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Fiud the horizontal and vertical asymptotes for the following fuuctiousf(r) 214V4ro9112r# f() 318 212. Fiud the slant aud vertical asymptotes for the following fuu...

Question

Fiud the horizontal and vertical asymptotes for the following fuuctiousf(r) 214V4ro9112r# f() 318 212. Fiud the slant aud vertical asymptotes for the following fuuction 4r" + r +7 f(z) = I2 _ I+]

Fiud the horizontal and vertical asymptotes for the following fuuctious f(r) 214 V4ro 91 12r# f() 318 21 2. Fiud the slant aud vertical asymptotes for the following fuuction 4r" + r +7 f(z) = I2 _ I+]



Answers

Find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. $$ R(x)=\frac{8 x^{2}+26 x-7}{4 x-1} $$

So we can see here we have our acts. Eco's to eight AC Squire last 26 acts vinyl sour over four X minus one So we can see the degree off our numerator, Rico, so too And the denominator You costa one. So it has a bleak ah, same toast and we are guarded by along the region. So here we how a tax choir bluffs your 26 acts Mina Salman. So here will be four ox minus one. So here you know how to act. So a tax choir minus eight axe and I know it's true. I started minus two ox. So here we're have 28 ax minus seven. So here we have Ah, savin on May to have 28 ax minus seven re Jiko suit zero. So our function our our ox year Rusty called to Truax process Allen. So you also be it probably

This question is asking us to find all horizontal and vertical symptoms, if any. Okay, What we know is that an irrational function vertical access codes for the zeroes of the denominator. So if we have explains to reset this equal to zero and we have four X plus seven for this equal to zero, they can actually solved. We get access to an ex is negative seven divide before these are vertical ascent tuts now horizontal ass and trucks are determined when we look at the degree of the leading term of the numerator and regards to the denominator so we can see the degrees are the same, they are equivalent. Therefore, we know the ratio of the coefficients is the horizontal accent. Oh, so to over four is 1/2 So 1/2 is why is in this context the horizontal acid took as we established X equals two and X equals negative seven. Divide by four are horizontal. Are vertical ascent tips

All right. We are looking to find the vertical, horizontal and oblique ascent tops for the following rational functions. So, here's the rational function I have. I want to look at any acid hopes that I've got. So even before I get to that point, I need to set up my factory factory is always the way to go for looking at acid tokes than anything else. So on the top I've got a try no meal. So I need to find two numbers that multiply to my outsides. So six times seven is negative 42 and at my middle, which is a positive 19. So, the possibilities for 42 there's lots one times 42 2 times 21. Uh three times and other stuff you can keep going but already have run into the one that works, which is two and 21. And then because we need to get to 19 1 and we're multiplying two, negative one of these has to be negative and in that case would be negative too, so negative two plus 21 is 19, so going through the process of back drain it um and you know what? I know it's not actually necessary to answer this question, but it's always good practice to get into this habit. Every time you see a rational equation. So doing that, you then look at your first two and your last two and try to take out something common. So six X squared minus two X. We can take out a two X. That would leave me with a three x minus one and then 21 X and then minus seven, I can take out a seven and that will leave you with three x minus one. I've done this right, my bracket should match I end up with three x minus one and two X plus seven. So my top factors 23 x minus one in two x plus seven. And my bottom factors to three x minus one. So your next step then would always be to do the domain. This question doesn't ask for it, but habits or habits. So in this case my domain, my bottom that would make it equal to zero would be one third. And then your next step is to simplify. So three x minus one and three x minus one, cancel. Ouch. Which means what I'm looking for here. My final graph would be two X plus seven. Uh Looking at that, that means there are no vertical isotopes because that canceled out. Uh But there will be a point of discontinuity at one third. There is no horizontal asuntos because that is a linear graph that we're going to see. So there is none of that, even though and then the only case that we would normally see, because the degree is higher on my top than my bottom, we would normally see bleak line. Um but we won't even see that in this case because I don't have a rational equation anymore. Once it's simplified, so I would say your best. But you don't have any asuntos. So instead of no vertical ascent, okay, you have no acid hopes whatsoever. There is a point of discontinuity. There is a hole in the graph though at X equals one third. And that's because of that domain restriction. But other than that, you're gonna see the graph of the straight line has a slope of two and an intersect of seven.

Because the appeal are doing well. So we've got our equation written out right here and were asked to find the vertical and horizontal acid to this function. So to find the vertical Assen toots for this function, when a set the denominator, the function equal to zero, we got X minus two times for explosive. Seven. Who is zero order? In order for this to be true, we need to make one of these factors on the left inside, equal to zero that will make the whole equation equals zero. So the two possibilities for that are X minus two equals zero and for its plus seven is equal to zero. So for X minus musical this year, if you add two to both sides to get X equals two, it's our first vertical ass in tow. For this for X plus seven equals zero. You subtract seven from both sides you get four X equals minus seven, which gives us the course vertical lassitude of solving for X. If you divide both sides by full get X equals minus 7/4 which is our second vertical got. This is our final answer for the vertical ass into its for this function? No, we want to find the horizontal ass and tits of dysfunction. So to do so, we need to compare the highest values of highest exponents of excellent numerator to the highest exponent of X in the denominator. So in order to compare that, we need to expand this thes numerator and denominator and convert it from factored for to expand it. So for the numerator, God moves playing this out we have two x squared then plus two x minus three x gives us minus X and then minus three times one minus three. Then again, for the denominator multiplying that out, we could. Excellence for X is for X squared minus three times for X is minus eight Action sometimes X plus seven x minus eight X plus seven X is minus X then seven times minus two is minus 14. This is our expanded form of our equations. Now we want to do is we see here that the highest exponents of X and the numerator is too in the highest exponent of X and the denominators too as well. So because those are the same are horizontal. Assen Tote is going to be the the ratio of the coefficients of these highest exponents of X. So this ratio is going to be to over four, which is equal to 1/2. That means that for this equation are horizontal ass into is why is equal to 1/2. And this is our final answer for the horizontal last into all right.


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