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Studying for a test? Prepare with these 10 lessons on Rational relationships.
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Let's do a couple more examples graphing rational functions. So let's say I have y is equal to 2x over x plus 1. So the first thing we might want to do is identify our horizontal asymptotes, if there are any. And I said before, all you have to do is look at the highest degree term in the numerator and the denominator. The highest degree term here, there's only one term. It is 2x. And the highest degree term here is x. They're both first-degree terms. So you can say that as x approaches infinity, y is going to be-- as x gets super-large values, these two terms are going to dominate. This isn't going to matter so much. So then our expression, then y is going to be approximately equal to 2x over x, which is just equal to 2. That actually would also be true as x approaches negative infinity. So as x gets really large or super-negative, this is going to approach 2. This term won't matter much. So let's graph that horizontal asymptote. So it's y is equal to 2. Let's graph it. So this is our horizontal asymptote right there. y is equal to 2, that right there-- let me write it down-- horizontal asymptote. That is what our graph approaches but never quite touches as we get to more and more positive values of x or more and more negative values of x. Now, do we have any vertical asymptotes here? Well, sure. We have when x is equal to negative 1, this equation or this function is undefined. So we say y undefined when x is equal to negative 1. That's definitely true because when x is equal to negative 1, the denominator becomes zero. We don't know what 1/0 is. It's not defined. And this is a vertical asymptote because the x doesn't cancel out. The x plus 1-- sorry-- doesn't cancel out with something else. Let me give you a quick example right here. Let's say I have the equation y is equal to x plus 1 over x plus 1. In this circumstance, you might say, hey, when x is equal to negative 1, my graph is undefined. And you would be right because if you put a negative 1 here, you get a 0 down here. In fact, you'll also get a 0 on top. You'll get a 0 over a 0. It's undefined. But as you can see, if you assume that x does not equal negative 1, if you assume that this term and that term are not equal to zero, you can divide the numerator and the denominator by x plus 1, or you could say, well, that over that, if it was anything else over itself, it would be equal to 1. You would say this would be equal to 1 when x does not equal negative 1 or when these terms don't equal zero. It equals 0/0, which we don't know what that is, when x is equal to negative 1. So in this situation, you would not have a vertical asymptote. So this graph right here, no vertical asymptote. And actually, you're probably curious, what does this graph look like? I'll take a little aside here to draw it for you. This graph right here, if I had to graph this right there, what this would be is this would be y is equal to 1 for all the values except for x is equal to negative 1. So in this situation the graph, it would be y is equal to 1 everywhere, except for y is equal to negative 1. And y is equal to negative 1, it's undefined. So we actually have a hole there. We actually draw a little circle around there, a little hollowed-out circle, so that we don't know what y is when x is equal to negative 1. So this looks like that right there. It looks like that horizontal line. No vertical asymptote. And that's because this term and that term cancel out when they're not equal to zero, when x is not equal to negative 1. So when your identifying vertical asymptotes-- let me clear this out a little bit. when you're identifying vertical asymptotes, you want to be sure that this expression right here isn't canceling out with something in the numerator. And in this case, it's not. In this case, it did, so you don't have a vertical asymptote. In this case, you aren't canceling out, so this will define a vertical asymptote. x is equal to negative 1 is a vertical asymptote for this graph right here. So x is equal to negative 1-- let me draw the vertical asymptote-- will look like that. And then to figure out what the graph is doing, we could try out a couple of values. So what happens when x is equal to 0? So when x is equal to 0 we have 2 times 0, which is 0 over 0 plus 1. So it's 0/1, which is 0. So the point 0, 0 is on our curve. What happens when x is equal to 1? We have 2 times 1, which is 2 over 1 plus 1. So it's 2/2. So it's 1, 1 is also on our curve. So that's on our curve right there. So we could keep plotting points, but the curve is going to look something like this. It looks like it's going approach negative infinity as it approaches the vertical asymptote from the right. So as you go this way it, goes to negative infinity. And then it'll approach our horizontal asymptote from the negative direction. So it's going to look something like that. And then, let's see, what happens when x is equal to-- let me do this in a darker color. I'll do it in this red color. What happens when x is equal to negative 2? We have negative 2 times 2 is negative 4. And then we have negative-- so it's going to be negative 4 over negative 2 plus 1, which is negative 1, which is just 4. So it's just equal to negative 2, 4. So negative 2-- 1, 2, 3, 4. Negative 2, 4 is on our line. And what about-- well, let's just do one more point. What about negative 3? So the point negative 3-- on the numerator, we're going to get 2 times negative 3 is negative 6 over negative 3 plus 1, which is negative 2. Negative 6 over negative 2 is positive 3. So negative 3, 3. 1, 2, 3. 1, 2, 3. So that's also there. So the graph is going to look something like that. So as we approach negative infinity, we're going to approach our horizontal asymptote from above. As we approach negative 1, x is equal to negative 1, we're going to pop up to positive infinity. So let's verify that once again this is indeed the graph of our equation. Let's get our graphing calculator out. We're going to define y as 2x divided by x plus 1 is equal to-- delete all of that out-- and then we want to graph it. And there we go. It looks just like what we drew. And that vertical asymptote, it connected the dots, but we know that it's not defined there. It just tried to connect the super-positive value all the way down. Because it's just trying out-- all the graphing calculator's doing is actually just making a very detailed table of values and then just connecting all the dots. So it doesn't know that this is an asymptote, so it actually tried to connect the dots. But there should be no connection right there. Hopefully, you found this example useful.