We will give some general guidelines for sketching the plot of a function.

Let’s get to the point. Here we use all of the tools we know to sketch the graph of :
  • Find the -intercept, this is the point . Place this point on your graph.
  • Find any vertical asymptotes, these are points where goes to infinity as goes to (from the right, left, or both).
  • If possible, find the -intercepts, the points where . Place these points on your graph.
  • Analyze end behavior: as , what happens to the graph of ? Does it have horizontal asymptotes, increase or decrease without bound, or have some other kind of behavior?
  • Compute and .
  • Find the critical points (the points where or is undefined).
  • Use either the first or second derivative test to identify local extrema and/or find the intervals where your function is increasing/decreasing.
  • Find the candidates for inflection points, the points where or is undefined.
  • Identify inflection points and concavity.
  • Determine an interval that shows all relevant behavior.

At this point you should be able to sketch the plot of your function.