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Studying for a test? Prepare with these 10 lessons on Rational relationships.
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Let's graph another rational function, because you really can't get enough practice here. So let's say we have y is equal to x over x squared minus x minus 6. So the first thing we might want to do is just factor this denominator so we can identify our vertical asymptotes, if there are any. So what two numbers when I take their product I get negative 6 and if I add them up I get negative 1? So they have to be of different signs. So one's going to be a plus and one-- let me write my x's a little bit neater than that-- so one is going to be a positive and one is going to be a negative. A 2 and a 3 seem to be pretty close, because they're one apart, and I'm going to subtract the larger number because when I add them, I get a negative. So x minus 3 times x plus 2 seems to work. That gets negative 6. Negative 3x plus 2x. negative 3 times x plus 2 times x is negative x, so that works. So this is equal to x over x plus 2 times x minus 3. And like we saw in the last video, since these expressions, since the x plus 2 doesn't cancel out with anything in the numerator and the x minus 3 doesn't cancel out with anything in the numerator, we know that these can be used to find our vertical asymptotes. The vertical asymptotes are when either that term is equal to zero or when that term is equal to zero, because at those points, our equation is undefined. So this is equal to zero when x is equal to negative 2, and this is equal to zero when x is equal to positive 3. And you could try it out here. If x is equal to negative 2 or positive 3, you're going to get a zero in the denonminator, y will be undefined. So vertical asymptotes at x is equal to negative 2. So there's a vertical asymptote, a vertical asymptote right there. Another vertical asymptote is x is equal to 3. One, two, three. There is our other vertical asymptote. Now let's think about horizontal asymptotes, or if there are any. So what happens as x gets super-positive or super-negative? And as we said before, you just have to look at the highest degree term on the numerator and the highest degree term on the denominator. Now, notice the highest degree term on the denominator is x squared, while the highest degree term on the numerator is only an x. So when x gets really large, what's going to happen? You could imagine, this is going to be like a million over a million squared, which is still one over a million. These terms over here don't matter much. But this term right here is going to grow faster than everything. This is an x squared term. As x gets large, it's going to get way larger than everything, including this term on the top, so it's essentially going to go to zero. When the denominator just gets bigger, faster than the numerator as you're going to approach zero. So we have a horizontal asymptote at y is equal to 0. I could draw it as a dotted line over our x-axis. So that right there is the line, y is equal to 0. Once again, we identify that looking at the highest degree term there. The denominator has a higher-degree term, so it's going to grow faster than the numerator. You could try it out on your calculator. And that's true whether you go in the super-negative direction or the super-positive direction. This thing is going to overwhelm this thing up here, the denominator grows faster than the numerator, which essentially we're going to approach zero. You're going to get smaller and smaller fractions. Just remember, 1/10 and then-- let me actually just-- as x gets larger and larger and larger, what's going to happen? Let me just show you on my calculator. Let's say x is equal to 10. 10 divided by 10 squared minus 10, and normally you wouldn't have to do this. I just really want to show you the intuition. Whoops! I'm not trying to graph. Let me exit from here. So if we have 10 over 10 squared minus 10, once again, you normally wouldn't have to do this. I just want to show you, give you the intuition. Let me put some parentheses there. Let me put some parentheses over here. So let me insert the parentheses there and put a parentheses over here. You get a small number. What happens if x gets even larger? Let me make, instead of a 10, let me make it all 100. Let me make these tens into hundreds, into 100. Insert 100 there, what do we get? We get even a smaller number. And if you try x is equal to 1,000, it's going to be be even smaller than that. That's because this term right here is growing faster than every single other term. That's why our horizontal asymptote is y is equal to 0. Now, the last thing we want to do, we've drawn all of our asymptotes, is just try out some points. So let's draw like a little table here. There's our table. When x is equal to 0, what is y? x is 0, we have 0 over all of this. 0 minus 6, 0 over negative 6 is just 0. When x is equal to-- I don't know, let's just try when x is equal to 1, what do we have? We have 1 over-- I'll write it here. 1 over 1 squared minus 1. Now that's just 0, so we have negative 6. When x is equal to negative 1, what do we have? When x is equal to negative 1, we have negative 1 over negative 1 squared, which is 1 minus negative 1. So that's plus 1-- right, minus negative 1-- minus 6. So what is this right here? This is negative 1, so this is going to be negative 1 over 2 minus 6 over negative 4. This is going to be equal to 1/4, so we're going to get a positive value. So we have-- let me draw this-- negative 1, we're at 1/4 right here. That's about right there. I'll do it in a darker color. We had the point 0, 0, and then at x is equal to 1, we had negative 1/6. So you could keep graphing more and more points, but it looks like as we approach this vertical asymptote from the right, we go to positive infinity. And that should make sense. Let's see, if we were to put in-- we're approaching negative 2 from the right. So if you were to put in negative 1.9999999, this term is going to be a very small positive number. This term's going to be a negative number. This term's going to be negative number. The negatives cancel out. You have a very small positive number in the denominator. 1 over that gives you a very positive number. Now, as we approach the other vertical asymptote from the left, we're going to go super-negative. My gut tells me that because when I tried x is equal to 1, I already went to a negative value. But you could imagine if you did 2.99999, right? Let me draw that a little bit better. You get the idea. If x is equal to 2.999, so we get really close to the asymptote, this is going to be positive, this is going to be negative, that's going to be positive, and this is going to be a small number. So you're going to have 1 over a very small negative number, which is a very, I guess, negative number. It's a negative of 1 over a very small number, so you're going to approach negative infinity. Now, let's try some point out here to see what happens. So what happens when x is equal to 4? When x is equal to 4, you have 4 over 16 minus 4 minus 6. What is that? That's 16 minus 10. That is 6. So this is equal to 4/6, which is equal to 2/3. So the point 4, 2/3 is here, so one, two, three, four. 2/3, just like that. So that gives me the sense, look, I have to approach this horizontal asymptote as we go further and further out. We're going to probably approach positive infinity like this. Let me draw it a little neater than that. Like that. You get the idea. Then over here, we're going to get closer and closer to our horizontal asymptote as we approach infinity. This should be a smoother-looking curve right around there. I'm making a mess here. This should be a smoother-looking curve. You get the idea, I think. Now, let's see what happens when x is equal to negative 3. So when x is equal to negative 3, we have negative 3 over negative 3 squared, which is 9 minus negative 3, so that's plus 3 minus 6. So what is this equal to? This is equal to negative 3 over-- this is 12 minus 6 over 6, right, which is equal to negative 1/2, So negative 3, negative 1/2. Negative 1/2 is right there. So we're going to approach this asymptote as we get really negative. And we're probably going to go straight down like that as we approach this vertical asymptote right there. And you could try more points if you don't believe me. But let us graph it just to verify it for ourselves. So our equation is x divided by x squared minus x minus 6. And let's graph it. And there you go. There you go. All right, looks pretty good. Our asymptote is zero, we go down. Vertical asymptote, bam! Go up there, then we go back down here, then we go just like that again. So once again, this looks just about exactly what we got. Obviously, the graphing calculator, it kind of a pitters out as you get close to these values and it does weird things, but it has the same general shape. We could actually close the range a little bit if we want, if we want to graph it. Let's make our x minimum value, let's make it 5. And let's take our x maximum value, let's make that 5 as well. We're kind of zooming in a little bit. So let's graph it now. Bam! bam! There you go! It's the same shape as we graphed right here. Hopefully, you found that satisfying.