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Use the guidelines of this section to sketch the curve.
$ y = x^4 - 8x^2 + 8 $
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Calculus 1 / AB
Calculus 2 / BC
Chapter 4
Applications of Differentiation
Section 5
Summary of Curve Sketching
Derivatives
Differentiation
Volume
Campbell University
Harvey Mudd College
University of Michigan - Ann Arbor
Idaho State University
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we want to sketch the graph. Why is it too excitable? Minus eight X work. Plus now in the shockers, they give us this laundry list of steps we should follow and graphing a function. So the first thing they say is we need to determine our domain. Well, in this case, our domain is just going to be all real numbers. Since it is a polynomial, the next thing they tell us to find is our intercepts. So let's go ahead and find our Why intercept persons that so either. Why intercept? Well, this is when X is equal to zero. So x report the next squared in both zero. So we just be like, Why is it now for our X intercept? We want why equal to zero. So get zero is equal to X to the fourth minus eight x squared plus eight. And you might notice that this here kind of looks like a quadratic even though it's to the power of four. Because we can write the X squared square minus eight x word pussy. So that means we can let x squared be equal to well, quadratic formula says negative be so plus or minus be squares or 64 minus for a C, which is going to be buried too. All over two A. You should just be two now this Here, I'll go ahead and simplify two or plus or minus this square root of eight. And since it's a goes to the X squared, we're gonna need to square root inside. So we're going to get squared. Anyone but just X is equal to plus or minus the square root of four plus minus the square root of So these would be are sore values for our ex intercepts. Ah, and we know that four mice to square the debate will be bigger. Zero just because the square root of 16 is four. So I'm burying this word. 16 per square debate even know that four would be larger. So all four of those are defined real numbers. Now the next thing they tell us is to look for symmetry. Intergraph. So Step three iss symmetry. And by that you mean so if we call why after box, we want to figure out what, uh, of negative exes. Well, this squared and the fourth term will turn negative X to just ex. So this would be extra four minus eight x word plus So this is equal to X. And so since f anegative exiting the other banks, this implies the function is even, um or symmetric across. Why access? So if we grab one side of this will be able to grab the others ball, since we'll just reflect everything over, All right, The next step based in you should do is find any acid trip. We know since this is a polynomial, there should be no asked. Oops. But we can use that similar laundry to figure out the end. So we're going to look at the limit as I proposed to Billy of athletics. Well, our largest term would be excessive force. So that's going to dominate eventually. So x to the fourth. There will go to infinity as execution pretty. And since extra Portis even should have the same behavior on bo signs which we also know from the from the symmetry of this. Okay, now, the next thing they want us to find is intervals where it is increasing end increasing. So, actually, so I'll go ahead and put 56 together five and six. You're gonna find intervals or it is increasing, slash decreasing and max slash men's. And I should say, local Max Ash mints. All right, so you figure out where it's increasing with decreasing, we're going to want to look at the first derivative. So why pride is equal to he derivative with respect to X Oh, excellent for minus X word plus eight and taking the grit that that will get you a parable twice so or x cute minus 16. Thanks now to figure out where this will be increasing and decreasing, we need to figure out where wide prime will be less than zero and where y prime will be strictly graven. So when it's less than zero, it is decreasing. And when the strictly greater than zero it is now, I'm not going to actually go through the whole process of doing this, so I'm just going to say what we get when we do solve it. But what we do is we would set that even 07 lost zero our son even greater than zero. The stumbling over my words and salt for that, and then where it's less Caesar would be just the other in trouble. All right, so for where this is going to be increasing our secrecy, we're going to get so are decreasing. Interval should be negative. Infinity, You negative too, You mia 0 to 2. And where derivative will be greater than that, it's going to be negative too. Zero and zero and then to to just make sure. Yeah. So now, looking at this, you might notice that we have these critical points that we would get from here. Also being negative to zero and two series would be are critical, please. Or in other words, you wear. This function here is equal to don't do that. Where this function is to zero people. Zero gives these critical points here. Or you could just look at these intervals and see where there changing. All right. So from negative infinity negative too. It's decreasing. So I left a negative to secrecy and then from negative to zero, it is increasing from zero to it is decreasing. And then from to onwards it is increasing. So what this tells us is explicitly the negative too will be a minimum zero for at zoo will be a max and at too will be a men which makes sense because of the symmetry of our problems. So that covers satisfied at six. Now what we want to do is like a con cavity and inflection points. So we're gonna want to look at second derivative, So I double price so e by the X oh will be found that spurts derivative waas or X minus 60 x. Now we can go ahead and take the ring back so really powerful against. Get 12 times at square minus 16 now too. Bye, our comm cavity. We're going to want to find where why Double prime is big, even zero and where. Why double time is less than and so why double time being bigger than zero tells us where its Khan came up and when it's less video tells us where its contrary down. And just like before you just set this function here equal is it er greater than zero and then go ahead and salt and doing that we will end up with So for con Kate should be negative. Infinity two negative too over the square root of three and it is also calm. Keep up on to the square root of three to Infinity, and the function is Kahn played down on that middle. So negative two over the sweater with 32 two over the square. And because we have this change in Kong cavity at the square root of two for two over this world with three both plus or minus, we get that are inflection. Points are going to be What's the minus the square root of where was a loving too two over the square root of 34 are values of X. All right, let's go ahead is about one more, and we can go ahead and use all this information to finally graph this. Now let's go ahead and first plot, so I'll put everything on the left. And then after that, we could just reflected across the X axis. So we have some exporters up so we can go ahead and plot first. So this first X intercept is one to be negative square root of for plus the square root of eight. This next one is going to be negative. Square root of four minus the square root of a and we have our why intercept at zero, uh, eight. So eight up here like that And we also know that are in behavior will be infinity on each side. So let's go ahead and put back. We know that we have a minimum at negative too, a maximum at zero. So at least we already know what that point. So it should look something like this up there, and and we'll also need to plot the con cavity at two over the square. So using this, we can go ahead and start. So we start from infinity and we're gonna go towards our 1st 0 and we would pass through. And then we know, since now you two supposed to be a minimum, it's going to curve ups at some point like this. And then we're gonna have to hit our next interval. Our next next year's up and this is going to go on until we hit our Why intercept? Because that's our next point. And we don't have any other critical values or anything like that running around. And now all we need to do is reflect this graph across, so it's gonna look something like this here again. And we got the right hand side by just reflecting left It's odd over Y axis due to the symmetry of the ground. So this is just what a sketch of it would look like. You can make a little bit more detailed, but since we just want to sketch and get idea of what it should look like, I think this is more than enough for us to do.
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