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Analyzing vertical asymptotes of rational functions

Video transskription
- [Voiceover] We're asked to describe the behavior of the function q around its vertical asymptote at x = -3, and like always, if you're familiar with this, I encourage you to pause it and see if you can get some practice, and if you're not, well, I'm about to do it with you. All right, so this is q of x, it's defined by a rational expression, and whenever I'm dealing with asymptotes, I like to factor the numerators and the denominators so I can make more sense of things. So the numerator here, what two numbers if I were... Their product is two and their sum is three, well that's two and one, so I can factor this as x + 1 times x + 2, if that's unfamiliar to you, I recommend you watch the videos on Khan Academy about factoring quadratics, and over x + 3, and when x = -1, or x = -2, it would make the numerator equal zero without making the denominator equal zero, so those are points where the function is equal to zero, but when x = -3, the denominator equals zero, while the numerator is not equal to zero, so we're dividing by zero, and so that's a pretty good sign of a vertical asymptote, that our function as we approach that value is either going to pop up like that, or it's going to pop down, or it's going to pop down like that, or maybe, or either way, or could pop up like that, or it could go down something like that, but we're going to have a vertical asymptote that the function as you approach x = -3, that the function is going to approach either positive infinity or negative infinity or it might do positive infinity from one direction or negative infinity from another direction and if this idea of directionality is a little bit confusing, well that's where we're, that's what we're about to address in this video. Let me just draw a number line here that focuses on these interesting values. So we care about x = -3, and then the other interesting values may be -2, - 1, it's all there, now what does it mean to be approaching x, what does it mean for x to be approaching -3 from the negative direction? And just to be clear, this little superscript right over here, that means we're approaching from the negative direction. So that means we're approaching from values more negative than -3, so those are these values right over here. We are approaching, we are approaching from that direction. Another way to think about it is we approach from the negative direction where on the interval x is less than -3, so let's think about what the sign of q of x is going to be as we're approaching -3 from that negative direction, from the left. Well, if we have something less than -3, and you add one, this is going to be negative. If you have something less than -3, and you add two, that's going to be negative as well, and if you have something less than -3, and you add three, well that's going to be negative as well. So a negative times a negative is a positive, but then you divide by a negative, it's going to be a negative, so q of x is going to be negative on that interval. So as we approach, we have our vertical asymptote. As we approach -3 from the left hand side, well q of x is going to be negative and so it's going to approach negative infinity. So as x approaches -3 from the negative direction, q of x is going to approach negative infinity. So at least this accurate, this is accurate, and this is accurate right over here, and you can validate that, try some values out. Try negative, let's see, if you did negative three point... q of negative three point, - 3.1, once again, that's on the left side, it might be like right over there. Actually, it'd probably be a little bit further. It might be something like this, like the scale that I've drawn it on. q of -3.1, if you wanted to verify it, it's going to be -3.1 + 1, which is -2.1, times -3.1 + 2, which is negative, - 1.1, 1.1, I could put that in parentheses just to make it clear, and then all of that is going to be over, well -3.1 + 3 is going to be, is going to be -0.1. So notice whatever we get up here that's a positive value, we're essentially going to, if we're dividing it by -0.1, that's like multiplying it by -10, so it's going to become a very negative value, and if instead of it being -3.1, imagine if it was -3.01, and this would be a 01 here. This would be a 01 here, and this would be a 01 here. And so this denominator, you're dividing by -0.01, it's going to be even larger negative value, so you're going to approach negative infinity. So it's going to be one of these two choices. Now let's think about what happens as we approach x from the positive direction, and that's what this notation over there means, this superscript on the right hand side, the positive direction, so we're going to approach x from the positive direction, and I'm going to pay attention in particular to the interval between -2 and -3, because then we know we don't have any weird sign changes going on in the numerator. So I care about the interval -3<x<-2. So I can draw an open circle here to say we're not considering when we're at -2, and of course we're not going to include -3 because our function isn't defined there, but over this interval, so x + 1 is still going to be negative, x +1 is still going to be negative. If you took a -2.5 + 1, it's going to be -1.5. x + 2 is still going to be negative, you're taking values that are more negative than -2, that are less than -2, so you add two to that and you're still going to be negative, and then let's see, if you add three, if you add three to these values, remember, they're greater than -3, or you could say they're less negative than -3, well then this is going to give you a positive value. This is going to give you a positive value, and think about it, q of, q of, I don't know, -2.99, what is that going to be equal to? If you add one to that, that is -1.99, and if you add two to that, times, times - 0.99 all of that over -2.99 + 3, well that's going to be 0.01, so you're going to get a positive value on top and then you're going to divide it by 0.01, that's the same thing as multiplying by 100, so you're going to get larger and larger values. You're going to approach infinity as you get closer and closer to it from the right hand side. So q of x is going to approach positive infinity. So this is the choice that is correct. This one is wrong, this says we're going to approach negative infinity, so that's incorrect, and we will go with that choice.