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Sketch the graph of the rational function without the aid of your GDC. On your sketch clearly indicate any $x$ - or $y$ -intercepts and any asymptotes (vertical, ho...

Question

Sketch the graph of the rational function without the aid of your GDC. On your sketch clearly indicate any $x$ - or $y$ -intercepts and any asymptotes (vertical, horizontal or oblique). Use your GDC to verify your sketch.$$M(x)= rac{x^{2}+1}{x}$$

Sketch the graph of the rational function without the aid of your GDC. On your sketch clearly indicate any $x$ - or $y$ -intercepts and any asymptotes (vertical, horizontal or oblique). Use your GDC to verify your sketch. $$M(x)=\frac{x^{2}+1}{x}$$



Answers

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x^{2}+2 x}{2 x-1}$$

No, we have given with here. Why equals toe FX equals two X squared minus six, divided with X plus two. Now we have the first fund, the domain, then a symptoms than zeros. Then why intercept? And then we have biscuits. Looker. So now first we're going toe, find the domain so domain will be minus infinity to minus two. Minus toe in union minus you, too. Infinitely. So, no, this is a moment. And now the working girl a simple will be here X equal to minus two. No, we're going for the solution. Zeros Excess square miner Teck sequences that wins X equals zero is a job like toe affects. And now, for the four point the public sm thought so public esteem told to find obligation towards delight. Numerator with denominator. So this will be X. So this will be excess squared minus two x. So this will be minus X squared. And now this with three x no multiplied with three minus three minus three x on. Now this would be minus six bless and this will be also place We were left with the question Why equals two X minus three is the equation off public. A simple no. We have found all the five points and now we have to find the graph. Now it's getting the graph. Now. This is a question. Why equals two X squared minus six X plus two. Now, this is a symptom, and this is required solution and the photograph?

So we're looking at X minus one over X plus one things going on here to find the X intercepts that setting the numerator equal to zero. Um, so I mean, just set X minus one equal to zero. That's a numerator s as you solve this and had won over the X intercepts. That one. Um, next thing I'll do is find the Y intercept, and that's putting zero in for X. So in the numerator, just right now. Zero minus one in the denominator. Get zero plus one. Get negative all over one, which is negative One, There's your y intercept. Uh, put box around both these groups was ugly. Fixed that x intercept at one y intercept in native one. Um, the Assam totes. Now me, The horizontal Assam totes is best to explain. The degree in the bottom is equal to the degree in the top, which tells me that you divide your leading coefficients, leading coefficients being the number front one and one. So, uh uh, when you take one divided by one you get why equals one? No, that's right this way as the horizontal ask him to. And the vertical lassitude is setting the denominator equal to zero. So that would be X plus one equals zero. So the vertical line for the bird class into his negative one eso as I graft. So there's change cultural quick across the X axis that positive one across the Y axis at negative one, we have a horizontal Assam towed out. Why equals one and we have a vertical aspirin. Duda y X equals negative one. So you can definitely see in this portion of the graph We have our curve if you want to verify. But look at the to the left of the vertical aspecto it is going to be above in this section. You can pick any point you want for X. Let's say you picked negative tool. You get negative three on top over, um, negative one and negative over Positive. Sorry, Negev overnight was positive. So it is going to end up in this quarter. Um, it probably makes sense to work it out and actually make it table values what your teacher might be asking you to do

Right. So, looking at two x over X plus one squared, the first thing that you want to do is find your ex intercepts. So as you set the numerator equal to zero and you divide by two, you get X equals zero is your ex intercept. Um, which also makes sense that you're y intercept will also be zero. Because if you were to plug in zero for both excess here and here, you have zero over something. Uh, 0/1. Um, but if you were to look at an exit receptive zero across that 00 the next thing I would do then is find the horizontal. Ask him to, uh, which you can see. That denominator has a bigger degree. So that tells you that why equals zero is your horizontal Assam talked and the last thing to discuss is a vertical ascent. Oh, not setting the denominator equal to zero. And that happens at X equals negative one? No. Ah, what else do we know? Well, um, may the best thing to point out is ah, uh, like, if you were to make a sign chart, um, doing negative one, which happens twice because of the being to the second power and zero depending if you are ever taught. The sign charge is ah plugging in zeros and from your negative ones and zero in for zero. And then negatives toe left Positives to the right. And what this tells you is, if you have a knack of times nine times a negative, you'll get a negative answer. So to the left of negative one, you're gonna get old negative answers in between positive his father Times negative is negative. So it's still gonna be negative to the left of X ago. Zero. But then to the right is gonna be a positive. And what happens is it goes up and then still goes back to the horizontal Assam touch. Um, now, unless you know how to use a sign chart, this might not make a whole lot of sense. The next thing I would say is, just created a table of values and see what happens when you clog in X, you know, values like negative three negative too. I would skip negative one because it will get zero, sir, you born because they will get you an undefined value. Ah, but maybe try like negative 0.5 and then values to the right 12 and see what you get. And then that will help you understand what this graph actually looks like.

Question number 49. So, in this case, the national function has given us Texas Square a little on many successful mediator fanatics. A square over. There's nothing but the difference of perfects waas, and it was used the formula. So that will be excess. Where one classics and one minus six. Let's talk about the article has improved to the point where the dominant receiver to speed up the process it's one minus X is equal to you. This means that we have cases. One plus X is equal to $0.1 minus X equals zero. This means that X is equal to minus one on ecstasy. So these are the true 40 pleasant girls, if you don't reward offers on Ellison Gold in the video of you both the American and a dominatrix. Same stupid. So it would just be the reassure. The coefficients under new burrito professionals One denominator, it's minus one. So the horizontal assumed is just why is he going to minus one? Okay, that's enough about as indoors then. Stoppable, have my intercepts supervise intercepts. We have xn zero when people executed zero. From here, we're definitely good affects at zero. So we have origin has a point from which the draft buses and one last thing is We've talked about the sign of effects. So we have. Ah, and I think that this first going weren't proficient to be positive because over here, the proficient of excess Negative. So we have produced some additional work. Have been saying that effect is you were excess. Where over one classics give you have X So we have one Plessix one minus six. Now if the of if we have to meet this altogether, then we divide minus and minus one. Bold, sir. So we have extra square with the same changes. So this intervenes. What? Here will be X minus one and this becomes lessons you because we have prepared a minus one? No, we console for this equation. So we have the who says minus one and the other groups us plus one on. We hear the answer. So it this positive van This is negative. So for this less than zero, we have toe, probably because we're you go will go like this. So this is positive and this is negative. So the region off the negative would be Oh, well say that this between minus one on one. It's negative on for the rest of the, uh, part. It would be positive, but for minus one and one when your dog it's *** dude, that does. Actually, when the original function is positive because we have more to play, minus one would states so So between minus and one origin function is actually positive. So this is there? Nothing for in for enough information about the crowd is larger? No. So we have these as Gordon in Texas, and then we have vertical ASM goals as X equals one. So this is X is equal to one, and this is X is equal to minus one. And we have horizontal as importers by is equal to minus one. So you have, er y is equal to minus one. That's name D. So you have be advisable. Minus one exercise will move on. Next is equal to minus one. So between minus one and one of the Clovis positive Onda, it passes through the origin as well. So the court would be like this. So this is for that on the remaining part would be negative. So, uh, I will start from Here we go like this. This is the tragic tree. Andi Slake weighs over here. So this is how the Cobell look like.


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