Increasing or Decreasing Functions  (f’(x) >0  or f’(x)<0)

First Derivative & Local Maximum Minimum  (f’(x)=0)

Behavior near Asymptotes (limits)

Behavior at plus or minus infinity

Second Derivative & Point of Inflection (f’’(x))

Algorithm of Curve Sketching

• Determine the domain and the range of the function f(x)
• Determine the critical points by calculating the first derivative, f’(x)=0
• Plot those points on the function f(x)
• Calculate the second derivative and determine f’’(x)=0
• Determine the concavity and the inflection points
• Apply x= 0 to determine the y intercept
• Find f(x)=0 (to determine the intercepts with the x-axis)
• Find the limits when x goes to ± or to an asymptote
• Sketch the curve

Additional Curve Sketching Techniques

• Use a variation chart to see the general form of the graph