There are different characteristics to look for when drawing a rational function graph. With a rational function graph where the degree of the numerator function is greater than the degree of denominator function, we can find an oblique asymptote. When the degree of the numerator is less than or equal to that of the denominator, there are other techniques for drawing a rational function graph.
I want to talk about rational functions where the degree of the numerator is bigger than the degree of the denominator, so remember the rational function is a ratio of two polynomials so let's call the degree of the top polynomial n and the bottom one m. If n is greater than m remember that there are no horizontal asymptotes but there maybe something called an oblique asymptote and the way to find out is to divide the the top polynomial by the bottom polynomial. Let's take a look at an example.
Here I have f of x equals x squared plus x minus 4 over x plus 2 and I want to find the asymptotes. Now the first thing you want to do is in order to find the oblique asymptote if there is one is divide these two so let's do that. We have x squared plus x minus 4 divided by x+2 and so what do we need to multiply by x to get x squared, x, so you multiply through and we get x squared plus 2x we change the signs and add and we get x-4, now what do we need to multiply by x to get minus x, -1, so we multiply -1 through and we get -1,-x-2 change the signs and add and I get a reminder of -2. What that means is that f of x is actually equal to this quotient x-1 plus this remainder -2 over this divisor x+2. This is going to be my oblique asymptote right here the x-1 so y=x-1 is an oblique asymptote.
Now if you, you can either fi- find the vertical asymptotes in the beginning or you can do it now because you're always going to have this denominator here x+2=0 when x is -2 so x=-2 is going to be your vertical asymptote. And those are the two asymptotes for this function, it's got an oblique as to y=x-1 slanted like this and a vertical asymptote and that'll be important when you're graphing functions where the degree of the numerator is bigger than the degree of denominator.