Sketching of Curves

When sketching the graph of a function — by hand or with software — only a part of the true (often infinite) graph can ever actually be shown. The derivatives developed through this chapter decide exactly which part to show and how to show it faithfully. The following seven-point checklist is used throughout:

1. The domain and range of the function.
2. The intercepts of the curve (if any) — set $y=0$ for $x$-intercepts, $x=0$ for the $y$-intercept.
3. The critical points of the function.
4. The local extrema of the function.
5. The intervals of concavity.
6. The points of inflexion (if any).
7. The asymptotes of the curve (if they exist). Worked pattern (Examples 7.69–7.72). Factorise the given expression where possible (this often gives the intercepts immediately); compute $f'(x)$ for the critical points and monotonicity; compute $f''(x)$ for concavity and inflection points; check the degree/behaviour for asymptotes; then assemble all seven pieces of information into a single coherent sketch — a smooth curve that rises/falls exactly where the sign of $f'$ says it should, bends the correct way per the sign of $f''$, and approaches its asymptotes (if any) rather than crossing them arbitrarily. …