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Use the guidelines of this section to sketch the curve.

$ y = 2 + 3x^2 - x^3 $

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Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

we want to sketch a graph of why is equal to two plus three Export minus X now in B chapter were given a laundry list of things we should do when we are trying to go about graphing any kind of function. The first thing they tell us is we should figure out what our domain should be. Well, since this is a polynomial, you know, polynomial ls have a right to have a domain of all round numbers. Which same thing is just infinity, too. Painted furniture. The next thing they suggest we find is our intercepts. So let's go ahead and find our why intercept first. So are why interceptors went X is equal to zero. So we get f of zero is equal to well, the X word and the X Q problem. Go away and we're just left with two. So we get our Why intercept to be, too, And now for the X intercept, we let why equal to zero, And I'm going to be completely honest and say I have absolutely no idea how to go about solving this without using some kind of a calculator or doing things that are waiting well not worth our time. But if we do the rest of the things in this problem, we will be able to figure out where our Exeter ships should be and how many we should have. So it won't be too big of an issue that, at least right now, we have no idea what this value or values of X should. The third step is to find symmetry. So this is to look at what of negative exits. So plug it in half of negative X week. It two, um, the square would cancel that negative out. So just get your X squared, and then the cube will make it negatives we get plus three X, keep hope not, plus three X, but just plus x cute. Now, this does not equal to FX, nor does it equal to negative f of X. So this tells us no symmetry. All right, so we've gone through the 1st 3 steps. The next part is to look for ass and tips, and we know that this year will have no accent hopes with it just being a polynomial. But we can at least use this to help us find our in behavior So let's go ahead and shat. What did limit as X approaches? Infinity? Oh, Ethel X is going to be and I keep saying f of X, which is just supposed to be our Why over here. So this is our necks. If it wasn't clear. So it's a cubic function so in we don't really need to care about the rest of the terms then and since has a negative coefficient. And it's odd then, as experts to infinity, the sugar to negative infinity. And likewise, when this goes negative energy, well, since it's an odd function or since it's a cubic function if positive any ghosts affinity, then this one should go to the opposite. So as exclusive infinity are in behavior ghost negative, Benny. And as X goes to negative infinity, rnb ever does too positive. All right, so we got accidents, or at least are in viator mildly acid. It's now we want to buy intervals of where this function is increasing, slash increasing. So that means we want to look at what? Why prime? It's so why prime is going to equal to so the derivative with respect to X of the polynomial there. So two plus three X squared minus excuse And to take the derivative of those who would just go ahead and use powerful, so derivative of a constant easier. The derivative three X squared. Well, that's one to be six X and then we subtract one from the power. So just have a power of one now and then how Ruan Excuse will be Pull the three out front and subtract one from the power and we get X squared. So let's go ahead and factor this so I can factor out of three X and doing that, I'd get X minus hurt to minus X. And since this is a quadratic with a negative leading coefficient, we know it should look something kind of like that there. So here we have X is equal to zero. So I set that equal to zero. We have X is equal to zero and X is equal to, and the reason why we care about this about a little is we know that wide prime will be strictly greater than or increasing on the interval. So it would be from 0 to 2, so 0 to 2 and then why prime will be strictly less than zero on the rest. So negative infinity to zero. Union zero to infinity, right next, we want to find our local Max is in our local Mims local max slash men, and we already did. Part of the job, because we know X is equal to zero and X is equal to two will be possible. Local Maximo commends Elisa are critical points so we can use the first derivative test to tell us whether these will be men's or Maxim's. So before we get to zero. So from negative infinity to zero, the function is decreasing. So to the left of X is equal to zero. The function is decreasing and then on zero to the function is increasing. So this tells us X is equal to zero will be a minimum, then for X is equal to so on. 0 to 2, we said this is increasing and then, from to to infinity actually comparable error down here. This should actually be two Jew, infinity, Nevada, Kostya. And so then on the other side of two, it's decreasing like that. So Miss at X is equal to would be a max. So we have our maximum. And now we want to look at our con cavity. So khan cavity slash inflection points. So we're gonna need to look at why Double prime. And we already found that why prime is this? And to take the derivative of six x minus the X word, we're also going to need to use power. So the body x oh six x minus the X squared. So the derivative six extra just six, the derivative of three x squared while we move to outfront could be six x and it's a track one from the power, too. And we just get one. All right, now we know that this here is a line that is decreasing. So wherever we find what makes the secret of zero, we know it will be a inflection point, since the left will be positive into the right of that 0.0.0 b negative. So which is going on factor that six out really quickly. It won my sex. So now if we set that equal zero, that implies X is equal to one, and maybe we'll just go ahead and use this to tell us where it's con cave up and calm kept out. All right, So why Double prime will be greater than zero. That's remember, this is con cave up on cable. Well, that will be to the left of our point of inflection. Just by looking at what six minus six x looks like. So this would be on negative infinity 21 And then it's going to be con cave down on the rest of the interval. So one to infinity. So now we have all the information we really need to sketch the scrap. So let's go ahead and plot are why intercept? So we know that supposed to be at 02 and then we also have no idea where our values for the X intercept you're supposed to be. But using our ass into information or in behavior, we know that on the left this function is increasing, and on the right, the function is decreasing. So something we should probably also find out is where are what values are local Max on a local bin are supposed to be. But we actually don't even need to do that because since X equals zero should be a mid, it should curb out like this at this point there and at X is equal to two. It should come up like that. So from this, we can tell that there should be no ex intercepts to the left of zero. So we could just go ahead and cut that. Have you got that? We should just go ahead and connect me todo oh, race instead. So Well, look, something kind of like Mom color. Go ahead, Paul Green look like that. And then we know it's gonna have to connect apparent to. And then after this point, since there's no more local Mitchell local Max is, we know it just needs to keep on going out until the end. So this year would be a nice little sketch of the graph. You couldn't go back in and figure out what its actual value here is. Uh, but for the most part, since we're just trying to sketch it, I would say this is sufficient

University of North Texas