>> Here's another one. The approach is exactly the same. And the trick here has to do with the horizontal asymptote. Now first of all, let's identify a vertical asymptote. The zero of the denominator is simply at X equals 1. X equals 1, that's right here. And now we look for the horizontal asymptote by looking at the degree of the numerator compared to denominator. So now, notice we have a second-degree numerator, a first-degree denominator. So the degree of the numerator is larger, and that means we don't have a horizontal asymptote at all. We have a slant asymptote. But let me show you how to find the equation for the slant asymptote. That equation is found by simply performing the division that's indicated here, and taking sort of the whole part of the whole polynomial part of that without the remainder. Here's what I'm talking about. I want to take this polynomial and divide by this 1. I could use long division like this. And go through the process. Or I could use synthetic division by thinking well if I'm dividing by X minus 1, then synthetically I'm dividing by 1. And now I use the 1, the 1 and the negative 3 here. And remember the process, bring down the 1, 1 times 1, 1. Together 2. 1 times 2, 2. Together negative 1. That's the remainder. And what I want is the polynomial described by those 2 items. And that polynomial is simply 1 X plus 2. So the line Y equals X plus 2 is a slant asymptote. So we can come back over here and, and draw that line if we want to. Now notice it's in the slope intercept form for a linear equation. And the, the intercept is a 2, and the slope is 1. Now for a slope of 1 that's a rise of 1, a run of 1. And from this point, rise 1, run 1, and so on. And so we have points out in this direction. And then in the other direction we can generate points as well. Now, the dotted line then describes our slant asymptote. And if we plot a few points, we'll understand the behavior of the graph around these asymptotes. Let's see what happens. I've made a table of values here. And zero goes to 3.
[ Writing on chalkboard ] Oops, zero goes to, I knew it couldn't possibly be right on top of the asymptote. Zero goes to 1, 2, 3. And the negative 1 goes to 1 and a half. Negative 2 goes to one-third. And you see the graph is just kind of real tight on the asymptote here and then it's just going to blast upward here. And, and we can go ahead and graph that. Because my other 3 points are on the other side of that vertical asymptote. So I know that on this side, this is what's happening. And I get closer, closer, closer as I go out this way. All right, 2 goes to 3. Then 3 goes to 4 and a half. We're getting tight on that asymptote already. 4 goes to 5 and 2-thirds.
[ Writing on chalkboard ] So it's kind of like this. Now notice that they look to be really close on that asymptote, and then the turn is made and it just, just kind of drops down like this. So the graph, oops, the graph looks like this. Let's take a look at this one on the calculator.