Give the equations of any vertical, horizontal, or oblique asy...

Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4.

$ƒ (x) = (x^{2} + 4) / (x - 1)$

Accepted Answer

Hey, everyone in this problem for the following function, we're asked to write the equations for the vertical, horizontal or oblique asymptotes. The function given is F FX is equal to X squared plus divided by X minus eight. We're given four answer choices. Option A, a vertical asom tode at X equals negative eight and an oblique Asymptote at Y equals X plus eight. Option B A vertical Asymptote at X equals negative eight and an oblique Asymptote at Y equals X minus eight. Option C A vertical Asymptote at X equals eight and a horizontal Asymptote at Y equals X plus eight. And option D A vertical Asymptote at X equals eight and an oblique Asymptote at Y equals X plus A. We're gonna start by rewriting the function we were given, we have F of X is equal to X squared plus divided by X minus eight. And let's start with our horizontal asymptotes. Now, when we're looking at whether or not we have horizontal asymptotes, we wanna compare the degree of the numerator of our function to the degree of the denominator. So let's look at the degree of the numerator, the degree of the numerator is going to be two, right? It's the highest exponent that we have in this case, that's two and the degree of the denominator. Well, that's just one. OK? We just have X minus eight. The heights exponent is just one. So we have the degree of the numerator is greater than the degree of the denominator, which tells us that we have no horizontal as to for horizontal atom to, we need the degree of the numerator and denominator to be equal or for the degree of the denominator to be greater. OK. So we have no horizontal asci to so we can already eliminate option C OK? Because it has a horizontal ascent tode in that second part of the answer. So we eliminate that answer and let's move on to our vertical ascent tode. Now, our vertical asymptotes are going to occur or the denominator is equal to zero. And when I say that, I mean, after simplify, OK. So we want to simplify as much as we can. We wanna factor and cancel any terms that we can and then look where the denominator is equal to zero. Now, in our function, let me go back up to it. We have X squared plus 16 in the numerator. We can't factor that any further. OK? We won't have real roots, we won't have nice real factors. And so we can't factor that. So we actually can't simplify any further. And so we're just gonna look at where that denominator is equal to zero. So we have a vertical ascent to at where X minus eight is equal to zero. And which tells us that X is equal to eight. OK. So we do have a vertical asom to, it's located at X equals eight. And let's move now to the oblique Asy. Now for the oblique asy tote, we also wanna look at the degree of the numerator and denominator. OK. We've already done that for a horizontal ascent to so we know that the degree of the numerator is two, the degree of the denominator is one. So the degree of the numerator is actually one more than the degree of the denominator. OK. So it's the degree of the denominator plus one which tells us that we have an ob Gaston. OK. So we do have an ob blue gas until we just need to find it. Now, in order to find an oblique accent to, we're gonna use long division. So we're gonna draw our little long division. Oh Here underneath our align, we have the function in the numerator X squared plus zero X plus 16. OK. It's important to include all of the terms even if the coefficient is zero. So we want to include this X term even though the coefficient is zero and we want to write the zero there and to the left of our long division, we're gonna have the denominator which is X minus eight. All right. Now let's perform our long division. We're gonna start and we're gonna take the first term underneath our division symbol X squared. And we're gonna divide it by the first term in our denominator X OK. X squared minus X or sorry, divided by X square divided by X gives us X. OK. We're gonna write that above. Then we're gonna take that value. We found X and multiply it by the entire denominator X minus eight A X multiplied by X minus eight gives us X squared minus eight X. OK. And we write it down below, underneath the function from the numerator. And then we're gonna subtract within these columns. So we have X squared minus X squared that gives us zero. OK. So you can see that we've now eliminated that X squared term. We're working our way to the right and we're eliminating turn by turn, then we have zero X minus negative A X just going to give us positive eight. All right. Now we're gonna do the same thing. We have positive eight X. We're gonna divide by the first term in our denominator. So we have eight X divided by X which gives us positive eight. OK. So now we have X plus eight above our division align. We're gonna take that value of eight, we just found and multiply it to the entire denominator eight multiplied by X minus eight. This gives us eight X minus 64. OK? We write it down below, we bring the 16 down and then we subtract within the calls. So we have eight X minus eight X that gives us zero. OK. And now you can see again, we've eliminated the X terms and then we have 16 minus negative 64 which gives us a, all right. So we've performed our long division. Now we need to interpret the results. Now when we're looking for an oblique ahem to and oblique atom to is going to have the equation of a line. And that line is gonna be given by the quotient of our division. So in this case, we have Y is equal to X plus eight that comes from the top of our division, the quotient. And now we've found everything we found that there was no horizontal Asymptote. We found that we had a vertical Asymptote at X equals eight and an oblique Asymptote at Y equals X plus eight. We compare this to our answer choices. We see that this corresponds with answer choice. D. Thanks everyone for watching. I hope this video helped see you in the next one.

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4.
$ƒ (x) = (x^{2} + 4) / (x - 1)$

Key Concepts

Vertical Asymptotes

Horizontal Asymptotes

Oblique (Slant) Asymptotes

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