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University of North Texas



Problem 16 Medium Difficulty

Use the guidelines of this section to sketch the curve.

$ y = \frac{(x - 1)^2}{x^2 + 1} $


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Video Transcript

you want to sketch the graph of X minus one squared over, Expert. Now, only when we are wanting to grab a rational function, one good thing to do is go ahead in factor. Everything and we already have it. In fact, form said expert plus one is not fact over the girl number. So we don't need to worry about that one. And now all we need to do is follow that laundry list of steps that is listed in this chapter for us to be able to actually. So the person that tells you to do is to look for our domain. And the only issues we would have with rational function is where our denominator is equal to zero. But expert plus one will always be greater than or equal to one. So our domain ends of just being all real numbers. Now we can go ahead and look for our intercepts. So let's first find our Why interested? So why intercept is where X is equal zero. So, doing that, we're going to end up getting zero minus one squared over zero squared, plus one. Why? So juror minus one is negative. One squared is one So we're going to have one over one, which is just one. So why is equal to one will be our Why? Interesting. And we're gonna find our exercise. So X intercept. Well, this is what why is equal to zero? The ex vice one squared over X squared plus one. Now, we will just set our new Marie to zero. Since no matter what we bump into, the nominee would never make the whole function equals zero, Which would just tell us. Zero is you. Two X minus one or X is a good one. Next thing they want to do is to determine if this function has symmetry. So rational functions are normally not known for being periodic, so we don't have to worry about. But we can check to see if this is even or odd poker for symmetry about the why access or the origin. So we'll look at of negative X and plugging that in. We'll get negative X minus one squared all over negative, X squared plus one and in fact, that negative one out and then squaring that negative one. We just get X plus one squared over X squared plus one and this year is not equal to that of X, nor is he a negative FX. So what this tells us is no symmetry earliest, no symmetry about the why access for about the origin. Next, What we want to do is to find our ass symptoms. So we at least know we're not goingto have any vertical substitutes since our denominator can ever equal. So well, go ahead and look at our in behavior of this function as he goes to a penny and negativity and we know is going to have the same behavior on each side due to the fact that in the new mayor we have a power of expert. And in the nominee, we have a power expert for the largest powers. So all we need to do is to take the coefficients from each and delightful. So in the numerator, if we were to foil that out, it would be just X squared. So the ink overshot that one and alien coefficient, and the dominator is one also. So we have a what was the last hope at why is equal to one or at the line wise? You know, the one I should say the next thing we want to find O. R r intervals of increasing and decreasing. And we want to find any local max for men's until these air sets five and six. But I'm gonna go head to him in one step. Since these do go together next, we want to. Five. What wide problem is unable to answer where is increasing and decreasing. So let's go ahead and do that on a new page. So why prime is equal to X minus one square over X squared plus one, not white crime. That's just why so stick the derivative of this. We're going to need to use the portable. So questionable says Low D Hi minus. Hi Hello, All over the square of what is below X squared plus one. We're going to square that now to take the drone of ah X minus one squared. We're going to need to use change rule and power rule. So powerful says move that two out front and we'll have that times X minus one. And then we will need to take the drift of the inside of ex vice one just one. And now to take the derivative of X squared plus one. What's gonna be two x for Explorer and the drift of one? Just one. Now, if we were to go ahead and do all that algebra and simplify and do a little bit of factoring will end up with two x squared minus one over X squared plus one squared for white crime. And we could go ahead and find where are critical about this They're gonna be so that just implies that X squared minus one is equal to zero or exes between plus or minus one. So we know possible maximum ends at exit with customized one, and we also know wide. So it's like that down now. So why prime is too X minus X squared minus one all over X squared, plus one squared. Now we want to find where the function is increasing first and this is where why Prime is strictly guarded zero. So going through the algebra. For that, we will end up with that. This is interval negative one to negative one union, one to infinity and then you can find where the function is decreasing by looking for lead. This function is strictly lessons here, and it's just going to be our remaining intervals here. He's gonna be negative 1 to 1. Now, we had those critical points there exceeded the negative one, and X is equal to one. So let's go ahead. Ends used the first derivative testes. If these air Max is or men's so between one and one, the function will be decreasing and to the left of negative one, it will be increasing. And after one it'll be increasing. So that tells us X is in the negative. One will be a local Max. Go, Max. And this will tell us sex, Physical one will be a local men. Then the next thing we want to do is to look at the com cavity of the function and look for any points of inflection. So we're gonna need to know what? Why? Double crime mysteries so defined by double prime, we're going to be too used. The potion rule again. The person I'm gonna do is factor out that, too. And that would just go ahead and write everything. So it's going to be low, be high and just to kind of safe since space. I'm just gonna go ahead and take the grim little expert by phone, which would just be so Q x and then minus Hi do you love and the derivative of our denominator We're gonna have to used changeable and question will get a tingle and powerful. So it was going to be too x squared, plus one times a drove of our inside puncture. If that helps, then we're going to be too square whatever we have in the denominator. So it was squared before. Now it's going to the fourth power and now we were to do all that algebra. We can simplify this down to. Negative for X Times X squared minus three. All over. Ex player plus one You. Now we want to set this equal to zero. And so we only need to set a new married with zero again. So we get X is equal to zero or X squared minus three is equal to zero by your apartment property. And we would get this. That X is closer. My sis work with three, which is approximately equal to plus or minus 1.73 So now we could go ahead and figure out where it's coming from. Down can keep up and we can deter. But if those points we listed are actual points of inflection or not, so we have negative or ex experience nicely all over. Expert Cross one. Keep your wife. Now it's going to be con taped up with y double planet strictly. And that is going to be the interval. Negative energy to the negative square. Three union with zero square with three. Then it's going to be Kong Kate down when white double time, it was strictly less than zero and it will just need the rest of the interval. So negative square with three to this work with error to zero and then you get square with debris too. So we had X is equal to negative square with three excessively zero and X is equal to the square. So we know the function will be concrete up to the left of the negative square with three and also between zero and the square with three and on the other interval is going to be calm down. So at each of these values, we do have a change of kong cavity. So all of these will be points of lecture. I'm just not a little bit. And this was the last thing that they said we should go about doing before we start graphing. So let's go ahead and plot some of the points we know already and then maybe decided we need to I want some more. So, uh, we have the points 00 or not. The point here is there, but excessive become one as our intercept and our why intercept is going to be 01 So we have those two points that we can plot. We know we're going to have a or his on plastic. Why is it too one we know at exited with a negative one, We're going to have a local max. So just put Max below this and at X is equal to one. We're going to have a minimum, so we know that this here will be a minimum point. Um, we also are going to need to use are in points of inflection. So this is going to be a regular square root of three. And over here will be square root of three. And so we know at those points are graphs should kind of blotting out a little bit and change cavity, so I could actually put those days. We don't know if we're gonna be above or below the words on class into yet. What? So let's just go ahead and start with this. Now let's start from R Y. Interesting. We know that we're going to start there and go ahead and go to our X intercept. So we're going to go ahead and touch that, and we know this is going to be a local minimum. So we should go ahead and balance up afterwards until they hit the square root of three. And then it's going to change from cavity. So we're going to approach horizontal acid hope from the bottom as we go to infinity? No. Or why is it too or excessive? Zero. On that interval, we know the function should be decreasing. So that means after this point it should be going up until we reach our maximum exited with a negative one. And then it's going to keep on going until we hit. Negatives were three. Work on cavity will change, and at that point there we will approach are horizontal Aston toke from above. So the accident need to go ahead and plot any more points. But sometimes it's nice to go ahead and say with this, Max is and maybe even the points of inflection. What values those? They're supposed to be up since we're just trying to sketch it. I think this is good enough for what we're trying to do.