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Graphs of rational functions: horizontal asymptote

CCSS Math: HSF.IF.C.7d

Video transcript

- [Voiceover] Let f of x equal negative x squared plus a x plus b over x squared plus c x plus d where a, b, c, and d are unknown constants. Which of the following is a possible graph of y is equal to f of x? And they told us dashed lines indicate asymptotes. So this is really interesting here. And they gave us four choices. We see four of them, three of them right now. Then if I scroll over bit over, you can see choice D. And so I encourage you to pause the video and think about how we can figure it out because it is interesting because they haven't given us a lot of details. They haven't given us what these coefficients or these constants are going to be. All right, now let's think about it. So one thing we could think about is horizontal asymptotes. So let's think about what happens as x approaches positive or negative infinity. Well, as x, so as x approaches infinity or x approaches negative infinity, f of x. F of x is going to be approximately equal to. Well, we're going to look at the highest degree terms because these are going to dominate as the magnitude of x, the absolute value of x becomes very large. So f of x is going to be approximately negative x squared over x squared which is equal to negative or we could another way to think about it. This is the same thing as negative one. So f of x is going to approach, f of x is going to approach negative one. In either direction, as x approaches infinity or x approaches negative infinity. So we have a horizontal asymptote at y equals negative one. Let's see, choice A here, it does look like they have a horizontal asymptote at y is equal to negative one right over there. And we can verify that because each hash mark is two. We go from two to zero to negative two to negative four. So this does look like it's a negative one. So just based only on the horizontal asymptote, choice A looks good. Choice B, we have a horizontal asymptote at y is equal to positive two. So we can rule that out. We know that a horizontal asymptote as x approaches positive or negative infinity is at negative one, y equals negative one. Here, our horizontal asymptote is at y is equal to zero. The graph approaches, it approaches the x axis from either above or below. So it's not the horizontal asymptote. It's not y equals negative one. So we can rule that one out. And then similarly, over here, our horizontal asymptote is not y equals negative one. Our horizontal asymptote is y is equal to zero so we can rule that one out. And that makes sense because really they only gave us enough information to figure out the horizontal asymptote. They didn't give us enough information to figure out how many roots or what happens in the interval and all of those types of things. How many zeros and all that because we don't know what the actual coefficients or constants of the quadratic are. All we know is what happens as the x squared terms dominate. This thing is going to approach negative one. And so we picked choice A.