Askjagden's Guide to Algebra: Finding Asymptotes
One of the most essential things to functions are asymptotes. What is an asymptote? An asymptote is a line that a graph approaches but never touches. This guide will show you how you can find an asymptote.
There are two types of asymptotes: horizantal and vertical asymptotes. Horizantal means a line stretching left to right; vertical means a line stretching up and down.
To find a vertical asymptote, you must find the zero of the denominator. If this is not possible, then there is no vertical asymptote. What do I mean by the "zero"? That is the value of something that will make the something zero. For example, what are the zeroes of x^2 + x? They are 0 and -1. Just plug them in; the output will be 0.
Let's find the vertical asymptote of f(x) = (x + 1)/(2x + 4). Let's set 2x + 4 equal to 0 and solve:
2x + 4 = 0
2x = -4
x = -2
However, there is a limit. Let's say t is a zero of the denominator, but is also a zero of the numerator. What happens now? For example, ((x - 4)(x - 2))/(x - 4): 4 is the zero of the denominator and the numerator. That means you will have to make the function discrete, which means all the set of points on a graph are not connected together. You might as well make the graph f(x) = x - 2, because (x - 4)/(x - 4) = 1, but here's what you must do: in the graph of y = x - 2, you must put a hole whatever the value is when x = 4. This means that there is no value for the graph there; it is undefined.
Finding horizantal asymptotes is more complex. What is a degree of a polynomial? The degree of the polynomial is the exponent of the highest term. Numbers such as 9, 6, pi, etc. have a degree of 0. If the degree of the polynomials of the numerator and the denominator are the same, then the coefficient of that term in the numerator will be divided by the coefficient on the denominator. For example, let's at a function: (x^3 + x)/(5x^3 - 3x^2 + 6). The horizantal asymptote is 1/5. If the degree of the polynomial on top is less than the bottom, then y = 0, or the x-axis, becomes the horizantal asymptote. If the degree of the polynomial on the numerator is greater than the one on bottom, than there is no horizanal asymptote.
Finding slant asymptotes is quite simple. Let's find the slant (or oblique) asymptote of f(x) = (-3x^2 + 2)/(x -1). Before we move on, let's find the vertical and horizantal asymptotes. x - 1 = 0
x = 1
The degree of the polynomial on top is greater than the bottom. Therefore there is no horizantal asymptote. The slant asymptote is the numerator divided by the denominator (if you are reading this, I shall assume you have studied algebra before; therefore I will not divide show you how to divide polynomials). Ignore the remainder; just gather the polynomial part. In this case, it is -3x - 3. That line is the slant asymptote.
Have fun with precalculus!
