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Use the guidelines of this section to sketch the curve.

$ y = \frac{x^3}{x^3 + 1} $

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we want two steps craft of why is it but to execute over execute plus one. So we're given in destructor this laundry list of steps we should follow an order to go about finding what the graph looks like. So let's just go ahead and start by following that. So the first thing they tell us is to find her domain. So we just want to make sure where our function denominator does not equal zero. So if you were to solve X cubed plus ones and zeros, you'd get exited with a negative one. So we just need you exclude that point. So it's gonna be an affinity to negative one union negative one to infinity. The next thing they tell us is to find our intercepts. Suppose go ahead of time, the Y intercept first. So that's going to be where X is equal to zero We're gonna get Why is it too zero cute over a zero, Q lost one just here. So our origin will be our whiteners up. And then to find our ex intercepts, we set the function equals zero. Get zero busy too. Execute over excuse plus one, and we know that nothing ever plummet. The Dominator will make the function equal to zero. But if we said the new Maria going zero, it will. So that just tells us X is equal to zero. Will be our ex interested, so only intercept will have his artwork. All right, so now let's go ahead and look for our symmetry of the function. So rational functions aren't really known for being for being periodic. So let's go head to check to see if it's even or odd so you can find symmetry about the why access or about the origin. All right, so here we can rewrite this as negative X cubed over negative X cubed plus one. Now, if I were to distribute that negative in the new mayor to the Nam there, we're going to get execute over. Excuse me, minus one. So this here is neither, um, epa, Becks. Nor is a negative aftereffects. And so, due to that, we know we're gonna have no cemetery or at least no symmetry along the buy access or about the origin. Next, they tell us to look for acid trips. So we know from chapter 2.6 that the in behavior of this, we'll go to where we just divide the leading coefficients since they are the same degree. So limit as X approaches, positive or negative nd of Ethel X is going to be Our numbers won in the denominator is also one. So we have a horizontal Jacinto at Why is it you want now We need to look for our behavior or are vertical ascent Oops. So here that would be where our denominators of zero, which is going to be at exes. He ignited one. So let's first look at negative one from the right of f o X. So wow, negative one from the right. Huge over negative one from the right, huge plus one. Now a number slightly to the right of negative one. It's still gonna be negative. And if we cube a negative number is gonna be negative still, and then if we go and cube, I got one again. It'll be slightly to the right, and it's still going to be smaller, or I guess, larger than negative one. So if we do one minus something larger than negative one, well, that's going to be positive. So here, this is going to go to negative in unity, and we can use a nice little trick, too. Make it so we don't have to go through that same process to find what happens to left. So excuse plus one is what I like to call cubic light around the access or around the vertical ascent. Oh, so since it's an odd degree, it should have opposite in behavior. So this year should be going to positive infinity to the left of X equals negative one and you could go through the same steps. I just think that saves a little bit of time. Our next. I'm gonna combine the next two steps because I think they go together. And we want to find where our function is increasing and decreasing and determine any kind of local max or men's that may exist. So that means we need to find out what? Why primates? Let's run another pitch. So we have. Why is it too excuse over X cubed plus one? And to take the derivative that's regretted it a questionable. So we have remember, questionable is low. Hi, minus high d low all over the square. What's below? And now we can go ahead into derivatives of excuse, which is when to be power rule to give us free x squared. And again we'll get the same derivative for three X plus one. Since the derivative of one is just going to be zero. Now, if we do that little bit of algebra there, This here should simplify too. The re X squared over X cubed plus one squid. All right. And now we can go ahead and set this equal to zero to try to find our critical points. So we'd sit through humor equal to zero, and that would just tell us X is equal to Theo. All right, so we're gonna need to know that information. So why prime here is going to be three x squared over excuse plus one squared. Now, this function will be increasing worldwide prime mystically larger, then. Sure. And I want a hit already solved that off on the side. And actually, even looking at this, we know that our numerator will always be greater than or equal to zero since his expert And in the denominator, since we have that square term there also, that will also be larger than or equal to zero. So we actually are increasing on our entire domain. Are all of the intervals are increasing? Actions say so it's gonna be negative. Infinity to negative one union negative One 20 union zero to and then I remember was distance excluding through because our derivative is actually equal to zero at that point, right? And so for decreasing we have nothing. So why prime less than zero nowhere. So for X is equal to zero, it is increasing to the left and increasing to the right. So that tells us this here is a saddle point. Or, in other words, you gonna have a change in con cavity there. So we also already know 1.0, inflection for this right now. The next thing is, we want to actually find the rest of our points of inflection and our comm cavity in the function. So we're gonna need to know why. Double time it. So go ahead and salt. So you have why double prime is equal to what we're going to use force once again. So it was going to be execute plus one square times the derivative what's in the numerator three X squared, minus than the opposite orders of three X squared times, the derivative with respect to X Oh, excuse plus one squared all over X cubed plus one So before it was squared. So now it's going to be to the fourth power. So the derivative of three X squared verbal abuse power rule to take that derivatives will be six X and to take the derivative of expert Plus once we're bringing me to use power as well as change. So first using power will get to X cubed plus one and then taking the Drew, though excellent plus one will be three x squared. All right, now, simplifying all that algebra there, we should end it with negative six x two X cubed minus one. All over. Ex huge plus one square. Uh, cute. What now? Let's go ahead and send this evil zero. So we confined our possible points of reflection. So that would tell us X is equal to zero or two. X cubed minus one is equal to zero. And that would have where X is equal to the Q Group of four over. So we have our two possible points of inflection out. So we at least already know X is equal to zero is appointed infection. But we also need to see if our other values and so first, let's just say this while you cared about your 0.79 All right, so why Double crime was equal to negative. Six sites times two x cubed, minus one all over X cubed plus one. Cute. Now this function is going to be Khan caged up when white double primates trickle Archie zero And that should give us negativity to negative one Union 02 The cube root of four or two and the function is going to be concave down when Wydell Prime District listen zero which is going to be the interval Negative 10 union que group of four to infinity So putting down are too points of possible inflection. So to be left of zero the function is going to be calm caved out to be right on zero The function should be Khan Cave Oh, and to the right. Oh, Hubert, before over too. The function is going to be fun. Cave down so we can see that all of these are points of in election. Now this is the last thing they told us we should do before we actually start crapping. So let's go ahead and grab. So we had a intercepted believe the order. So we'll go ahead and put that down. We could go ahead and put our Assam totes down as well. So we know at why is equal to one. So why is a good one? People have a warzone possibility and at X is equal to negative one we're going to have are vertical ass into and we know to the right of negative one, we should be approaching negative infinity and to the left of negative one should be approaching positive infinity. So we have no Maxie Germans. But we do have points of inflection at zero and Cuba before over too. So let's go ahead and plot that. So this year will be you brood of three. Okay, Our Cuba before over, too. And we know there should be a collection point. So we went ahead and plotted everything. So go ahead and start to the left of negative one. Since we have no points of interest over there, that's just going to tell us that we will keep on going and get very close to offers on plastic tip and now to the right. Well, we know we need to start from the city and keep on going until we get our origin. And then there's going to be a change and con cavity until we reach about the Q crew of four over two. And then it's going to change from cavity again like that's actually maybe let me try to talk about a little bit better for the change. So so here it's changing cavity, and then it's going to stay like that until about here on Dhe Just got Just keep on getting closer and closer so maybe you can draw the cacophony. Change at you grew a portable to a little bit better, but it should look something along those lines. Now you could go in and may need save water that by value occurs. But for me it I don't think it matters too much since we're just trying to sketch the graph. So I would say this is all we really need to do. We can stop here for the sketching of the graph

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