There's a specific list of instructions to do when graphing a rational function.
- Factor the numerator and denominator
- Find roots
- Find y-intercept (to help get the general shape of the graph)
- Find vertical asymptotes
- Make positive-negative chart (determine which parts are positive or negative)
- If largest degree of denominator > largest degree of numerator- x -> 0
Divide the numerator by the denominator to get an oblique/horizontal asymptote.
For example, (x^2 + 8x + 16)/(x - 3) can be divided using synthetic division: so it's
x + 11 + 49/(x-3)
If the form can be written as:
asymptote + (x-a)(x-b)/( (x-c)(x-d) ), then you know that at x = a, or x = b, the function crosses the asymptote.
Basically that's it, there are a lot of checks in place, so if you mess up, it's easy to catch. Just to check, you can plug in a few points.