>> In this section we'll be talking about rational functions and their graphs. And those graphs are some of the most interesting in mathematics, because they're, they're, there's a great variety of graphs that come from rational functions. Now rational functions are composed of a polynomial numerator over a polynomial denominator, like this example. Now because we have a polynomial denominator, we have variables involved. And those variables can take on values that will cause the denominator to become zero. And if the denominator becomes zero, the entire fraction is undefined. That presents a rather interesting situation graphically, as we'll see in just a minute. We're gonna begin our discussion and our investigation of the graphs of rational functions by using the point plotting method. Now we know that if we replace X with 3 here we have a Y value that's undefined. Now I have just given a number of other points that we can plot here in our table. I've plotted those points and some others here on this coordinate plane. Now notice that the graph is curvy right over here. And it's kind of curvy in the same way over here. And it turns out that the, the last points down here on this curve does not connect with a point way up here. So the graphs of rational functions are, very often, not continuous. Well let's see, the graph then looks like this. It's kind of a curvy thing going on. This is called a branch, by the way. This is a branch of the graph. Here's another branch. [ silence ] >> Now it turns out that these branches are approaching lines, which are called asymptotes.