There are different characteristics to look for when creating rational function graphs. With rational function graphs where the degree of the numerator function is equal to the degree of denominator function, we can find a horizontal asymptote. When the degree of the numerator is less than or greater than that of the denominator, there are other techniques for drawing rational function graphs.
I want to talk about graphing rational functions when the degree of the numerator is the same as the degree of the denominator. Let's talk a little bit about what the horizontal asymptote is going to be in that instance. If f of x is ax the n plus da da da over bx to the m plus da da da then and these are our leading terms for the numerator and denominator, then the numerator has degree n and the denominator has degree m and so if those degrees are the same, then your rational function f of x is going to approach a over b the fraction a over b which is the fraction of the leading coefficients as x goes to infinity and as x goes to negative infinity and that means that y equals a over b is going to be your horizontal asymptote then it's really easy to find the horizontal asymptote. Now one thing you should know if the degree of the numerator is larger than the degree of the denominator there is not a horizontal asymptote. Let's take a look at an example. It says find the horizontal asymptote. So all you have to do is first ask yourself are the degrees the same and if they are then the horizontal asymptote is going to be leading coefficient over leading coefficient so the horizontal asymptote is y=-4 over 1, -4, y=-4 that's our answer. Now in this instance the degree of the numerator is bigger than the degree of the denominator so there's no horizontal asymptote I'll abbreviate it ha and in this instance the degrees are both 1 they're the same so we look at the leading coefficients again 3 and 1, so y equals 3 over 1 y=3 that's our horizontal asymptote. So it's really easy to find horizontal asymptotes when the degree of the numerator is the same as the degree of the denominator. It's always easy to find horizontal asymptotes. If the degree of the numerator is less than the degree of the denominator, it's always y=0. We'll come to the case when the degree of the numerator is larger later.