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