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University of North Texas

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Problem 11 Medium Difficulty

Use the guidelines of this section to sketch the curve.

$ y = \frac{x - x^2}{2 - 3x + x^2} $

Answer

see solution

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Video Transcript

we want to sketch the curve. Why is even X minus X where? Over two minus three X plus X squared. So, Inspector, they give us this laundry list of steps we should follow in town. We must get out and what we do that any time we have a rational function, I always think it's the first Go ahead and factory since we're gonna need to figure out are vertical acid pits as well as our ex intercepts as well as if we have any Cole's anywhere. So let's call it a factory. The numerator we're gonna get X one minus X over and that are denominated were affected to the X minus one and X minus two and now one minus sechs divided by ex mice. One. Well, if we pull out a negative one does what counts out. But we just keep up with negative one so we could go ahead and right. This is negative. X over X minus two, and we should say that ex cannot equal to one. Since when we plug one end to the denominator of the original function, we get zero. We can't divide by zero. So now the first thing they tell us to do in this trucker is to determine what our domain should be. Well, we know one value. We can't have his one. And if we set our denominated here equal to zero, we also can't have that. So it'd be executed too. So our domain is going to be negative. Infinity two one, union one to do union to to infinity. The next thing they tell us to look for is our intercepts. So we could go ahead and find our exercise. So intercept is going to be when zero is equal to negative X bill over X minus two and recall that are denominator can ever be zero. Our denominator can never make this function equals zero, I should say so I can just get the numerous equals zero. And I would tell me zero is equal to negative X, which is safety. It's just sex. And this here tells us that our why intercept is also going to be just why is he could zero. So then we can go ahead and go to our next up being. We want to determine if there's any symmetry. So rational functions aren't really known for being periodic, so we should check to see if it's even or odd so we can find symmetry about the why axe or words. So we look at it from a complex, there's gonna be negative. Negative X over negative X minus two. Now, these negatives were gonna count slow and we go ahead and factor that negative about of the denominator to write says negative X over X plus two. Now, this here, uh, does not equal to Apotex, nor does equal to negative f of X. So this tells us no symmetry. Early Snow cemetery about you. Why access or our origin? All right. The next thing they want us to find is our asset, toots. So ass end tips Or so we want to look for burst. Just rn behavior, which is gonna be the limit as X approaches and many of f of x, which I'm just going to say is our function. Why here? And we know that when we have a rational function will go back to chapter 2.6. We can just divide the leading coefficients when they have the same degree in the top and bottom. So this would just be negative one over one isn't too negative One. And not only does this hold for many, but it also holds for negative. Summer's gonna put Customize it right there. All right, now, we could go ahead and look for our vertical ass into since infinity to negative. Many will tell us are horizontal awesome. So now let's go ahead and look at the limit as Ex purchase. Well, we know that her denominator can't. He would, too, once we simplify it. So we know we have a class until that excessive, too. So let's first look at as we approach to write of ethics. So this went to be negative too, from the right over to from the right, minus two. Now two from the right is gonna be positive. And when I multiply that by negative, it'll be a negative. And if I subtract two from something slightly larger than two, that's gonna be positive. So I have a positive over negative. So this year is going to go to negative infinity and we can do the same thing for the so bucks is going to be so. It's negative. Two over two from the left minds, too. Now, once again. Two from the left is gonna be positive. Multiply that by negative. It's gonna be negative. But now, if I subtract two from something slightly less than two, that's gonna be negative. So negative number negative is going to be 1000. So this is gonna go to plus so we know our behavior around our vertical ascent. Now, the next thing that talks to find is our intervals of increasing and decreasing. And I'm gonna combine this with part six, which is looking for a local. Max is admits, So we're gonna need to figure out what wide prime this. So let's go ahead and do this on another beach. So why is he into negative X over X minus two? And we should include that ex con article one? Because if for some reason we end up with a critical value of X is a good one, we will know to exclude that as well as willing to exclude it from our intervals. All right, so to take this trip that we can apply questionable. So remember questionable says low D high minus hide below all over the square. What's below, where the D's are of representing derivatives. So you get X might sue times derivative of egg of X, minus them in the ops, or so that negative X derivative with respect to X of explains to all over what we have in the denominator, I swear. Now the derivative of negative X is going to be negative. One, the derivative of X is one in a dream, too. Is zero So during exercise to just be one Now, these two magnets here will counsel, and we could also distribute that negative one into X minus two and doing all that went out with to minus X plus X over X minus two squared Did notice these negatives Here are these accidents Will can't slow and I want to over X minus two, that's where. Now notice that two is better than zero and X minus two squared. We'll be strictly greater than zero as long as X is not equal to. So we know that this here is strictly larger than zero. Now this here can never equals there. We said that, but we still get a critical value of win for this functions undefined, which would be ex is too. But in our original function. We know that it is not defined there, so we can go ahead and ignore that. And we know that we actually have no critical values. So now we had wide prime is equal to two over X minus one square. Let me just make sure that strict Yes. And we said that this is going to be strictly greater than zero, which implies that this is always increases so strictly increasing or increasing on the interval Negative in there 21 union 1 to 2. Union to to remember We can't include importance where our function is undefined. And that means we would never be oh, decreasing at in equal. All right, now we can go ahead and check part seven, which was Kong cavity and inflection points. So we're going to need to find out what why double crime is. So let's go ahead and take that derivative. So why double pride is going to be derivative of two times now I'm going to rewrite this using a little bit about as X minus two to the native second power so we can go ahead and light power, war and chain. Will that suppose having to do the question well for this. So that's for says. We move with power to make it too often X minus two. Now it's going to be too negative. Third, and then we need to take the derivative of our inside function explains to Well, we already said the drill of exploiting students. One, we took the first derivative so we can revive. This is negative for over ex fighters too. You all right? So let's go ahead and write that over here. So negative for over X minus two. Cute. Now we want to find where it's comfortable conk it down. So we have why double prime is strictly larger than their or con Kate up when? Well, we would need our denominator to be negative. So that would be from negative Infinity two, too. But we have to exclude one says the seminar domain and one to not want to have any but one toe. Then we know that white double time will be strictly less than zero on the rest of the interval. So from two to in vanity, all right, now we can go ahead about sketching describe, so you might notice that we have the change of Kong cavity from two, um, from up after two, two down. But since that's where our Berta class into this, we don't really need to be worried about the changing on cavity or be an inflection point. All right, so that was the last thing we need to do before we start Graffin. So let's go ahead and burst. Plot our intercept, which is 00 and then we have no symmetry. Our Athens hopes we'd go in partners. So we said at X visit to we're going to have a vertical acid so X is equal to to be left of two. We should be going towards positive and into the rite of to We should be going towards negative. And we also have a vertical or horizontal awesome go at. Why is it was a negative one. So why is it going to make art now? We can go ahead and actually ground, So I'm gonna just start from the rite of our control for a vertical axis. So there's no points of inflection. No, Max is airmen's or any intercepts after excited, too. So we could just go ahead and get close to our Where's all fastened to now, starting from the other side of our vertical ascent. Oh, well, we have the origin that we have to pass through, and then after that, we have no Max is for men's or any other intercept, so we'll just keep getting closer. Two our interceptors. So this would be a sketch of or graph?