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Studying for a test? Prepare with these 10 lessons on Rational expressions, equations, and functions.
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In this video, we're going to see if we can graph a rational function. A rational function is just a function that has an expression on the numerator and the denominator. It has a polynomial in the numerator-- Let's see, we have x squared over-- and another polynomial in the denominator --x squared minus 16. We could obviously graph this by just trying out a bunch of points and then connecting the dots. That's what a calculator would do for us, a graphing calculator. But what we want to do is, before we try out some points to kind of fill in the gaps, I want to understand the basic structure of this graph first. And to understand that, I want to see what happens as x gets really big, so x gets really big, or x gets really, really small, as x goes in the negative direction. Or another way we could think about it, I want to understand what happens as the magnitude of x, or the absolute value of x, becomes really big as it approaches really, really bigness, or as it approaches infinity. So when the size of x approaches infinity. Which essentially is saying, as x goes really far in the positive direction, or x goes really far in the negative direction, what is going to happen to the value of this function? So let's get out a calculator. Won't use the graphing part of it just yet, but let's just try out some values. What happens when x is equal to 10? It's going to be the same thing as when x is equal to negative 10, because when you put a negative 10 here, you square it, you get 100, just like 10. Same here, negative 10, you square it, you get the same thing as a positive 10. So whether you go in the super high positive direction or the super low negative direction, as you approach positive or negative infinity, you're going to approach the same thing because you're squaring the values. But let's try out some values. If I get 10 squared divided by 10 squared minus 16, I get 1.19. Now what happens if x gets a little bit bigger? This is x is equal to 10. What happens when x is equal to 100? We have 100 squared divided by 100 squared minus 16. I'm getting even closer to 1. When x was 10, around here, we're getting y is 1.19. When x is 100, 100 squared over 100 squared minus 16, y is 1.0016. Just for fun, let's try 1,000. So it's 1,000 squared divided by 1,000 squared minus 16. And we're even closer to 1. So as the size of x gets larger and larger and larger, our y gets closer and closer to 1. And that would also be true if this was a negative 10, because negative 10 squared over negative 10 squared minus 16 is going to be the exact same thing. Because the negative, when you square it, is going to be a positive. It's going to be the same thing as 10 squared, same thing over here. So whether x gets really big or x gets really small, we're going to be approaching y is equal to 1. You could try it with a million if you want, and you're going to get a number even closer to 1. So as the size of x approaches infinity, the absolute value of x, or the distance from the origin, approaches infinity, y is approaching 1. Or another way to think about it is, the graph of this function is going to approach the line y is equal to 1. So let me graph the line y is equal to 1. So I'll do it in a dotted line because this isn't the graph of our function, but this is a line that our function is approaching. So that is the graph of y is equal to 1. Now, this idea of a function, or the graph of a function, approaching a line but never quite touching it. So this is going to get closer and closer and closer to this line, y equal to one, in that direction, but never quite getting close enough to it. It'll approach 0, its distance from this y equals 1, but it'll never quite get there. This line that the graph is approaching is called an asymptote. And it'll be even more clear once I actually graph the function. We're going to work up there. And since it's a horizontal line, we call this a horizontal asymptote. This is what our graph approaches as we go in the positive direction, or really far in the negative direction. Let's think about some of the other interesting things about this, about this function right here. Well one thing that might pop out at you is this is a difference of squares. This is x squared minus 4 squared. So we can rewrite this as x squared over x plus 4 times x minus 4. So what's going to happen here, as x approaches either positive 4, or x approaches negative 4? Well, first of all, try those values out. If x is equal to 4, what is going to happen? This expression right here, this term right here, is going to be equal to 0. And we're going to be dividing by 0. We cannot do that. Similarly, if x is equal to negative 4, we'd be dividing by 0. This expression, right here, is going to be equal to 0. We can't do that. We could say that this function is undefined at x is equal to plus or minus 4. It can't equal those values because we'd be dividing by 0 in either one of those circumstances. Now, what happens as we approach those values? What happens as x approaches negative 4? Let's just do that one for fun. What happens as x approaches negative 4? Let's say we're approaching it from the negative direction. So let's try it out in our calculator. So let's say we want to go from the negative direction. So let's start with negative 4.1. So if we have negative 4.1 squared divided by negative 4.1 squared minus 16, what do we get? We get 20.75. So we get this number, whatever. Let's see if we get even closer to negative 4. So let me just get that entry there. So let's get a little bit closer to negative 4. So instead of negative 4.1, let's do negative 4.01. So let me insert a negative 4.01. And then over here, this is negative 4.01, and see what it is. Now we went to 200, so we're getting to larger and larger values. Let's try negative 4.001. Let's try that out. Whoops, that's not what I wanted to do. I wanted to do that. So let's try. No that's not what I want to do. Let's see. So we want to go to, instead of 4.01, I want to do 4.001, and over there, negative 4.001. And what do we get? We get 2,000. So as we get closer and closer to negative 4 from the negative direction, we're approaching larger and larger, super larger numbers. And you can try it, if it's 4.0000001, it's going to get to smaller and smaller numbers-- or sorry, larger and larger numbers here. If you do 4.001, it's probably going to be 20,000. And then if you add another 0 here. So as we get closer and closer, it's getting to larger and larger numbers. So we could say, as x approaches negative 4, we could say y is approaching infinity. It's getting to a larger and larger and larger value. But we can't ever quite get to x is equal to 4. It's undefined there. That will make the denominator here equal to 0. So what we want to do here is, we can never quite equal x equal negative 4. So let me see, x is equal to 1, 2, 3, 4. We can never quite get to x is equal to negative 4. Let me draw x is equal to negative 4 as a dotted line, right there. That is x is equal to negative 4. We can never quite get there, but as we approached it from the negative side, as we had 4.1, then 4.01, we went to larger and larger values. And we also know that as we went on the left-hand side, as we go to larger and larger x values, that y will get closer and closer to 1. So you have a general sense of what this part of the graph will look like. This part of the graph is going to look something like that. As x gets to super negative numbers, it gets closer and closer to 1, as x gets closer and closer to negative 4 from the negative direction, it's going to go closer and closer to infinity. You're going to get closer and closer to a very-- It's going to get larger and larger, I guess, is an easy way to say it. Now, just like x equal negative 4, x equals 4 will also be a point where the graph is undefined. So let me graph that here. 1, 2, 3, 4. Right here. Right over here. x is equal to 4. And, once again, what happens as we approach x equals 4, let's say from the positive direction? So as x approaches 4 from the positive direction, what's going to happen? So this is like trying out x is equal to 4.01, or x is equal to 4.001, or x is equal to 4.0001. So we're just getting closer and closer and closer to x is equal to 4. Well, these values are the exact same values that we just tried on our calculator, except they are the negative version of them, right? And we already saw that, just the way that this function is set up, the negative numbers, they get squared, so whether you take the negative or the positive x values, it's going to be the same thing. This graph is symmetric. When x is equal to negative 5, it's the same thing as x is equal 5. When x is equal to negative 10, it's the same thing as x equals 10. So the same thing is going to happen. You could try it out with your calculator, if you like. If you try out these values, you're going to see, as we get closer and closer to 4, we're going to approach larger and larger numbers. These same numbers over here. So the graph over here, as we get closer and closer to 4, we're going to approach larger and larger numbers. And then here, as x gets larger and larger and larger, we saw over here, we had these horizontal asymptotes, y gets closer and closer to 1. So just like we called this a horizontal asymptote, these values-- or you can even view these vertical lines: x is equal to negative 4 and x is equal to 4 --we call these vertical asymptotes. These are lines, asymptotes, once again, they are lines that the graph approaches, but never quite touches. So that's what's going on here. And then we can think about what's happening to the graph inside of here. So you could think of it in a couple of ways. You could say, well, what happens as x approaches 4 from the negative direction? So let's try that out, from the negative direction. So what happens if you do 3.9 squared divided by 3.9 squared minus 16? You get negative 19.25. Now what happens if we do 3.99? So let me put another 9 here. So we're going to get closer and closer to 4, and we're going to do it from the left-hand side as we approach 4. So insert another 9 here. So even more negative. So let's just do one more. So we're going to be even more negative. So let me make it 3.999. Get even closer to 4. You're getting even more negative. And this is also going to be true if we did negative 3.9, or negative 3.99, or negative 3.999, because when we square it, the negatives and the positives are the same thing. You square negative 1, you get a positive 1. So as we approach 4 from-- you go 3.9, 3.99, we get closer and closer to 4 --we're getting more and more negative numbers. We approach negative infinity. So as we approach-- let me just graph it here. As we approach from this direction, we're going to get smaller-- want to not touch our asymptote --it's going to look something like that. As we approach it from the left-hand side, we're getting smaller and smaller numbers. And that's also going to be true as we approach negative 4 from the right-hand side, right? As we get negative 3.9, 3.99, 3.999, we're going to drop down. It's going to look something like that. And now that we have a general sense of what the graph is, now is a good time where we could maybe plot a few points here. And the easiest one is, what happens when x is equal to 0? You have 0 squared over 0 squared minus 16. So the point when x is equal to 0, we're going to have 0 over, well, negative 16, which is just 0. So the point 0, 0 is on this curve. And then we could try some other points if you like, but the general shape here is going to look something like this. You could plot more points if you really want to nail down exactly what the curve is doing in between, but here is the general structure. And we tried out a lot of values with the calculator. And I did that because I really wanted to show you why it's dropping down like this. And if you think about it, it makes complete sense. As you get closer and closer, let's say you get closer and closer to 4. Either way, as you get closer and closer to 4, this is going to become a smaller and smaller and smaller number, because this is the difference between x and 4. So this is becoming a smaller and smaller and smaller number. Then, when you take 1 over that, right? You can essentially view this as x squared over x plus 4, or times 1 over x minus 4. If this is becoming smaller and smaller, this whole thing, 1 over a super small number, is a super large number. So as you can imagine, you're going to get larger and larger, and depending on whether you are approaching from the positive or negative, so whether this is a super small negative number or super small positive number, that's going to flip the sign. But either way, the magnitude-- So we're getting to a very large magnitude in the negative direction because the difference between x and 4 on this side is negative, right? 3.9 minus 4 is 0.1. Take the inverse of that, it's 10. So we're getting negative numbers here. You take the inverse, you're going to get super large negative numbers. So I really want to give you that intuition. But the general way of being able to graph these type of things, your first thing you want to do is identify the horizontal asymptotes. What happens as we get very-- the magnitude of our x is very large, so super positive values or super negative values. You could try it out on a calculator, if you like. You literally, if you try out the value of a million or a billion, it will kind of give you the answer. But the way you could also think about it is, as x gets really large, you could view that this thing, these terms right here grow so much faster-- I mean this is just a constant term. This term doesn't matter anymore. If this is a million and a million, who cares about the 16? So as x gets really large, you could say that y is approximately x squared over x squared. These two terms dominate. You don't need to worry about the 16 anymore. And of course, this is equal to 1, which is exactly what we got when we plugged in really large numbers. So, in a problem like this, where you have the same coefficient, or where you have the same degree on the numerator and the denominator, you look at the coefficient of those terms. So in this case, the coefficient is 1 and 1. So our horizontal asymptote is going to be 1 divided by 1, or y is equal to 1. If this was 2x squared over x squared minus 16, our horizontal asymptote would be y is equal to 2. We would approach that line, up there. If it was a negative 2, our horizontal asymptote would be y is equal to negative 2. So that's how you identify the horizontal asymptotes where you have the same degree in the numerator and the denominator. If the denominator has a larger degree, then the denominator is going to get larger much faster than the numerator, and your asymptote is going to be 0. I'll show an example of that in the future. And obviously, if your numerator has a higher degree than your denominator, it's going to grow way faster than your denominator, and you won't have any asymptotes. You'll just keep growing, or keep going in the negative direction. And that's actually the case with all of the polynomials we've seen. You can do them all as being over 1. In which case, there was no horizontal asymptote. Now the vertical asymptotes you identify by essentially just factoring the denominator and figuring out where does it equal 0. Those are the points where the function is not defined. And I'll show you in the future, there are some special cases where they won't be vertical asymptotes, and I guess that special case is, for example, if you had-- Well, I won't show you the special case right now. I'll show you that in a future video. But in general, if you factor the bottom terms and they don't cancel out with anything on the numerator, then you're going to be dealing with a vertical asymptote. If I had another x minus 4 up here, if my numerator was x squared times x minus 4, and then these canceled out, and my expression simplified to this, the equation would still be undefined at x is equal to 4, because you would give the 0 in the denominator. But since that x minus 4 cancels out with the x minus 4 in the numerator, it would not have been a vertical asymptote. We'll look at that in the future. But this equation wasn't that. So the general rule of thumb for identifying the vertical asymptotes, factor the denominator, figure out where the denominator equals 0, and if those terms don't cancel out with any terms of the numerator, then those are vertical asymptotes. And then to figure out the behavior, I guess, within the asymptotes, you can plot some points. You can try out some points. You can actually substitute values for x and figure out what y is. Now just to validate that we hopefully got the right answer, let's actually graph our rational function. So let me turn it on. Let me graph it. And we say y is equal to x squared divided by x squared minus 16. And let's see what we get. Nope, I just want to graph it. My range is off. Let me do my range. Let me see, x minimum value I want for x, let's say it's negative 10. My maximum value I want for x is 10. x scale is 1. y minimum value, I want negative 10. y maximum value, I want 10. Then y scale, I want 1. Now let me graph it. There we go. Look at that. Just like what we drew. We have an asymptote, as x gets really large, or x gets really, really small, that asymptote is y is equal to 1. We have our vertical asymptote. It graphed it because it tried to connect the dots, but it essentially graphed our asymptotes for us, but that wouldn't actually be part of the graph. But as we approach 4 from 0, I guess we can say, we go super negative. As we approach negative 4 from 0, we get super negative. Because in either of those situations, as you approach 4 from this side, this term is going to be negative. As you approach negative 4 from thid side, this term, right here, is going to be-- Well, this term right here is going to be positive, but then this term right here is going to be negative. Negative times a positive, you could play with it as you like. But we approach negative infinity in either case. And then as x approaches infinity, this thing asymptotes away. So hopefully you found that fun.