 A rational function is a quotient of two polynomial functions. The denominator should have a degree of one or more. So f of x equals 3x minus 7 over x squared plus 8x plus 5. That's a polynomial divided by a polynomial. And so this is a rational function. g of x equals 5 divided by 2x minus 8, while 5 is a polynomial of degree 0. 2x minus 8 is a first degree polynomial. So this is a rational function. h of x equals x over 5. Since we want our denominator to have a degree one or more, this is not considered a rational function. And similarly, h of x equals 1 divided by the square root of x. Square root is not a polynomial. So this is not a rational function. The simplest rational function is going to be the simplest 0 degree polynomial, 1 divided by the simplest first degree polynomial, x. So our simplest rational function, f of x equals 1 over x. And we can graph it. And we plot a few billion points and we get something that looks like this. And notice our graph gets very close to the y-axis and very close to the x-axis. We say that the x and y axes are asymptotes and y equals 0 is a horizontal asymptote and x equals 0, well, this one's vertically. So let's call this an up and down asymptote. Let's call it a vertical asymptote. Now these terms are geometric terms. So the question is, what is very close algebraically? And so we might make the following observation. We get close to the y-axis. If we go to the far, far, far, far right or to the far, far, far, far left. Now points that are close to the y-axis have y-values that are close to 0 and points to the far, far, far, far right have very large and positive x-values. So we write it this way. y goes to 0 as x goes to infinity where the infinity symbol represents the idea that however big a number we have, we'll make it larger. Now more commonly, we like to think about x as our independent variable. x does something and y follows suit. So we more commonly write this as x goes to infinity, y goes to 0. Similarly, if we go to the far, far, far, far left our y-values get close to 0. x-coordinates will be large but negative and so we write y goes to 0 as x goes to negative infinity and again, we like to put our x-values first as x goes to negative infinity, y goes to 0. What about x equals 0, our vertical asymptote? Notice that to get close to x equals 0 we must have a very large positive or negative y-value. So we could say x goes to 0 as y goes to infinity and we could say x goes to 0 as y goes to negative infinity but we don't. And the reason is that we like to think about y as a function and here we see that this x is going to 0, y does two completely different things like that. So we'll introduce a little bit more notation. If x is close to 0 but slightly more than 0 then we're in this portion of the graph and so y is a very large and positive and so we might write as x goes to 0 but stays a little bit more, that's our plus, y goes to infinity. And likewise, if x is close to 0 but slightly less than 0 then y is very large but negative and so we write this as x goes to 0 but stays a little less, that's our minus, y goes to minus infinity. Now you might wonder if we're being inconsistent and the answer is yes, we should write as x goes to infinity, y goes to 0 but stays a little bit more and similarly as x goes to minus infinity y goes to 0 and stays a little bit less and we should do this but in practice we don't. Now let's combine these ideas with our horizontal and vertical translations so the graph shown is a horizontal and vertical translation of y equals 1 over x let's find the equation. So notice that our graph does have two asymptotes x equals 3 and y equals negative 2. Now since a vertical and horizontal translation will also move the asymptotes we can draw the graph of y equals 1 over x and its asymptotes. So y equals x has a vertical asymptote of x equals 0 so first we'll shift our graph horizontally to the right by 3 units and that puts the vertical asymptote in the right place. We also have a horizontal asymptote of y equals 0 so to get that in the right place we need to shift down by 2 units and so this graph is produced by a horizontal shift of 3 units to the right producing the graph of y equals 1 divided by x minus 3 then we apply a vertical shift of 2 units downward producing the graph of y equals 1 divided by x minus 3 minus 2.