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University of North Texas

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Problem 13 Medium Difficulty

Use the guidelines of this section to sketch the curve.

$ y = \frac{x}{x^2 - 4} $

Answer

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Video Transcript

we want to sketch the graph of why is he going to x over expert mind sport Now in the structure they give us this laundry list of steps we should follow to sketch a graph. And one of the things we should always do for any rational function is to go ahead and factor in your area. The Dominator, Since we need to be ableto find any kind of holes or vertical ascent Oops, as well as our ex intercepts. So memory doesn't factor, since just but the denominator is different square. So it's gonna be X plus two X minus two and X plus two all over X. All right now, we can't cancel anything else. Let's go to the first step, which is our domain. Well, we know we need to make sure that our denominator does not equal to zero. So that's gonna be an exit with the two and negative too. So the domain is gonna be negative. Infinity too Negative to union. Negative too. 22 Union to To infinity. Let me go ahead and drop this down. What? The next thing they tell us that the court is our intercepts. So let's go ahead and find our X intercept First X intercept. Well, that's going to be where the function is equal to zero. Or why is it so? Zero is equal to X over X squared minus four. And I'm not factoring that it on there because remember, no matter what we plug into the dominator, you never make this equals zero. So Soling for zero, we would just get X is equal to zero and since exited desirable. That would also tell us our Winer said it would be at the same point. So bye intercept is also going to just be why is zero? So the origin is our only interest. The next thing they suggest we find is any kind of symmetry we might have. So national functions aren't really known for being periodic. So what we want to check him said, is whether the function is even or odd. And we do that, remember, by looking at of negative X, so we'll get the negative X we get negative X and negative. X squared is just going to be ex squared minus four. Now notice that this here is equal to negative of necks. So this here implies that we have an odd function. So we have symmetry about our origin. All right, the next thing we need to chart core is our ass in Toots. Now we know that the limit as X approaches. So plus and minus infinity of our function at the backs, which is really just why so since our denominator has the larger power going back to dr 2.6, this tells us we'll have a war zone toss until at, why? Easy, which is zero. All right, now we need to figure out our vertical acid vote since we just found our horizontal using that information. So the limit as X approaches. So we said our Berta classic. Oh, it's going to be Where are the dominators? Zero. So X is equal to an X equals negative. So start with negative, too. So negative, too. From the right of Ethel bex. So let's go ahead and plug that in. This could be negative too. From the right. Oh, negative too. From the right. Minus two times. Negative too. From the right. Plus two. Uh, let me go screw that over a little bit now. Negative, too. From the right is still going to be negative if I subtract so negative too from the right is gonna be negative again. And then if I subtract something that's still gonna be negative and then if I do negative too. To the right, that will be something like negative 1.99 If I add that to two would be positive, so is gonna be overall positive. So that tells us that this should go to positive infinity. And we don't even need to do the same steps are looking for as we approach it from the left because we know both. These are Ligier terms. That means they should have opposite behavior on either side. So just to save a little bit of time, we know that this here should go too negative, and Bennett and you could go to the same logic that we didn't know before, but just to save a little bit of time. Now, the next thing we want to do is look at our other interesting, which is going to our other verte class, which is gonna be as X approaches to. So let's look for right again. Let's start so it's ready to from the right over two from the right, minus two times two from the right, plus two and using a similar logic that we did before. So the new mayor is going to be positive. Slightly to the right. We'll still be a positive number now. Something slightly to the right of two will be like 2.1 and subtracting, too. From that well, that would still be positive and tune from the right will be all positive number. So addict, something possibly possibly as well. So this here will approach positivity and using similar logic that we did before. We can find out that as we approach to formula, this should be negative. Infinity already. The next step is we want to find our intervals of where this function is increasing and decreasing and any kind of local Max's ornaments. So this is That's five and six, and I think they should be combined into one step. So we want to find what, why prime ms. So let's go ahead and do that on another page. So if y is equal to X over X squared my sport now take the derivative of this. We're going to need to use questionable. So question rule is low d high minus high d low all over the square. What's below? And these just represent derivative. So going in filling that out, we will get the following expression and X squared minus for And then we need to square that now the derivative of X is going to be one, and the derivative of X squared should be to excusing the power rule and in the derivative of, um, negative forward, just be zero. So that should be two X and we can go ahead and rewrite this. Ah X squared minus four minus two x squared All over X squared minus four, I swear. Then, in the numerator, shove negative X squared minus four. Unless factor that negative out. And I will leave us with X squared plus four in the numerator over X squared minus four squared. Now notice that expert plus four will always be strictly greater than zero. And expert minus four squared should also always be greater than or equal to zero. So that means since we had that negative out here, this function will always be less than zero or decreasing. So go ahead and use that. So let's go ahead and write. What? Why? Prime is real fast. Well, wide prime was negative, X squared plus four over X weird minus four square. And we already said that this year is going to be strictly less than zero. So our interval of decreasing will be everywhere that the function is just defied. So it's always decreasing since why Prime Mister Crewe? Us these air. So it would just be negative infinity to make it up to union. Negative too, to do union to to infinity. All right. And we have no critical values for this since the numerator can ever equal zero. So the function can never actually equals zero and are derivative is undefined at X is equal to two and negative too. But those are not in our domain, so we don't need to check those for critical ones. All right, The next step is we need to find our con cavity, so call on cavity and any kind of inflection points. So that means we need to figure out what? Why double prime Ms actually go ahead and move this suicide over here instead. So let's just go ahead and little barrier there. So, like I'm saying, we need to go find what? Why? Double promise first. So let's go over to hear again and start taking the derivative. So why double prime? So we can factor that negative out front and then just apply bullshit rule again. So we are going to do low d high, so X squared minus for squared times, the derivative of what's above, which is X square plus four minus. Then the opposite order. So x squared plus four times the derivative with respect to X of what we have in the denominator x squared minus for squared and then all over what we have in the denominator squared. So X squared minus four squared But squaring that would give us to the power. And so don't forget this You negative out here now, taking the derivative ot to these well expert plus four is just going to be two x and to take the drilled of X squared minus sportswear do enough to use Powerball and chain will. So the derivative of the outside, his first going to be two x squared minus four and then the derivative on the inside is going to be two x part. And just so we don't have to do all that algebra when we simplify this, we should be left with two x x squared plus 12 over X squared minus four. Keep. So let's go ahead and use that. Actually, before we do that, let's figure out where this function is equal to zero. So again the denominator can never eat with zero, so we don't need to worry about that. So we just need to worry about the numerator. Well, X squared plus 12 can ever equal to zero. And we know two x can. So we get to X zero or exes just is there. So this is one possible point of inflection. All right, so it was the hand right down that derivative or second, Really, we just found you're going to be two x x squared lost well over X, where minus four Q. And just for the sake of brevity, to find where this option strictly larger than zero, I already went ahead and solved. That is, well, why Double prime greater than zero tells us what the function is comfortable and that's going to be on negative, too. 20 Union too. Then, fans, Why double prime theatrically less than zero. Well, this stuff this is going to be conch aid down. And this is just the rest of the using interval from our domain. So negative many to negative to union with zero toe too. So we found that ecstasy. Coke zero is a possible point of inflection. So let's go ahead and see if we have a change in Kong cabin here. So to the right of zero, we will be conch eight down and to the left of zero, We're going to be con cave. So this here tells us we do have a point of inflection at X is equal to zero. All right, now, let's go ahead and actually grab this function. So we know we have our intercept at only the origin. We know the function is odd, so it should be symmetric about the origin. And now we can go ahead and plot our ass in toots. So we know we haven't asked. Toe X is equal to negative too, and X is equal to two. So excited, too accident negative too. And we have a horizontal ask himto at why is equal to zero. Um, so we have no local men's or maxims from 0.5 and six, but we know at zero at X is equal zero wish of a change in con cabinet. What now? For horizontal Herbert glasses tips. So let's start there to the right of negative too. We said that this should be going to positive infinity until the left of negative, too. We should be going too negative infinity. And we also said to the right of two we should be going to positivity until after to wish you were going to negative infinity. Now we can go ahead and start grabbing, so I'm just gonna start in this middle section, so we're gonna start to the right of negative too, and we need to go to our origin. And at that point, it's one to switch con cavity. So to the left of this, it's complicated. But to the right, is Kong cave down now to the right of two? Well, we have no points of interest, so we're just going to go until we hit our horizontal awesome too. And we will need to approach it from above as we can see there and now you can go ahead and go to the left of negative two. We have no points of interest after that. So we're just going to go toe was awful, Jacinto and approach it from below, so you could maybe plot some actual numbers to make it look a little bit prettier. But since we're just sketching it, I think this is a good place to stop.