PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
There are different characteristics to look for when graphing rational functions. When graphing rational functions where the degree of the numerator function is less than the degree of denominator function, we know that y=0 is a horizontal asymptote. When the degree of the numerator is equal to or greater than that of the denominator, there are other techniques for graphing rational functions.
I want graph rational functions and we're going to start with rational functions where the degree of the numerator is less than the degree of the denominator so Let's talk a little bit about horizontal asymptotes.
If a rational function is f of x equals p of x over q of x, p and q are going to be polynomial functions. Let's say that the numerator has degree n and the denominator has degree f, now if the degree of the numerator is less than the degree of the denominator, then the function f of x goes to 0 as x goes to infinity and it also goes to 0 as x goes to negative infinity and that means y=0 is a horizontal asymptote. Now it's very important that you know that if the degree of the numerator is not less than the degree of the denominator if it's bigger than or equal to the degree of the denominator, then y=0 is not the horizontal asymptote, so let's take a look at some some rational functions.
Now this one, f of x 1+3x over 1-x, the degree of the numerator and the denominator are both one the degrees are the same and so y=0 is not the horizontal asymptote here.
What about g of x, the degree at the top is 2 the degree at the bottom is going to be 3 right? The degree of the denominator is bigger than the numerator so here y=0 is, so yes but no here.
And finally here the degree of the numerator is bigger than the degree of the denominator, so no, y=o is not horizontal asymptote so again the only time y=0 is the horizontal asymptote is when the degree of the numerator is less than this degree of the denominator and this is the only example of a rational function that has that property.