Curve Sketching

Suggestions for sketching the graph of a function

  1. Find the x-intercepts and y-intercepts if they are easily found.  Mark those points on the graph.
  2. Sketch any vertical and horizontal asymptotes.
  3. Find the x-values where f '(x) is zero or undefined.  Use the squiggly line test or sign test to see when f '(x) is positive and negative.
  4. Find the x-values where f "(x) is zero or undefined.  Use the squiggly line test or sign test to see when f '(x) is positive and negative.
  5. Plot the relative extrema, the points of inflection, and any other important points.
  6. Sketch the graph, paying close to where the function increases or decreases and is concave up or concave down.

Problems to try:  Sketch each graph analytically.  Then graph it in your calculator to see if you are correct.

  1. 

  2.  

  3.  

  4.  f(x) = x2/3 - 2x4/3

  5.  f(x) = x4 - 13x2 + 36

  6.  f(x) =      sin x      
                 1 + cos x