We’re going to analyze simple rational functions, of the form:

• constant in the numerator & linear term in the denominator (e.g. f(x) = 5/(x+2) )
• linear term in the numerator & linear term in the denominator

Here’s how we can identify the following features of a rational function f(x) and its graph:

• domain: solve for where the denominator equals 0 (exclude those points from the domain)
• x-intercept(s): solve f(x) = 0 (in the case of a rational function, this means solving for where the numerator = 0)
• y-intercept: calculate f(0)
• vertical asymptote: for these simple rational function, the vertical asymptote occurs where the denominator equals 0 (so the same x-value that is not in the domain)
• horizontal asymptote: depends on which type of rational function we’re looking at:
  1. If f(x) = constant/(linear term), then the horizontal asymptote is at y=0
  2. If f(x) = (linear term)/(linear term), then look at the ratio of the leading coefficients (i.e., the ration of the “x” coefficients)