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

Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(2x+6)/(x-4)

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 we're given is F of X is equal to four, X plus 12 divided by X minus six. We're given four answer choices. Option A, a vertical Asymptote at Y equals six and an oblique Asymptote at Y equals four. Option B A vertical Asymptote at X equals negative six and a horizontal Asymptote at Y equals four. Option C A vertical Asymptote at X equals six and a horizontal Asymptote at Y equals negative four. And option D A vertical Asymptote at X equals six and a horizontal Asymptote at Y equals four. Now, let's start by rewriting our function. So we have F of X is equal to four, X plus 12 divided by X minus six. And let's start with our horizontal as to, and when we're looking at our horizontal atos, what we wanna do is compare the degree of the numerator to the degree of the denominator. And what we see is that the degree of the numerator is actually equal to the degree of the denominator in this case. OK. They are both equal to one, that highest exponent is one. OK? We have an X term in both the numerator and denominator. All right. So the degree of the same of the numerator and the denominator, what that tells us is that we do have horizontal ascent tode and the horizontal axon to is going to be at Y is equal to the ratio of the leading coefficients. So we have the leading coefficient of the numerator divided by the leading coefficient of the denominator. Now you'll notice I wrote this as Y equals, this is our horizontal Asymptote. It's always going to be Y equals something. OK. If you think of drawing a horizontal line on your graph, that has an equation Y equals something. So we always have Y equals when we have a horizontal Asymptote, not X equals. Now the leading coefficient of our numerator, that's gonna be the coefficient corresponding with the highest exponent term. The highest exponent term here is X, it has a coefficient of four. So we have four in the num writer. Same thing in the denominator. The highest exponent term is X, it has a coefficient of one. So we get four divided by one for a horizontal Asymptote, which is just equal to four. So we have a horizontal as at Y is equal to four. If we have a horizontal Asymptote, that means we don't have an oblique Asymptote or an oblique Asymptote, we need the degree of the numerator to be one greater than the degree of the denominator. But we found that the degree of the numerator and denominator are the same, which gave us a horizontal Asymptote. OK. So we don't have an oblique Asymptote. So let's work on the vertical asom too. Now, for the vertical as to what we want to look at is where the denominator equals zero. We have to be a little bit careful. We need to simplify the equation. First, you guys, we're gonna simplify, we're gonna cancel any terms that can be canceled. And then we're gonna look where the denominator is equal to zero. So in this case, our function F of X is equal to four X plus divided by X minus six. Now, in our numerator, we can factor for, OK. So we have four and then we have X plus three left over. And in the denominator, we still have X minus six. We can't simplify um or factor that any further. OK. And nothing cancels here. We have four multiplied by X plus three in the numerator divided by X minus six. And so we still want to look where that denominator is equal to zero for our vertical AY, that means that X minus six is equal to zero, which tells us that we have a vertical Asymptote at X is equal to six. So we found that we do have a horizontal Asymptote. It's located at Y is equal to four. We also have a vertical Asymptote located at X is equal to six and we do not have an oblique Asymptote. So if we compare this to our answer choices, we see that this corresponds with option 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)=(2x+6)/(x-4)

Key Concepts

Asymptotes

Rational Functions

Finding Asymptotes

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