Hey, everyone. When working with vertical asymptotes, we found that they affect the domain of our rational function, and we could even calculate them in a way that's really similar to finding the domain. Now with horizontal asymptotes, they're instead going to affect the range of our rational function because now we're dealing with horizontal lines that go across our y-axis. Since we don't have a way of finding the range of our function, you may be worried that this is going to get complicated. But don't worry because it just depends on only two things that we can determine by just looking at our function, and that's the degree of both our numerator, the top of our rational function, and the denominator, the bottom of our rational function. So with that in mind, let's go ahead and just jump right into finding some horizontal asymptotes.
Looking at this first function here, I have f(x)=1/x and we want to be able to determine what our horizontal asymptote is. Now, as I said, we'll look at the degree of the numerator and the degree of the denominator. Here the degree of my numerator is 0 because there's not even a variable there, and the degree of my denominator is 1. So, 0 is less than 1, and whenever the degree of your numerator is less than the degree of your denominator, you are always simply going to have a horizontal asymptote at the line y=0, which is what we can see on our graph is already happening here.
Now let's look at another function here: I have f(x)=2x+3x. So, looking at the degree of my numerator and denominator, the degree of my numerator is 1 and the degree of my denominator is also 1. Hence, 1 is equal to 1. So here, since the degree of my numerator is equal to the degree of my denominator, I actually need to consider one more thing: my leading coefficients. So, I'm going to take the leading coefficient of my numerator and then divide it by the leading coefficient of my denominator. And that will give me my horizontal asymptote y=2, which is exactly where that horizontal asymptote is on our graph.
Now, I'll discuss another function: 4xx2-xx3-5x+9. Here, the degree of our numerator is 2, and the degree of our denominator is 3. So, 2 is less than 3. Accordingly, we have a horizontal asymptote at y=0, following our original examples.
Next, suppose we have f(x)=2x23x2+x-1. Here the degrees for numerator and denominator are both 2. Thus, the degree of my numerator is equal to the degree of my denominator. Then, by dividing the leading coefficients, I end up with a horizontal asymptote at y=23.
And finally, a word about the graph behavior: sometimes, the graph of your rational function may actually intersect the horizontal asymptote and fully cross it, then approach it from the other side. While this may seem to contradict the typical behavior of asymptotes, it's just one of those nuances you should be aware of. It can happen, so don't be startled if you see a graph crossing a horizontal asymptote.
With this knowledge, you're now better equipped to identify horizontal asymptotes and understand their role in the behavior of rational functions. Let's get some practice!