Glossary for Lesson 24: Asymptotes

Rational Function

A rational function is any function that is a ratio of two polynomials. That is, a rational function is of the form \frac {p}{q} , where p and q are both polynomials and q\neq 0 . Here are some examples of rational functions: y=\frac {3x-1}{x^2+6x+8} y=\frac {1}{x} y=\frac {4x^3-9x+12}{x+8} y=\frac {(x+4)(x-2)}{x(5x+2)(x-2)} All polynomial functions are technically rational functions, since any polynomial p(x) can be written as \frac {p(x)}{1} , and 1 is a polynomial of order zero. 

Vertical Asymptote

A vertical asymptote is a vertical line that a curve will approach but never touch.  On either side of this line, then, the curve will to grow toward \infty or toward -\infty .  The function y=\frac {x+2}{x^2-2x-15} has two vertical asymptotes, x=5 and x=-3 , as you can see in the graph below. 


A more formal definition of a vertical asymptote might sound something like this: The line x=c is a vertical asymptpte of the graph of a function f if, as x\rightarrow c from the right or from the left, f(x) \rightarrow -\infty or f(x) \rightarrow \infty

Horizontal Asymptote

A horizontal asymptote is a horizontal line that a curve approaches either as x gets bigger and bigger, approaching \infty , or as x gets smaller and smaller, approaching -\infty .  Many rational functions have horizontal asymptotes.  Whereas curves may not cross their vertical asymptotes, they may cross their horizontal asymptotes at values of x that are relatively close to 0.  The graph of y=\frac{4x+2}{x^2+1} shows us how what this could look like:
ybleii49f3.png A more formal definition of a horizontal asymptote might sound something like this: The line y=d is a horizontal asymptote of the graph of the function f if, as x\rightarrow -\inf or x\rightarrow \inf , f(x) \rightarrow d