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Graphing a Rational Function

There's a specific list of instructions to do when graphing a rational function. 

  1. Factor the numerator and denominator
  2. Find roots
  3. Find y-intercept (to help get the general shape of the graph)
  4. Find vertical asymptotes
  5. Make positive-negative chart (determine which parts are positive or negative)
  6. If largest degree of denominator > largest degree of numerator- x -> 0
  7. Otherwise:

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.

Converting between Cartesian and Polar

De Moivre's Theorem