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Problem 44 Medium Difficulty

(a) Find the vertical asymptotes of the function
$$ y = \frac{x^2 +1}{3x - 2x^2} $$

(b) Confirm your answer to part (a) by graphing the function.

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Video Transcript

here. We have a function Y equals x squared plus 1/3 x minus two X squared. And we want to find the vertical ascent. Oops uh vertical as um tubes are vertical lines where the graph of the function will get closer and closer and closer to that line but never touch it. The best way to find vertical assam tops are defined values of X. That would make the denominator Equal to zero. And so if we want to find values of X that make the denominator three x minus two, X squared equal to zero. Let's rewrite um uh this denominator three x minus two X squared in factored form. And so we can rewrite our Y function Y equals X squared plus one over. Uh We're going to factor out in X. And that means we're going to be times you could buy a 3 -2 x double checking our factoring by distributing this multiplication by X X times three is three X. Subtract subtract X times two X two X squared. Ok. So why does equal this expression here? Now we're going to find a vertical assam taub's when the denominator equal zero. So when does x times 3 -2 x equals zero. Well, using the zero product property, uh if you multiply two terms and you get 01 of those has to be zero. So if x times in 3 -2 x gives you zero. Either x equals zero or The 3 -2. X Factor could equal zero. Well X equals zero. There's nothing we have to do with that. That's one of the values of X. That would make our denominator equal to zero. So that's going to help us find a vertical ascent. Tobe solving three minus two X. Our other factor equal to 03 minus two. X equals zero. We can add two extra both sides. Uh we get three adding two extra both sides of the equation. We get three equals two X devoted dividing both sides by two. X. Being ties by two, divide both sides by two. We get X equals three halves. And so when X zero and when X is three halves, the denominator is going to be zero. And that means where you are going to have a vertical ascent Ope Uh huh. We'll have vertical assam. Stopes. Let me do a better job than that. We are going to have vertical assam. Tokes. When X is zero at this point here we'll have a vertical line. Let's go ahead and draw that the vertical assam Tope vertical as um Tope when X zero is going to be the line at X equals zero. All right. It's not gonna let me draw a line on top is so I'll just pencil again. So this is the vertical assam. Tope Corresponding to when X0. And so the equation of this vertical ass and took this red line is X equals zero because X equals zero everywhere on that line. And so. X can never actually equal zero in dysfunction because effects equals zero. The denominator is zero and you can't divide by zero. So for every point on the Y axis, the X coordinate is zero. So for every point on the y axis you will never see the graph of this function touch. Uh This red line. Uh The line whose equation is X equals zero. This same line which is the Y axis. It will get closer and closer to it but never touches. So this is a vertical ascent up. The other vertical assam. Tope is going to come in at X equals three hash. It should allow me to grab this one. Um So this other vertical assam taub is X equals negative three hairs. And of course of course X equals negative three hits. Means we are hitting the X axis right here at negative three tips. So to find vertical assam tops for function uh Find the X. Values where the denominator would be zero. This denominator will be zero when X zero and when X. Is three hips. So this function, why is going to have to vertical to vertical as um toasts one at X equals 01 at X equals negative three halfs. Okay, X is positive three half. So please please uh understand that. This is a positive three halves. Of course it is in the correct location positive X. Is positive three years. Okay, X was equal to a positive rehab. So we expect to see two vertical ascent. Opus one at X equals 01 at X equals treehouse. When we graft dysfunction. So here's the graph of our function. You can see it's in three different sections. Now, one of our one of our assam tops we expected to see was that X equals zero. Well here's uh the graph of the line. What uh X equals zero. This blue line. So as uh the graph gets closer and closer as we get closer and closer to zero, the graph gets closer and closer to the blue line, moving closer and closer to the blue vertical aspirin top line moving down. But even though it looks like it touches it, it never does. Um So this blue vertical line is our ASM tope X equals zero And I don't want three. I want this to say 3/2. So if we graph the line X equals three halfs. Okay, that's the green line you saw just pop up. This was our other vertical ascent to whose equation is X equals three halves. So two vertical assam taub's X equals zero. X equals X equals three halves. Uh The graph gets closer and closer to that vertical as from top line but we'll never touch it