Curve Sketching:

An 8-step menu for polynomial and rational functions

  1. Asymptotes
    • Vertical: The values of x for which the given function is not defined.
    • Form: 
    • Horizontal: The values of y which the function approaches as x becomes very large.
    • Form: 
    • Oblique: The values of y which the function approaches as x becomes very large
    • Form: 
  2. Domain and Range
    • Domain: The values of x used to generate the curve.
    • The values of x which are not vertical asymptotes.
    • Range: The values of y used to generate the curve.
    • The values of y which are not horizontal asymptotes.
  3. Symmetry
    • Corresponding value of x and y reflected about the x-axis, y-axis, or the origin.
    • Symmetry about x-axis: let  and test it.
    • Symmetry about y-axis: let  and test it.
    • Symmetry about the origin: let  and  then test it.
  4. First Derivative
    • We obtain Critical Points and examine the behavior of the curve. We determine the intervals along the x-axis where the curve increases and decreases by examining the slopes of the tangent lines.
  5. Second Derivative
    • Maxima/Minima: We use the 2nd Derivative Test to determine which of the critical points are maxima or minima.
    • Points of Inflection: The values of x where the curve changes concavity i.e. the values of x where the curve stops being concave down and begins being concave up (and vice versa).
    • Concavity: We examine the intervals along the x-axis where the curve is either concave up or down.
  6. Intercepts
    • x-intercepts: The values of x where the function intercepts the x-axis.
    • Let  and solve for x.
    • y-intercepts: The values of y where the function intercepts the y-axis.
    • Let  and solve for y.
  7. y-values
    • The corresponding values of y for the maxima, minima, and points of inflection.
  8. Sketch