Hey, everyone, whenever we worked with polynomial functions and we graphed them, we had to consider the end behavior of our graph. And this gave us great information so that we could easily graph our polynomial function. Now, with rational functions, we're going to have to consider the behavior of our graph around things called asymptotes. Now, asymptotes are lines that our curve or our graph is going to approach and get really, really close do but not quite touch. Now, just that word Asymptote might make the graphs of rational functions sound like they're going to be complicated. But here we're gonna graph a rational function together and I'm gonna walk you through identifying asymptotes and the basics that you need to know about them. So let's go ahead and get started. Now, the first thing that I want to do here is just go ahead and graph a rational function. Now, this function F of X equals one over X is a really basic and common rational function that you're gonna see. So let's go ahead and start with graphing the number one and plot those points on our graph. So if I plug in one to my function that just gives me one divided by one, which we know is just one. So plotting that point on my graph, I have the 0.11 then if I take two, this is just 1/2. So I can also plot that on my graph at 21 half, then if I go to three, this is just 1/3. So we're getting even smaller here on our graph. And I, if I take it all the way up to 10, that's going to give me 1/10 or 100 it's gonna take me all the way up to 100. So we see that these numbers are getting progressively smaller. 1/100 is a much smaller number than one. So what we're going to see happen on our graph is that we're gonna get really, really close to the X axis but not quite touch it. Now, we're going to see the same thing happen on the negatives. But let's go ahead and verify that I know that if I plug negative one and I'm going to get negative one, so I can go ahead and plot that point on my graph at negative one, negative one, then if I go to negative 10, I'm gonna get negative 110. So we know that it's getting smaller in that negative direction. So we're gonna see a similar thing happen over here. Now it's getting really, really close to that X axis, but not quite touching it, which is exactly what we said an Asymptote is. So what I have here is an Asymptote at my X axis, which I represent using a dotted or dashed line. Now, I'm gonna go ahead and highlight it just so that it's easier to see here. But here I have a horizontal Asymptote because it's just a horizontal line at Y equals zero. Now, whenever we look at this Asymptote, we say we see that our graph is getting really close to it but not quite touching it. And whenever we worked with the end behavior of polynomials, we describe the behavior using arow notation. We said that as X approached infinity, F of X approached something for this particular graph, it's approaching infinity. And as X approaches a negative infinity, here F of X is also approaching infinity. Now we want to do the same thing with our rational function. So here as X approaches infinity, which is on this side of my graph, I see that F of X is instead approaching my Asymptote at zero. So F of X is approaching zero. Then on the other side, as X approaches negative infinity, I know that the same thing is happening. I'm still approaching that Asymptote at zero. So we can describe our end behavior still with error notation. But now it's getting close to that Asymptote rather than going to negative or positive infinity. So let's go ahead and graph the rest of our function because we still don't know what's happening in the middle here. Now, if I take a look at some smaller numbers in between zero and one, like say 0.1 that will give me one over 0.1 which is just 10, then if I get even smaller, even closer to zero and I take one over 0.01 that gets even bigger at 100. So what we're going to see happen here is that our graph is going to get really high up and get really close to my Y axis but not quite touch it. And the same exact thing is going to happen over here. So I end up having two different curves on the graph of this rational function. Now, whenever we consider that it's getting really close to the Y axis that tells us that we're dealing with another Asymptote. So I have an Asymptote here on my Y axis, which I will, I will again represent with a dashed line. I'm going to go ahead and highlight it again so that we can see a little bit better. And this is a vertical Asymptote at the line X equals zero. Now, we see that our graph is approaching this Asymptote from either side. And whenever we worked with polynomial functions, if we were to consider X approaching zero and wanting to describe the behavior of our graph, we would just get a single point in this case zero. Now, on our rational function, we see that our graph is approaching that line from their side. So that means we need to be able to describe both of those behaviors. So we're going to split the behavior into two different sides as X approaches zero from the right side, which we denote as X approaching zero with this little positive. That just means from the right. And then we consider X approaching zero from the left side, which we denote with this little negative sign. So this just means from the left. So in order to describe this behavior, if I look at it approaching zero from the right side, I see that my function is getting really, really high, it's getting to be a really large number. So it's approaching infinity. Then as X approaches zero from the left side, I see this going the opposite direction. So F of X is instead approaching negative infinity. Now, this is only we're only looking at zero here because that's where our Asymptote is and our asymptotes are not always going to be at zero. They just happen to be for this function. So this could really be any number I could consider X approaching two or five from either side or any other number. It just depends on your function and where your Asymptote is. And there can actually be a lot of different possibilities with asymptotes because a rational function may have no asymptotes, we may have just one Asymptote like we c in this function A here or we can even have multiple AYP totes. But they're always going to be represented with these dashed lines. So if we look at function B here, I have a vertical and a horizontal asym tote kind of shifted from what we saw in our function up there. And then I could also even have multiple vertical asymptotes. It really just depends on your function. Now that we know the basics of asymptotes. Let's go ahead and get to calculating them. I'll see you in the next video.

2

Problem

Problem

Sketch the graph of the function $f\left(x\right)=\frac{1}{x^2}$. Identify the asymptotes on the graph.

Hey, everyone now that we know what asymptotes are, we need to be able to determine where they are on our graph when given a function. So whenever I'm given a rational function, I need to be able to determine if there even is a vertical Asymptote. And where exactly that vertical Asymptote is. Now, luckily finding vertical asymptotes is super simple and almost identical to finding the domain of our function. When we set our denominator equal to zero and solved for X, we're gonna do the same thing to find vertical asymptotes set our denominator equal to zero and solve for X. But before we do that, we're just gonna write our function in lowest terms. So we're gonna do two things that we already know how to do in order to find our vertical Asymptote. So let's go ahead and walk through this. So like I said, to find our vertical asymptotes, we want to go ahead and put our function in lowest terms. So looking at the function that I have here, I have X plus two over X plus two times X minus three. So to put this in lowest terms, I'm gonna go ahead and cancel my common factors, which here is X plus two. So this leaves me with the function one over X minus three. And now my function is in lowest terms. Now, I'm simply going to set my denominator equal to zero and solve for X to get my vertical Asymptote. So if I take X minus three and I set it equal to zero, I'll end up with X equals three. And that's my vertical Asymptote. And I'm done, that's all we need to do. In order to find those vertical asymptotes, let's go ahead and look at a couple more examples and find some vertical asymptotes of different functions. So looking at my first function here, I have F of X is equal to 2/2 X plus six. So remember the first thing we wanna do is write our function in lowest terms. So let's go ahead and do that here and factor. So my didn't, my numerator cannot be factored. It's just a constant, it's just two. But in my denominator, I can go ahead and pull out a greatest common factor of two and that leaves me with X plus three in my denominator there. So writing this in lowest terms, I'm gonna go ahead and cancel these twos leaving me with one over X plus three. Now I can go ahead and set my denominator equal to zero. So setting this denominator equal to zero, isolating X, I'm just gonna subtract three on both sides leaving me with X is equal to negative three. So that's my vertical Asymptote of this function. And I'm done, let's look at one more example here, here I have F of X is equal to one, one over X squared minus nine. Now, looking at this function, we want to go ahead and write it in lowest terms. But because my numerator is just one that's not going to cancel with anything in my denominator. It's already in lowest terms. So I can just go ahead and take my denominator and set it equal to zero. Now, solving for X here, I want to add nine to both sides canceling that nine out leaving me with X squared is equal to nine. And then I can go ahead and square root both sides to isolate X leaving me with X is equal to plus or minus the square root of nine, which we know is just three. So this tells me that I actually have two vertical asymptotes, one at X equals positive three and one at X equals negative three. It's totally OK to have more than one vertical Asymptote that's gonna happen sometimes and it's perfectly fine. Now that we know how to find vertical asymptotes. Let's go ahead and get some more practice.

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Problem

Problem

Based only on the vertical asymptotes, which of the following graphs could be the graph of the given function? $f\left(x\right)=\frac{x^2-4x}{x^2-x-12}$

A

B

C

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concept

Determining Removable Discontinuities (Holes)

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3m

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Hey, everyone, whenever we found vertical asymptotes, we would factor our function and then cancel out any common factors in order to get our function in lowest terms before setting our denominator equal to zero and solving for X. But having that common factor actually causes something to happen in the graph of my function called a hole. And we need to be able to identify where these holes are on our graph, which we can do by simply taking our common factor. And instead of canceling it out, just setting it equal to zero and solving for X. So here I'm gonna walk you through how to find any holes in your graph starting from exactly where we do with vertical asymptotes and just taking a different path from there. So let's go ahead and get started now with vertical asymptotes, like I said, we would cancel out our common factor and then we would our denominator equal to zero and solve for X. So for this function here, I end up with a vertical Asymptote at X equals three which we can see right here on our graph, a vertical Asymptote at X equals three. Now, whenever we find the holes in our graph, we're going to start at the same place. So here I have this factored function. But instead of canceling out that common factor to put it in lowest terms, I'm going to take that common factor and set it equal to zero. So here my common factor is X plus two and setting that equal to zero, I then want to solve for X. So solving for X, here I end up with X equals negative two. And that tells me that I have a hole in the graph of my function, right at X equals negative two, which I'm going to represent with an open circle on my graph. Literally just draw a, on the curve at that point and that's where our hole is. So looking at this graph, now I have both a vertical Asymptote and a hole for the same function. Now you might also hear holes referred to as removable discontinuities. And that's just the more technical term for it, but it's referring to the same thing. So now that we know how to find the holes in the graph of rational function. Let's go ahead and look at some more examples here. So let's look at this first function of F of X is equal to X plus three over X squared plus four X plus three. So here we want to go ahead and factor this the same way we would if, if we're trying to write it in lowest terms. So my numerator can't be factored here. It just stays as X plus three. But my denominator using the AC method could be factored to X plus three times X plus one. Now, I wanna take my common factor and I don't want to cancel it. I want to take it and I wanna set it equal to zero. So here, my common factor is X plus three setting that equal to zero. I'm gonna go ahead and solve for X which I can do by subtracting three on both sides. So I end up with a hole at X equals negative three. And I'm done, that's where the hole on my graph will be. Let's look at one more example here on this side, I have F of X is equal to X squared plus one divided by X minus one. So let's go ahead and factor this. So my numerator can be factored into X plus one times X minus one. And I'm going to divide that still by X minus one because that denominator can't be factored anymore. Now again, I want to take my common factor and I don't want to cancel it. I'm gonna take it and set it equal to zero. So here X minus one equals zero. And then we want to solve for X. So adding one to both sides leaves me with X is equal to one, which is where the hole in my graph is at X equals one. And we're done. So now that we know how to find the holes in our graph, let's get a little bit more practice.

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Problem

Problem

Find all vertical asymptotes and holes of each function.

Hey, everyone, when working with vertical asymptotes, we found that they affect the domain of our rational function. And we could even calculate them in a way that's a really similar to finding the domain. Now with horizontal asymptotes, they're instead going to affect the range of our rational function because now we're dealing with horizontal lines that go across our Y axis now because we don't have a way of finding the range of our function. You may be worried that this is going to get complicated, but don't worry because it just depends on only two things that we can determine by just looking at our function. And that's the degree of both our numerator, the top of our rational function and the denominator, the bottom of our rational function. So with that in mind, let's go ahead and just jump right into finding some horizontal asymptotes. So looking at this first function here, I have F of X is equal to one over X. And we want to be able to determine what our horizontal Asymptote is. Now we're going to look like I said, the degree of the numerator and the degree of the denominator. So here, the degree of my numerator is zero because there's not even a variable there. And the degree of my denominator is one so zero I know is less than one. And whenever the degree of your numerator is less than the degree of your denominator, you are always simply going to have a horizontal ascent toe at the line Y equals zero, which is what we can see on our graph is already happening here. Our line horizontal Asymptote is at Y equals zero. And that's it. So when your degree of your numerator is less than the degree of your denominator, that's all. Now, let's look at one other possibility here. So looking at my function here, I have F of X is equal to two X plus three divided by X. So looking at the degree of my numerator and denominator, the degree of my numerator is one and the degree of my denominator is also one. I know that one is definitely equal to one. So here the of my numerator is equal to the degree of my denominator. Now, whenever that happens, I actually need to look at one more thing on my function and that is my leading coefficients. So I'm going to take the leading coefficient of my numerator and then divide it by the leading coefficient of my denominator. And that will give me my horizontal Asymptote. So here, the leading coefficient of my numerator is two and the leading coefficient of my denominator is this invisible one here. So horizontal Asymptote is two divided by one, which is just the line Y equals two, which we can see is exactly where that horizontal Asymptote is on our graph. So now that we know how to find them looking at the degrees of our numerator and denominator, let's go ahead and find some horizontal asymptotes of a couple more functions. So looking at this first example, here, I have four X squared divided by negative X cubed minus five X plus nine. So looking at the degree of our numerator, I have a two and then the degree of my denominator I have is three. So two is two less than or equal to three. Well, it's definitely less than. So looking back at our original examples, we know that if our numerator is less than the denominator Y is going to be equal to zero and that will give me my horizontal Asymptote. So here I simply have a horizontal Asymptote at Y equals zero. And I'm done, let's look at one more example. So here I have F of X is equal to two X divided by three X squared plus X minus one. So here looking at those degrees, I have a degree of two in my numerator and I also have a degree of two in my denominator. So the degree of my numerator is definitely equal to the degree of my denominator. Now, remember whenever that happens we need to take a look at one more thing here, our leading coefficients in order to find our horizontal Asymptote. So the leading coefficient of my numerator is two and the leading coefficient of my denominator is three. So taking those and dividing them, I end up with a horizontal Asymptote at Y equals two thirds. And that's where my horizontal Asymptote will be on my graph. And that's all. So we know how to calculate horizontal asymptotes. But I want to mention one more thing that you might see pop up in your studies. So sometimes whenever we have a horizontal Asymptote, the graph of your rational function may actually intersect it and just fully cross it and then approach it from the other side. Now, I know that this goes against what we said, our definition of asymptotes are, but it's actually just crossing it and then approaching it from the other side. Now, you don't need to worry about why this happens or figuring out when it will, you just need to be aware that it can happen. So don't be afraid when you see a graph and it crosses the horizontal Asymptote, it's totally fine, it can happen. It won't always but just be aware that it can. So we now know how to calculate horizontal asymptotes and how they may appear on our graph. Let's get some practice