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Give a graph of the rational function and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes and labe...

Question

Give a graph of the rational function and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes and label them with their equations. Label point(s), if any, where the graph crosses a horizontal asymptote. Check your work with a graphing utility.$$ rac{4}{x^{2}}- rac{2}{x}+3$$

Give a graph of the rational function and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes and label them with their equations. Label point(s), if any, where the graph crosses a horizontal asymptote. Check your work with a graphing utility. $$\frac{4}{x^{2}}-\frac{2}{x}+3$$



Answers

Graphing Rational Functions Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer.
$s(x)=\frac{x^{2}-2 x+1}{x^{3}-3 x^{2}}$

All right. So for problems, 7 45 were given this function. And we need to find its intercepts Assen toes and to sketch a graph of it. So first, we're going to find this intercepts now for its Y intercept. You're going toe sets of the access to zero. So it's gonna be a zero plus for over a zero spread nice two times their own lines. Three, Which doesn't give us negative 4/3. So our y intercept just gonna be zero negative 4 30 And how far are X intercept we set? The function equals zero. So it's gonna be X plus for over extra nice to ex Yukos zero. Multiply both sides by the denominator and you just get X plus for e cigarettes. Zero an extra, be negative four and those. Our X intercept will be negative for zero. And now for the Assen toes. So we'll start with the vertical Assen toes first and vertical ass in total, meaning that the function is undefined at that point. Yeah, and usually a function is undefined when the has a zero adds the denominator. So let's see if it's possible for the denominator X equal to zero. So we're gonna set it in on their equals zero. So, like us. And we noticed that is a fact. Herbal quadratic equations. So because factor and we noticed that the denominator is equal to zero in access eagle toe negative one or three. And these will be our vertical ascent owes so and now for our horizontal Assam toes. Well, there are, like, three rules that you basically just have to remember on. Basically, the girl has fallen. So really, I get up there, uh, won't. Uh okay, So basically they go us phones. If the new Meridor's degree or the highest exponents is a smaller than the denominators degree than the ascent Otis, the X axis If the numerator is degree, is larger than the denominators degree, then there is no horizontal ass until you have a slant asking toe instead. And if the numerator and the denominator is degree, are the same than the Assam toe Izak, why a sequel toe the leading coefficient of the numerator divided by the leading coefficient of the Denon near. So let's check the degrees off our function. So the highest exponent of the numerator, it's one and the highest exponent of the denominator is too, since there's an X squared there. And as you can see, the degree of the numerator is smaller than the degree of the denominator and thus will go with roll number one, which means the horizontal asking tone It's gonna give the X access or why thing with zero? And was that weaken go on with our ground some of the plot that accedes and, well, first of parts of the points and the S M toes that we know. So we've got that intercepts the Y interest of being zero negative 4/3. So it's going to be here and the eccentric Septus Negative 40 So it's gonna be here and now for the Assen toes. So the vertical ass symptoms are at excessive growth. The negative one and three. So was one be here and another one was convening some around here and the horizontal ass until is as described before that before the X axis. So we're just gonna talk the X axis, and with that was storage graphing our curves. So we first a craft like a where we know there is definitely gonna be a curb, so starting this in middle section and we see that the Y intercept is below the horizontal access. I mean below the horizontal acid toe, which means there is definitely going to be a curve down here. So reach the wines intercept, and there's no, like the X intercepts in this middle section. Which means this curve is gonna stay below the horizontal Assam tone on gay. Something like this and on the rights the curve is gonna be of here. Andre will be slowly approaching the Aston toes. And the reason why it's up there is because, well, that's considered to grass. The wire, your secrets, right? Sorry. He's the wise. You were the one over X craft and the wise you got one over exploited graft. So the Y over because toe one of our X craft looks something like this if excessive guilt of one, then why signals of one if excessive for the negative ones? And why a secret in the negative one. So I think, as you can see, if you divide a number by negative numbers and the results were also being negative. But for the one where X squared graph is gonna look something like thus because if excess of one, then why is going to get one? But if excess negative one will negative one square, this positive one, which means why it's also gonna be positive. One. It's like a second. See if he asks if the critical Assen toe has some are the multiplicity, then the curves are going like swamp spaces between, like the open down size, not try described that. But if the vertical Assen toe hasn't even multiplicity, then the curves are going to stay on the same side of the horizontal house in tub. But in our case, all of the vertical Assen toes only occur ones. So there are all the multiplicity, which means the curves working, the switch between like a glove or below the horizontal acid, which means on the left side. The curve is also convenient up here. But it's going across the horizontal Assen toe since there's an X intercept like this, and then this guy goes slightly back of and then slowly approach, but never actually touched the horizontal massington. And yeah, this is a viewer graph

For the following problem. We want to make a sketch of the graph F If X is equal to X square plus two x minus three. You better buy X squared minus work and to do show we need to find the X intercept. It's why intercepts and it's acid jokes as well as the end behavior of the function to start up, we're gonna find its X intercept. And so any X intercept happens when FX Physical Desert. And so, in this case, we're gonna set everything on this On the right hand side of the equation X square. Let's do X minus three, divided by X squared minus four. Equal to sure we're gonna multiply everything through by the term X square minus four. So the left hand side of states that x square plus two X minus three and the right hand side of states of zero. Now we need to solve this equation right here. X square plus two X minus three people. Does Europe introduce show? We're gonna factor this as X plus three an X minus one. And now we're gonna solve each one of them separately. So we'll have experts. Three physical does era and X minus one difficulties Euro. Now we're gonna mess Obstruct three on both sides for this equation and we're gonna add a plus one on both sides for the second equation to solve for X. So the first question says that X is equal to minus three is a zero, and the other one states that X equal to one is another zero. So we have the points negative three commas era in a one comma zero as our X intersects. Now we're gonna work our Y intercept, and so are y intercept happens when X is equal to zero. So we're gonna evaluate F effects and X is equal to zero. So this term's cancel over here. So negative three divided by nearly four, which is just three over four. So why Interceptors have zero? Come on, three over four. Finally, we're gonna work our asset jokes. So for the vertical ascent oats, we want to set the denominator X squared minus for equal to zero. If we add a plus for both sides, Insults for X, we have that X square circle to four. Now we're gonna take the square root on both sides, and so X equal to plus or minus two. So they start to ask him toasts to critical mass in tubes. In our horizontal asking chokes happened as ex approach infinity in a sex approach Negative infinity. And so when X approaches infinity or negative infinity the only terms we need to take a look at all right Peace to excess square. So we can say that if effects it's approximately x square The bed of my ex A square which is equal to one and so s x goes to infinity are negative Infinity FX will just approach one which is just a line Why sickle to one? And so this is all information that we need to make a sketch of the graph. So you have summarized everything over here the ex intercepts, winter sip and all of the three acid jokes. Someone start by drawing the horse until s core isn't a lesson. Toot, she said the line Why sickle to one which is this point right here He's dotted line will represent the line bicycle to one with green I'm gonna mark all of my ex my vertical ASM totes which are X equal to two. She's just one right here. And an ex icicle. Negative, too. Now I'm going. I'm going to plug this. Three points were here, so we'll have to zero at one comma. Zero are y interested at zero comma, three forts and our another X intercept on negative three comma zero now went into the Germany. How does the craft looked like? Introduced. So we're gonna take a look as the behavior of the function as he approaches the essence jokes, these vertical acid trips and the horse until one. So for the 1st 1 we have the line X equal to chew. And so us X. So we're gonna use limits over here, So we'll have that X approaches. Infinity. We know we want to know what happens to FX. So just a reminder. F if X is equal to X square plus two X minus three Invited the X squared minus four. And so that's six approaches. I'm sorry. It's not infinity over here, but in this case, we're gonna want to find the behavior of the function a six approaches to from the wrecked. So a sex approaches to from the right will be this point over here or this one. So a six approaches to from the right. We're looking at X battles that are greater than two. And because of that, we'll have a denominator. Sorry, The Dominator X squared minus four will be a positive small value because we're looking at Bally's. That approach to from the right. So we're looking at something like 2.0 Sierra Sierra one, and so the square of thes it's gonna be created them for. So this valley over here, the denominator is gonna be positive. And because we don't have any zeros on this section of the graph, there's one right here. We can assume that either these values positive for these values negative. However, these a square right here it's at least four for the lease for this section six separate, just two from the right. So we can safely say that this FX as it approaches to from the right, it's also gonna be positive. So we'll have a positive number. You have a positive number, however, we'll raise it that two point Syria 01 square. It's gonna be greater than for. But the difference between these two values is gonna be relatively small, so I mean I write this. So the difference between 2.1 square minus four, it's gonna be close to zero. It's gonna be some decimal point. Something close to 10 to the negative five or something like that is just an example to this case. We're gonna be dividing a positive big valley. You got about a small, positive value. And when we do that, we're gonna approach, if effective, approach infinity. So is gonna increase a lot. And as we have already said, SFX Sorry. Ass X approaches Infinity. FX will approach one. So we're gonna write these in this form. So in this case, in this section, if affects is approaching infinity, we're here with this line, and with this line, we're sitting that ffx approach us one. Now we're gonna take a look at at X approaches to from the left. So US X approaches Jew from the left. It's over here. We're gonna take a look at what happens as except purchase two from the left for ex valleys. There are in between two in one because we know that we have a zero over here. So we're gonna take a look just a this section right here. So for this section, look, have FX. We're gonna make the same analysis over here is equal to for this case. Let's take the 0.1 point 9999 over here. We can see if, Lucy, that these value over here it's a steel gonna be positive because we have this Jew times this value one point on and on and on the square is gonna be close to four. So we can say that our moment numerator is gonna be old, so positive. However, 1.9999 square he slays them for because these numbers listen to and so the difference between these two valleys. So the difference between these two values, it's gonna be a lesson. Zero. It's gonna be almost zero, but it's gonna be a negative value times something like 10 to the maybe five again. So we have a negative sign over here and something really negative small. So we have something positive and something negative. So we have a negative, we're gonna be approaching something negative. And because this valley on the denominator is really close to zero again, we're gonna be a dividing a positive big number by a small negative number. So we're gonna be approaching negative infinity. So this line over here will purchase of infinity. We can assume that these two points connect each other. And now we're gonna take a look. Someone a real right. I want to take another words. So we're gonna rewrite equation if effects is equal to X square minus two X plus three. I'm sorry. Plus sorry. Plus two X minus three. So this must be a negative sign. 83 divided by X squared minus four. Now, we don't want to take a look at what happens that X is equal to native to. And so, in this case, as ex approach approaches negative, True from the right and so far from the right will be this line over here. And in this case, we'll have our FX sequel chair. So we went to make the same analysis. So let's take a point over here. Let's take the point. Native 1.999 I serve volley for X, and so this value of was substituted on the denominator. We know that this is gonna be a negative value and because it is negative. We already know that this is gonna be lesson for so excess square we're here. It's gonna be a list them for. And we're hearing these values at least five. And so our nominator it's gonna be negative because we have this tunic valleys that are greater than this X square for something like negative 1.999 So we have a negative balance. Now our denominator over here, we also have a negative value because excess squares listen for the difference between exes Square and four, it's gonna be something negative. And it's gonna approach Ciro. So we'll have negative divided by negative. And so in this case, F effects will approach something positive. And as X tends to get closer and closer to negative, too, from the right, this difference will become so small that we divide when we take the product of this division FX shotgun approach. Infinity. So now this section we look something like this, Finally, we need to determine what happens. That's except purchase Negative too. Oh, sorry. As X approaches negative too, from the left. So we're gonna do the same analysis. So in this case, we're looking at Bally's effects that are in between negative, too. And three. So for X in between that are greater than negative three. But listening to. So let's assume that we have true native to 0.1 That's her Bali for X. So our new mediator in this case, if we square these value, we're still gonna have a negative the nominator, because these value we're here, it's gonna be greater than the sex is square. So we can say that to X minus three. It's gonna be a crater. I'm sorry. So the magnitude of two X minus three it's gonna be greater than the Magnitude Defects Square, and so in this case will have a negative value. We're here because this to X minus three is gonna be greater than the sex is square. And we have a negative right here. This native number will be larger than this positive number. We'll have a negative value. Now let's take a look at our denominator. So negative 2.1 square. It's gonna be greater than for. And so the difference it's gonna be a positive value. There s but it's close to zero as well have positive over here. So if effects of purchase something negative because we're dividing negative by a positive Bali it was The difference between X and two becomes a smaller from the right. Sorry from the left, we're gonna be approaching a negative is a small value. It's small, positive value. So when we divided native value by a small positive value, FX is gonna approach negative infinity. And so this section of the graph right here it's gonna look something like this. Finally, we know that if effect is gonna approach one, it will never be one. And because of that, we're gonna connect this craft like this. As X approaches, Infinity effects will approach one. And so, in this problem, we made this sketch of the graph F of X is equal to x square plus two x minus three, divided by X squared minus for which is this crap over here? My obtaining the ex intercepts? Why intercepts awesome jokes and taking a look at its in behavior and also the behavior off the craft as it approaches and its vertical aspirin jokes

So here's the grapple way that sequel. So two X men 6/4 minus x So we have the question Why that this equals toe bull X minus six over there. Four minus x. Who said the That it does your This is probably be so we have the data beteen off caution. So we have deal crime My nose be meal prime over B squared soil Doing How of these? Four minus X the baby But the book toe X minus six piece toe My boss, we are from the bees for a week, you'll be remedies toe X minus six and the labour people the East negative born. So we have over four minus x Good. So we have four times to the tee Sequels toe eat. Here is two X minus six. So we have negative express toe so we have good new. So this tribute. So we have eight minus toe x my nose class because we have one years were who x my most six all over four a minus x splayed. So we have to Ober four minus x squirt. But this slight brain here is like brain, my friend so setting while from 20 there Isno rial number So little shun. So here we have no extreme my point. So you have stick the ASAP thought so. Subsidies that we have to set up the nominator to zero so that we have the practical are Scinto. So you have four minus x Q zero So x icicles for he sees the vertical ass in throat. So in horizontal a simple we use the hospital. You're so the limit No X box to impede That is steady But ive off the numerator that is still and the didn't didn't did inevitable denominated its negative one so and its limit It's getting toe So the no more sandal up as in top piece Why is he goes to negative toe? Do you have to take the crab? Closest order center costs Ask implode. So we have toe subjected toe Eichel stew negative too. Did Ah, we've medication. So you we have got eight No store X not because Toto X minus six. So here we have. So I said no. I think that crows sing and this is stopping up

So you have the occasional equals eight Auburn wanted the X squared mints for So we have the body off. Basically shown here. We have toe everybody with So we have toe yet? It's terrible. TV must ah, power function. Didn't believe a power bang shown that this equal store the is she was my prime Callisto. And you and minus one deal Peace equations Liberty from the Given. They are ugly. Ben Equation, the retributive off power relation that is here. So let and the snake at one you'll ease. Exclude minus flor for the u E f equal store, two x So we have the question here, so we'll fry. Um, she's the it the and we have the end that this negative one that you East exclude Bind us water then and questioning at the one when he found that this negative sending you if she calls topo X orifice equals toe prime with this equals store negative. A time store call 16 knicks. Yeah, a lover X squared buying this supporter squared. So this is the derivative off this aggression setting set by my prime. But the Sequels to Ciro. So we have surely sickles toe negative. 16 x all over X squared. Find this for squared. So we have Syria or that the sequence tonegative 16 x So we have the value off extractive equals zero. This is the stationary point off off the ground. So let that the our lead a nominee Tor expert mile ish for not equals zero So at least because the zero there is a s said off leather up So you have expert at the secrets before so the ask him but Mexico's toe toe and negative So the system horizontal ass implode So we have another is the horizontal ask him toe to used Ah, with that degree by both side of the numerator and denominator by experts So we have it X squared Lauber one minus four X squared. So here we have the limit off x Boxing Day D c closed toe hit over X squared browan my nose Moreover, expert So we simplify to become hero in this 1 g zero. So the reason it'll ask him told if zero r it this way that this equals 20 So crossing we said That's why calls to zero that this might take waas toe eight over X minus squared. So we have No, this is seriously greater. Done. Eight or eight. I rest on it. Serious listener are eighties created no. Zero. So we have no cross seeing horizontal, A simple


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