Vertical Asymptotes How mathbff
Hi guys! I'm Nancy. and I'm going to show you how to find the vertical asymptotes of a rational function. If you don't know what I mean by rational function just ignore that, you're probably in the right place. Actually all this stuff is a lot easier than it sounds. So don't worry. It's gonna be fine. Let me show you.
OK, so say you need to find the vertical asymptote or vertical asymptotes, if there's more than one. First of all. what is a vertical asymptoteé It's basically an invisible vertical line that your graph approaches and gets really really close to but never actually touches. Never quite reaches. So how do you find the vertical asymptoteé Well, if you have a rational function, the fraction form type,
which is the most common kind for this, you can use the same 3 steps always to find the vertical asymptote. OK. So here are the 3 steps you can always use to find the vertical asymptotes of a rational function. The first is to factor the top and bottom. If you can and all that you can. I'll show you what I mean.
The second is to cancel any common factors from the top and bottom to simplify. And the last step is to take the bottom, the denominator and set it equal to zero to find the vertical asymptotes. So let's try it for this function. The first step is to factor the numerator and denominator. if you can. And all that you can.
So you can't factor the top. in any way, but the bottom you can factor. It's a quadratic. And if you want more help on how to factor quadratics I have a tutorial for that. So you can check that out. But basically the idea is you want to break this into. 2 separate factors that start with x.
So x plus a number times x plus another number. OK, and what you want are 2 numbers here. 2 numbers here that. multiply to 6 and add to 5. And those 2 numbers would be 2 and 3. Since 2 times 3 is 6 and 2 plus 3 is 5. So in here we have 2 and 3. And that's all that you can factor for this rational function.
Finding vertical asymptotes and holes of rational equations
HELLO, Mr. Tarrou. We are going to have aseries of tutorials and I don't think I am going to finish them all today. We are going tobe finding vertical asymptotes, horizontal asymptotes, and slant asymptotes. Becauseultimately this section in PreCalculus is going to be coming up with, or graphing, rationalequations. And rational equations have a lot going on with their graphs. You are not goingto just plot a a couple of points and see what those look like. So the first thing youare going to do is learn as much as you can about the graph before you start trying toconsider doing a t table of points. All right, the first thing and the easiest thing to findis if there are any vertical asymptotes. Now
that happens when you have an equation thathas an x in the denominator, at least that usually happens when you have an x in thedenominator. This is because you cannot divide by zero. So any situation where x is in thedenominator and you are able to make that denominator actually equal zero, somethingis going to happen there. It will usually be a vertical asymptote. Let's take a lookat a couple of examples. This is the most. This is like a parent function. Rational functionscannot get any simpler that just simply y equals one over x. So lets make a table ofvalues here. We already know that we cannot divide by zero. So the domain is all realnumbers except zero. But graphically what
does this look like. Well, you could and weare going to make a table of values. We have x values of negative two, negative one, zero,one, and two. And you can come up with others points as well. You probably will need morethan these. Let's plug those in. If you plug in negative two you get a y value of negativeone half. If you plug in negative one, one divided by negative one is negative one. Iincluded zero but we already know that is undefined. So that just simply does not work.And we have the values of one, one divided by one is one, and one divided by two is onehalf. Let's see if that is enough to get an idea of what is going on here. Well I cannothave x equal to zero. I cannot divide by zero
so this graph will have a vertical asymptoteat x equals zero. But what happens after thaté Well at negative two we are at negative onehalf. And at negative one we are at negative one. At positive one we are at one and atpositive two we are at positive one half. So (1,1) and (2,12). Is that enough to letyou see what is going one. Probably not. Because you may never have seen vertical asymptotesin your graphs before. So we might need a couple of more points, so let's add a fewmore. What happens as you get closer to zero from the left and the righté Like what ifhappens if x is equal to negative one halfé So as we creep a little bit closer to thevertical asymptote. Well, that is going to
be y is equal to one over x is going to beone over negative one half. And as I have explained before, like how this top dividedby the bottom. You cannot really divide fractions, that is going to flip up the bottom. One dividedby negative one half is the same as one times negative two which is negative two. Ok, soas I get closer where x is negative one half my y is negative two. If I connect these pointsyou can see that there is a little bit of curvature happening. That graph is going to,if you can put in negative one fourth.negative one tenth, you are going to see your y valuesbecome more and more negative and actually go down to negative infinity. We will be talkingabout horizontal asymptotes but not in this
tutorial. I know that this graph is going toflatten out and approach the x axis from the bottom, but we have not talked about horizontalasymptotes yet. For now I am going to just say take my word for it. What happens if xis positive one halfé Well again. Y is equal to one over x. That is now going to be oneover one half. When you have a fraction over another fraction you flip that bottom fractionup. You really can't divide fractions. Like one divided by one half means that you needto change that division to multiplication and flip that second number. Remember fractionsfrom many many years ago. You flip the bottom up and you get two. At one half we are attwo. Again you can see this curvature happening