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Problem 66 Hard Difficulty

Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.

$ y = \dfrac{1 + 5x - 2x^2}{x - 2} $

Answer

see graph

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Video Transcript

So here are domain is restricted to ex cannot equal to because of the denominator. Can I able to no have intercepts it? Zero negative. 1/2. And we have two more at approximately negative. 0.20 and 2.7. You're symmetry. None sm toots, we have a vertical one. As we said previously, denominator can't be zero. So we have a vertical one at, um, X equals two. And we do have a slant asking too. It's so for that using polynomial division coming to reorder this just so that I have my highest terms than the beginning in my lowest terms at the end. So this is my enumerators reordered. Okay, that's inside that have been signed. And the next one is to my denominator on the outside. So x squared. I mean accent. Ah, negative two X squared. Negative to x. Negative two x Times X would normally be negative Two X square, but we're going to change. Assigned to cancel the first term two x squared Seem goes here. All right. Next term, Negative for X. We had changed slight. We're gonna bring down the one cancel. We're left with X plus one Sorry, X in Texas coast one one times X, You're in this thing, Jane, This thing cancel. All right. We can no longer continue polynomial division because this has the higher power thing here. So this Mike Oceans is my slam task into. And now I'm gonna look for the first derivative test in order to see um, increasing decreasing intervals. So I I sold for why Prime using the quotient will so negative to x squared plus eight x minus 11 allover the denominator squared set that equal to zero. This is inconclusive, and you'll see that the second derivative is also a very tricky So in this case, we can just look at the ass end totes to help us grab, and they really do help. So we have won the X equals too. So let's say this is too. Is it just a rough sketch right now? Let's let's sketch in negative two x plus one. So let's say this is about one Mr of Sketch. You did too. Hurry. And now we can grow for intercepts. Zero negative 1/2 would be. Let's say this is one. So this would be Is your negative half okay? and is going to approach but never touch. Likewise, it's gonna approach, but never touch. And over here well, we have another intercept. We have an intercept. Uh, we really got this one. We got this one. So this is about negative. Zero negative points. Your 20 It's about here where you got that. And then we have 2.70 So 2.70 Merced's about here. Now we know it's going to approach, but not touch the ass in tips, so it has to look something like this.