 So now let's put everything together so that we can graph a rational function. So we are going to graph rational functions in a completely different, in the same way that we've graphed everything else. So we'll find the x and y intercepts, we'll identify other graph features of significance, and we'll find the behavior between the features of significance, and then finally sketch the graph. Now for rational functions, so for every function, the other features of significance the things we're interested in will change so that we might be looking at the vertex of a quadratic equation, for example. For rational functions, the other features that we care about are going to be the vertical and horizontal asymptotes. So let's take a look at the graph of y equals x minus 2 over x plus 5. So we'll find the x and y intercepts, so the x intercept occurs where y is equal to 0. So we solve 0 equals x minus 2 over x minus 5, multiplied by the denominator, that'll drop that out, x minus 2 equals 0, x equals 2. Remember that the x intercept is a point, so it's going to be located at the point where x is equal to 2, y is equal to 0. So that's the point to 0. Likewise, the y intercept is the point where x is equal to 0. So I'll solve, let x equal to 0, y equals 2 fifths, my coordinates x equals 0, y equals 2 fifths, and I have the x and y intercepts. Because this is a rational function, we have to identify the forbidden values. Again, those will be the things that make the denominator equal to 0, x equals 5 is bad, so we want to not let x be 5. And we'll draw our intercepts and our forbidden values on our graph. Now, notice that because the x and y intercepts are points, I've actually graphed them as points, this x can't be equal to 5. Well, I can't draw that as a point because if I put a point at x equals 5, I'm putting a point where I shouldn't be. So what I'm going to do is I'm going to express this notion x can never be allowed to be 5. I'm going to use a vertical dashed line. So there's my line x equals 5, the dashes say do not touch this line. Now, the next thing we'll want to do is we'll want to find the behavior of the function between the features of significance. In this case, between the x intercept and the forbidden line. And so if I'm in between those two, x is going to be someplace between x equals 2 and x equals 5, the value of my function is going to be negative. So I'll draw a point to represent those facts. So someplace down here, I have a negative value of y. And this point represents the fact that between here and here, my y values, my function values are going to be negative. Now it's a rational function. So the asymptotes are also going to be significant. So I'm going to divide the one by the other to get 1 plus 3 over x minus 5. And as x goes to plus or minus infinity, y is going to get close to 1 plus 3 over a large positive number. That's going to be a small positive number or 3 over a large negative number. That's going to be a small negative number. So that says that as x goes to plus or minus infinity, y is going to approach 1. And so that says that y equals 1 is going to be a horizontal asymptote. So I'll sketch that in. As x goes to positive infinity, 3 over x minus 5 becomes a small positive number. So that says that y is going to be close to 1 plus a small positive number. Y is going to be slightly more than 1. And so I do want to draw a point to represent this fact. If I'm way out here to the right axis close to positive infinity, my y values, y will be slightly more than 1. My y values will be a little bit higher than this line y equals 1. So I'll draw up a point there. As x goes to minus infinity, 3 over x minus 5 becomes a small negative number. And so my y values, y is going to be 1 plus a small negative number, is going to be slightly less than 1. And as x goes to negative infinity, I'm going off to the left someplace. Way over here to the left. And I want to draw a point to represent this fact. So x is close to left, way left. And I'm slightly below this line y equals 1. So I'll draw up a point there to indicate that. As x approaches this forbidden line, if I'm slightly less than 5, if I'm slightly to the left of this line, then 3 over x minus 5 becomes a large negative number. The denominator is a small negative number. And so the quotient is large negative. So y is also going to become a large negative number. And we'll ignore this fact. No, we'll draw a point to represent it. So y is going to become a large negative number. As x gets close to 5 from below, y becomes large and negative. So I'm a little less than 5, but I'm way, way, way, way down here. And I'll draw a point to represent that fact. And if I'm on the other side, if x is getting close to 5, but staying a little bit above it, then 3 over x minus 5 is a large positive number. And so y is going to become a large positive number. And I'll draw a point to represent this fact as well. So again, I'm a little bit more than 5. I'm approaching 5 from above. y is very large, very positive way up here someplace. And so I'll draw a point to represent that fact. And now I have a bunch of points of significance on the graph. And I'll connect the dots, but remember that I'm not allowed to let x equal 5. This dashed line, this vertical dashed line, is a forbidden line, so I have to avoid it. And I'll connect the dots, and I'll do a little bit of smoothing out to produce my final graph.