Section 4.4: Curve-Sketching Techniques

Graphing Strategy (Final Version)

1. Analyze f(x).
   • Find the domain of f.
   • Find the intercepts.
   • Find asymptotes.
2. Analyze f'(x).
   • Find the partition numbers for f' and the critical numbers of f.
   • Construct a sign chart for f'(x).
   • Determine the intervals on which f is increasing and decreasing.
   • Find the local maxima and minima of f.
3. Analyze f''(x).
   • Find the partition numbers for f''.
   • Construct a sign chart for f''(x).
   • Determine the intervals on which the graph of f is concave upward and concave downward.
   • Find the inflection points of f.
4. Sketch the graph of f.
   • Draw asymptotes and locate intercepts, local maxima and minima, and inflection points.
   • Sketch in what is known from steps 1 - 3.
   • Plot additional points as needed and complete the sketch.

Oblique Asymptotes

If a graph approaches a line that is neither horizontal nor vertical as x approaches ∞ or -∞, then that line is called an oblique asymptote.