x-intercept. Though that part's the same, but for rational functions, we also have to deal with how does a function, how does a rational function behave near its vertical asymptotes? Which are kind of like the opposite of x-intercepts. The vertical asymptotes aren't where we cross the x-axis. The vertical asymptotes will be where we cross infinity, if you like to think of it that way. And so honestly, the way I like to think about it is through classic video games. A game that I really liked when I was a kid was the game of asteroid, which by all means, look it up and you'll probably find a free version of it somewhere online, or you can download it for your phone here. But asteroid was this great game which you had a very high tech spaceship. It looked like in a sausage triangle. In which case you had these asteroids that would appear on the screen, you would shoot them, and they'd break apart the asteroid. It would then break up into like maybe two smaller asteroids. You get points for doing this, for which then you turn and you shoot the asteroids. You blow that one up and it just keeps on happening until it disappears. And then new ones would appear on the screen. And this would just be a game that you just continue on until you die and see if you get a high score. Well, as these asteroids floating around, one of the best ways to protect your ship is to shoot the asteroid, but also as these asteroids are floating around, sometimes you have to dodge the asteroids for which your spaceship then would have a thruster on the back, right? You could turn the player with their joystick can turn where you're aiming. Then you could also hit a button to thrust it. This would cause your spaceship to fly around, right? So you could dodge asteroids and then shoot them at the same time. And so what happened though, this is the part I'm bringing up this game right here, is that when you flew off the edge of the screen, you would appear on the other side of the screen. So like if you flew off the top of the screen, you would just come back from the other side. This is a wraparound feature. And many classic video games that were two dimensional have this wraparound feature that like when you, on asteroids, you flew off the screen and wraparound the other side. The original Mario Brothers also had this feature. And so when one is thinking of graphing rational functions, I want you to kind of think of this same wraparound feature that if we have a vertical asymptote, if our function is approaching infinity, right? It might hit the edge of the screen that like when we graph this on our graphing calculator, but then it wraps around from the other side, right? We get this idea kind of like wrapped around the screen. And so hitting the edge of the screen is like kind of like going towards infinity, right? And so if we have a function for which it wraps around, we often draw these graphs like this. This is what I'm gonna mean by when I say you're going to touch infinity, excuse me, you're gonna cross infinity. So you crossed your screen and wrapped around the other side. This is an example of crossing infinity. Another thing that can happen with a rational function when you approach its vertical asymptote is that you can come and you can touch infinity, but then you come back, right? So this is an example we'd say that we touch infinity. We don't cross the screen, we just touched it and kind of bounced off infinity and came back. And so when a rational function approaches its vertical asymptote, one of two things happens. You either cross infinity, which is to suggest there's some type of sign change. You switch from positive to negative or negative to positive, or you touch infinity. In which case, when you touch infinity, you have no sign change whatsoever. You either were positive and you stayed positive or you were negative and you stayed negative. This is very resembling to what we saw when we talked about a polynomial function approaching its x-intercepts. Our options were we either cross the x-axis or we touch the x-axis. We had a cross or we had a touch. And in both situations, if you cross the x-axis, you went from negative to positive. There was a sign change going on there. If you just touch the x-axis, maybe you went from positive to positive, there's no sign change. And this had to do with the fact that when you had an even multiplicity, you touched the x-axis. When you had odd multiplicity, you crossed the x-axis. This same analog is gonna be true when you cross or touch infinity. If you cross infinity, that is because you have an odd multiplicity. There's gonna be a sign change. Odd powers, when you stick in a negative, will pop out a negative again. Because of the odd's powers and even powers behave differently with negative numbers, when you cross infinity, there's a sign change. And this will represent you have an odd multiplicity to your vertical asymptote. And when you touch infinity, that's because you have an even multiplicity of your vertical asymptote. Well, what do I mean by multiplicity of a vertical asymptote? Well, the vertical asymptotes are gonna come from like these x-minus c's. There's gonna be some linear factor in the denominator of your rational function. It has some power. We call that multiplicity, the multiplicity of the rational function. So we played a previous game about who's my graph? For which we had a polynomial function which we wanted to pair it up with a graph. That's kind of like maybe like the ABC's channels, the bachelor, right? Well, to keep things well balanced, the bachelor has the bachelorette. So we gotta play the inverse game here. So instead of saying who's my graph, we wanna play the game. Who's my function? Who's my formula? This time, we actually have our bachelorette in front of us, right? Actually, I think in the original video, I said the graphs were the men, right? So now we're playing the bachelor this time, right? So now we have a graph and we wanna come up with the formula of who's compatible with this graph that we see in front of us here for our game show. So some things we can look into is we can see x-intercepts, for example. So we have an x-intercept right here of negative two and we have an x-intercept right here of positive one, two, three, four, five. So we have an x-intercept of five right here. So what this tells me about my rational function, so we're building a function r of x right here. So rational function is the numerator and the denominator. I'm going to get factors in the numerator based upon my x-intercepts. Since x-2 is a root, is an x-intercept, I have to have the factor x plus two in the numerator. And since x equals five is an x-intercept here, I have to have the factor of x-5 in the numerator. The numerator is what's gonna give you the x-intercepts of your rational function. But notice what happens. There's a cross going on right here. We crossed infinity from negative to positive. But at negative two, we actually went from negative to negative. So this was only a touch at infinity. And so that tells me that this is gonna have, that we're gonna have an even multiplicity of negative two and we're gonna have odd multiplicity at five. So to keep things simple, I'm gonna take x plus two squared and x minus five to the first, for which I'm not gonna write the one there at all. Okay, what about my vertical asymptotes? You'll notice that my vertical asymptotes, I have a vertical asymptote at x equals negative five and I have a vertical asymptote at x equals two. Much like the x-intercepts, I can only have these vertical asymptotes if in fact I have some factors in the denominator because the vertical asymptotes will come about from dividing by zero. So I'm gonna have to have an x minus, x plus five in the denominator from this guy right here and I'm gonna have to have an x minus two in the denominator as well. But look at the behavior as we approach infinity here. On the left-hand side, you're going up towards infinity and on the right-hand side, you're going down towards negative infinity here, right? So the labels we used before is as x approaches negative five from the left, y approaches infinity and as the other direction here is as x approaches negative five from the right, y will approach negative infinity. You see the switching of signs. So we go from a positive infinity up here to a negative infinity down here. That sign change indicates it's odd but this will move it by crossing infinity. My little spaceship here, it's thrusting towards infinity, right? It wraps around to the other side. We crossed infinity and therefore this is indicating us, we have odd multiplicity right here. On the other hand, as we approach positive two from the left, it's negative infinity. As we approach positive two from the right, it's negative infinity still. It's negative and negative. There's no sign change. This is what we mean by we're touching infinity, right? We are crossing infinity with the odd case. We're touching infinity, which means we have an even case going on right here. So the x minus two needs to be even. So I'm gonna just to make it simple, I'm gonna take x minus two squared and then the x plus five needs to have an odd power. So we'll just make it one to be the simplest case right here. And so that actually does pretty good, right? The numerator takes care of the x intercepts, the denominator takes care of the vertical asymptotes. But we're not quite there yet. The last thing to consider is actually going to be this horizontal asymptote. Notice we have a horizontal asymptote and y equals two. Okay? How do we get a horizontal asymptote again? Well, it has to do with whether the function's balanced, top heavy or bottom heavy. Top heavy means the denominator has a bigger power. Bottom heavy means the denominator, excuse me, top heavy is the numerator we had a bigger power. The bottom heavy meant the denominator had a bigger power. The balance was the same. So at the moment the top has degree three and the bottom has degree three. So this is a balanced rational function. And so it will have a horizontal asymptote and the horizontal asymptote will be the ratio of the coefficients on top and bottom. So you have two over one. So I need to slap in a coefficient of two right here. And that would give me, that would then give me a rational function whose horizontal asymptote is two. And then it's other behavior we need the X intercepts and horizontal asymptotes, vertical asymptotes would be all the same. And so then, so I've been able to come up with a formula to describe, to describe your rational function right here. Now, one other thing I do wanna mention is that what if our graph, what if we wanted to like a remove point? Like what if we stole the point, this point right here? What is that gonna be? One, two, three, four, five, six, seven, eight, nine. What if we steal the point X equals nine? How do you deal with that? Turns out that's a very easy thing to deal with with our rational function right here. Give me a little bit more space. If you need to get, if you need to remove point right here, what you're just gonna do is you're just gonna slap on like an X minus nine on top and then an X minus nine on the bottom. And so notice that when you reduce the fraction, X minus nine would cancel out, but in the un-reduced form, you do have an X minus nine in the denominator, which would mean that nine is outside the domain of this function. And that's exactly where you get a remove point. So if you have a remove point, just put the same thing on top and bottom and so that it cancels out like we have in this situation here.