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University of North Texas

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Problem 10 Medium Difficulty

Use the guidelines of this section to sketch the curve.

$ y = \frac{x^2 + 5x}{25 - x^2} $

Answer

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Video Transcript

we want to sketch the graph of why is he gonna expert prosthetics led by police by minus X squared. Now in this truck where they give us a laundry list of steps so we can follow, two actually grafted. So let's go ahead and just follow me. Look, before we do that intend of a rational function that was a good idea to after the new Miranda down there to see if we can get anything that cancel out so we can know where we have possible. Hold that. And we also need to factor the top bottom to find our Berta blaspheme tips as well as our zeros of function. So go ahead. With the merger we can factor out of next year, we have X Plus five and in orbit on there. That is the difference of squares. That would be X plus five and by minus X. So know this. These experts fives are factors so they can cantaloupe and we'll be left with X over five minus X. And we should say that ex cannot equal to five are negative. All right now working with X over by my sex along his exits on equal to negative five is a much easier functional Brooklyn than what we had so down. The first thing that goes is to determine our domain. So, in rational functions as were the denominator is not conducive. So we already found one of them being negative five. And our other value would be where five minus zero or just five. So our domain, in this case, it would be a negative entity to negative five Union negative 5 to 5. Union five to Infinity. The next thing they tell us to find is our intercepts. So let's go ahead and bind our Why intercept first. So why intercept? This is when excessive reserve so have why is equal to zero over five minus zero, which would be easier over five, which is just zero. Now, To find our X intercept, we're going to go ahead and set y equals zero. So love zero is equal to X over five minus x and now are the matter what we've dug into the denominator. We could never make this function go to zero. So we were just said it equal to the numerator, and that would imply that X is equal to so are only interested in this case is going to be 00 right next. They want us to determine if there's any symmetry and the problem. So we know rational functions won't really be periodic. But we can at least check to see if it is even are odd. But we want to look at of negative X and doing that, we would get negative X over five minus negative X, which is going to the negative X over five plus X. And you might notice that this year does not equal to flex nor negative FX. So that tells us we're not symmetric about the buy access or the origin. Now, the next thing they want us to find is where our acid jokes are. So Assam gets so we can look at first are in Peter. So the limit has extra purchase affinity of the fact that I'm just gonna call backs, um, our function one. So, since this is a rational function, we know that since they have the same degree in the top and bottom, just divide the coefficient. So it's gonna be one over negative one or negative one as exclusive. Any and we get the same thing or going negative energy is what I'm gonna do is go ahead and move it to be over. I'm just gonna put a plus for minor. So going to plus or minus infinity Well, give us a horizontal ass into that. Negative. What? Now we can go ahead and check or vertical awesome tips. So we know we're gonna have a vertical Jacinto. Where once we factor and simplify this function here the denominator 00 So where X is equal to five. So we need to look at what happened as we approached dysfunction from the right and the left of five. So, uh, ex. So this is going to be five from the right over by minus five from the right and five from the right will be a positive number. Something a little positive sign right here and then subtracting five minus something to the right off. I will be negative. So this here, we know we'll go to negative infinity and looking at the other side. So the woman as our approaches or as we try from a lot of effort, Lex no, there's gonna be by from the left and then five minus five from the west. So once again, bye from the left is still going to be a positive value. But now, when I do five minus by from the left will be taken number slightly smaller than five and subtract it from buy. Well, that's going to be positive. So this year is going to go to positive and all right, so we have our ass in tips. You know, our, for example. And we know how are vertical ass into this Going to behave the next thing we want to find. So this was steps five and six, where function is increasing and decreasing and local Max Freshman's. So to do this, we're going to want to find why Prime? So let's just go ahead and do this on another page. So we have. Why is equal to X poker 5.6. So if I want to take the derivative of this something to use question rule, so way I remember is low d high. My high, low all over the square. What's below, where the D is just supposed to be derivative. So five minus x times the derivative O X minus and them switch so ex ive i t x oh, minus X all over the square of what is below. Now we know the derivative of X is just gonna be one. The derivative of five will be zero since Scott's in the derivative of negative X will be negative one. So this year is negative one. Now we do you left with make five minus X plus X over five minus X squared. Now these negatives cancel out here and we're going to be left with five over five minus x weird. And we should keep in mind that this here ex cannot equal to negative. And that's just because why here is not defined a negative five. What? So something we need to do is look for critical values and we can't set this function equal to zero. Since we have nothing in the new mayor, that'll make it zero. But we can possibly check what makes the denominator question zero. So zero is equal to five minus X, which would tell us X is equal to five. So this is a possible critical point, but we know that our original function why is not defined at executed five. So in this case, we actually have no critical points. So now let's go ahead and put this over here. So we end up with bye over X minus by squared and doing this, we want to find him first. So let's look for interval of increasing. Well, that is where why Prime has been destroyed. Larger than zero well X minus five squared will always be positive or equal to zero and five always be positive. So this function here is actually always been strictly greater than zero since those on there could never make this. So this will be negative. Entity. June Negative five Union negative. 5 to 5. You again. Bye To remember, we don't include negative five and by sense, they're not in our domain. And so this tells us for where this function is decreasing, there are no interval of decrease. And so then also, because of that, we could go ahead and say no local max slash mints. What now? The last thing they say you should do is a for con Cappy as well as our inflection points. So that means we want to look at what? Why double prime is so let's go back over to that page and try to take the derivative of the first derivative? No. First I'm gonna go with it. And back to that five he I d x of And remember, we can rewrite one over five. Mind sex. Weird. Um, as one over X minus X. Yes, the first X minus five. Since we can switch the by my sex since it's being squared and put that to the negative second power like that and to take the derivative that we could go ahead and use power and change will. So we will end up with bye times e. So the power I multiply all front will be negative too. And then x minus five now to the negative third power. And then we need to take the derivative of what we have on the inside, which is expired. Driven of X minus five is going to be one since derivative of X is one driven. Five is Europe. And now we can go ahead and combine these get negative 10 over X minus five Q. So let's go ahead and write this on the other page. So we get negative 10 over Y minus five x minus five. Cute. So for Kong Kate up. All right, we want why double time to be strictly greater than doing so. I went ahead and just solved this already, and that would give us the interval. So, in other words, we want the denominator to be negative. And that would be when X is less than five. So that's gonna be negative. Five two negative five. Union Negative. Five, 25 And remember, we don't include negative five. Um, do to us canceling out of X plus five term. All right, then define Kong Kate down. It's going to be worldwide about crime. A strictly less than zero. Which would just be when x is greater than five by two. All right, so this was the last thing they told us we should do. We actually start crapping. So now let's go ahead and begin grabbing. So let's go ahead and plot our first. So, you know, at X is equal to Bye. We should have a vertical assam kip and we know asked X is it And at why is equal to a negative one. Why we're going to have a horse now. Let's slot our intercept, which is going to be 00 So we know we have no symmetry about dysfunction. Um, we know we have no local minimum axes, and we also know we're going to have no points of inflection. So we already have everything I wanted that we need to know. And we could go ahead and actually start applying stuff, right? So for ask himto we know to the right of five, we're going to be coming from negative infinity. So we're gonna come from negative thing like this, and then we need to approach our horizontal awesome trip. Since there's no points of interest to the right of five, then on the other side of executed five, we should be coming from positive. And then our point of interest is going to be the origin since that is intercept and then we're going to approach are horizontal asking too. So you could go ahead and maybe plot a couple of actual points on this bus. Since we're just trying to get in touch of this graph, I don't think we need to be too accurate and just kind of talk about the important points like this here