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Use the guidelines of this section to sketch the curve.

$ y = \sqrt{x^2 + x} - x $

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we want to sketch a graph of the Earth. Why is equal to the square root of X squared plus X minus. So the first thing that they tell some structure to do is to find our domain. And we know that to do that we want to make sure that the inside over square root is non negative, so you can go ahead and salt for that and doing that will give that our domain is negative. Infinity two negative one, including a union zero to infinity and to remember, just offer this. We would just factor the inside, and then you can solve that. However you would normally saw for a polynomial being a polynomial inequality. So we have to make. The next thing they tell us to do is to find our intercepts. So let's go ahead and find our X intercept first. So for the Exeter said, we're going to go ahead and set the function equal to zero So X, where pro sex my sex is a reserve so we can add the X over square each side and then just a little bit about true foot. In the end, we should get executed and This also tells us that are why intercept will be This is well, so this is going to be our X and R Y intercept. But next, we can go ahead and look for any symmetry in the problem. Well, this doesn't look like one of the functions that we would only have any kind of periodic behavior, so we can just check to see if dysfunctional, even on possibly So plug it in. Negative expended here we would get So do you. Negative, X squared minus X plus X. We're just gonna be expired. My sex most likes and we have that square. Now this here does not equal to FX, nor is it negative f of X. So we know busier is going to be no symmetry or at least no cemetery about the origin or b y access. The next thing they want us to go about finding are are asking tips. So in this case, we're not goingto have any murder Kal acid trip, since there's really no place where we're really divided by their or anything like that. But we do have some in behavior. Um, other than just going toe. So only a little bit more space sector do this. So let's go ahead and find the limit. His ex a purchase. Let's do negative infinity first. Just because this one will be the easier of the to define minus X. All right, so on the inside of our radical is going to go to infinity since on the inside eventually X squared dominates X so we could just pretend like exit there and x squared would go to inventing and then square routing event It would be infinity. Now X is going to go to negative infinity and the negatives Arable counts out and we just get that as excuse the negativity this will go to And now you might remember And Chuck her 2.6. We've encountered something like this before. So the first thing I'm going to do this go ahead and multiply by the conjugal of dysfunction. So I'm gonna divide by X squared plus x presents and also multiply the liberator. Why? It's complicated. So x squared plus X squared plus x. So this now can simplify to what we have differences squares in our numerator, which just leaves us with X squared. Well, thanks by a sex worker all over X squared plus sex. Well, sex. And in the new mayor, these X squared Sze comes up and now what I'm going to do is multiply the top and bottom of this by one over x so multiplied by one multiplying top of bottom by one of her ex. So in the noon, where we're just gonna be left with one and one over X to pull that inside of the square root, we're going to need to square it. So that's going to give us one plus one over X and then just X becomes one again. Now, if we were to go ahead and, like any good for extra divinity, that goes to zero and we're going to be left with one over one plus, well, the square root of one, which is just one. So we end up with our in behavior as exclusive, and they should be one. How so? Remember, if we were to just do this right away, we would end up with defended by symphony. And that was a form which says we should do a little bit about territory like this. All right, so we now know are in behavior is going to be so the limit as experts in Sandy of f of X is going to be 1/2 and as extra purchase negative, Any function just diverges too positive. All right, So now we can go ahead and do the next part, which is defined where our function will increase and decrease. And while we're at it, we can also there one step six, which tells us where our local max and men's are. So we're going to need to find out what why prime is equal to. So let's do this on another page as well. So why'd Prime is equal to? So I'm going to, actually. Why is it so? I'm gonna rewrite the square root of the half power so we get X squared plus X to the 1/2 power my sex. Now taking the derivative of this, we're going to need to use power rulers. Wallace change will. So first, we're going to get 1/2 X squared plus X now to the negative 1/2 power, and then we need to take the Kuroda of the inside because of change will. And that's just 1 to 2 x plus one and then minus the derivative of X, which is just going to be one. So we can go ahead and rewrite this as two x plus one over two times the square root of X squared plus X minus one. Now divider possible. Max's immense. We're going to go ahead and set this function equal to zero. And if you were to go ahead and try to solve this, you're going to find that it never actually equals to zero. So we don't need to go about doing that. And the only other thing we need to check for possible critical point is where our denominator is equal to zero. Because we want to check where y prime it's undefined. But that would just tell us X is equal to zero and ex visit to negative one as well as all the other parts that aren't in the domain. So usually we would check to see if these points are important. But these are the end points of our domain. So they really can't be Max Instruments, so we could just go ahead and ignored those as well. So this case, we're going to really have no critical about these right. So why prime we get is going to be so we found two X plus one over too explorer plus square root of minus one. Now, to figure out what the function is increasing, we need to find where, like Private, strictly larger than zero. And I went ahead and already solved this beforehand. And that is going to be from zero to infinity and why prime will be strictly less than zero. What were decreasing on the interval? Negative ability to negative one. The next thing they tell us to look for is calm, Cappy and any possible inflection points we may have. And for that we're gonna need to know what? Why Double climates. So let's go ahead and do that off on the side here. So to take the derivative of two x plus one over two times the square root of expert sex, well, I need to use questionable. So let's go ahead and do that. So it's going to be so I'm on the first girl that 1/2 and now is going to be so the square root of X squared plus x times, the derivative of what we have in our new might have sewed two X plus one and then minus them in the opposite orders to expose one times the derivative of the square root of expert what sex and then all over what we have in our denominator squared. So that would just be X squared plus X. And normally we would have an absolute value here. But since on our domain we know those values should be always zero or bigger than zero, we can just let's take these derivatives So we know that the derivative of two X plus one is just going to be and the derivative of the square root of X squared plus X. Well, we already found that out in the first step. So this is just going to be two x plus one over two times the square root of expert plus X No using little bit of algebra. And to simplify this, we should end up with negative one over or times X squared, plus X now to the rehabs power. Now this here is strictly less than zero always so that tells us our function is always going to be calm. Cave down. What? So why Double prime was equal to negative one over for X squared plus x 23 So we just said that dysfunctions Kong came down When y double prime, it's strictly less than zero. And this is just going to be on our entire domain, which is negativity to negative One union zero to remember for this we would not include are in points within the come cavity being down since they would one be undefined on our Interpol or on our secondary, but also on the end points. They really can't be con cave upper down since we don't know what happens next. And so that tells us it is calm, keep up or wide. Old pine is strictly larger than zero on no interval. Now we've done everything that they suggest we do in the chocolate before we actually start graphing this function. So let's go ahead and start. So let's put our intercepts down first, which only ends up being zero in this case. And one other thing we should actually find the hole would start is what our value for F of negative one is going to be. So we need to figure out what up with negative ones. So let's just go ahead and do that over here on the side. So of negative one, there's going to be Well, it's going to be negative. One squared, minus one minus negative one. So negative one squared is one minus 10 square root of 00 And those negatives here accounts out, so we just get one. So at least we needed to know where we were starting above or below our interceptor. So at least have that starting point as well, which is important. So we have a horizontal institute at Why is it you want, huh? So let's go ahead and put that here. So this going out, why isn't it to 1/2? And actually, I'm good and screwed all this up a tad, because otherwise it's not going to look very interesting. All right, so now, um, and we know are in behavior on the other side is just going to go to infinity, so we really don't need So let's start from our ex Anderson. What is acceptable? So we know this should be increasing on zero to infinity, and it should be Khan cape down all it needs to approach our horizontal acid. So it's going to look something kind of like this. It just kind of keep it in closer and closer. Now on our other domain. We know we need to start from the point negative 11 and the function is going to approach infinity and it should be con cave down on this animal. So it's going to look something like this on that interval goes well and it's just going to be go off to infinity. So this year would be a nice sketch over graft. Might want to go in and get a couple more points just to actually get the draftable look a little bit closer to what it actually is. But this is what I would go ahead and just stop, since we just want a sketch of it.

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