Graphs of rational functions: horizontal asymptote | Algebra II | High School Math | Khan Academy

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? They tell us dashed lines indicate asymptotes.

So, this is really interesting here, and they give us four choices. We see four of them—three of them right now. Then, if I scroll a little bit over, you can see choice d. I encourage you to pause the video and think about how we can figure it out. Because it is interesting, 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. One thing we could think about is horizontal asymptotes. So, let's consider what happens as x approaches positive or negative infinity. Well, as x approaches infinity or x approaches negative infinity, f of x will 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, or 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 one.

Thus, f(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.

Now, let's see choice a here. It does look like they have a horizontal asymptote at y equals 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 at negative one.

So, based only on the horizontal asymptote, choice a looks good. Choice b has a horizontal asymptote at y equals positive two, so we can rule that out. We know that our horizontal asymptote as x approaches positive or negative infinity is at y equals negative one.

Here, our horizontal asymptote is at y equals zero. The graph approaches the x-axis from either above or below, so it's not the case that the horizontal asymptote is y equals negative one. We can rule that one out.

Similarly, over here, our horizontal asymptote is not y equals negative one; a horizontal asymptote is y equals zero. So, we can rule that one out as well.

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 determine 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 function is going to approach negative one, and so we pick choice a.