Graphing

Comprehensive Curve Sketching

Curve sketching synthesizes every calculus tool into a coherent picture of a function's behavior. Rather than plotting random points, you build the graph systematically so each feature—where it rises, falls, bends, or approaches infinity—is justified by algebra or calculus. The key principle: no single test tells the whole story, but together they eliminate guesswork.

The Systematic Checklist

Follow these steps in order:

1. Domain — identify where $f$ is defined
2. Intercepts — find $f(0)$ (y-intercept) and solve $f(x)=0$ (x-intercepts)
3. Symmetry — check if $f(-x)=f(x)$ (even) or $f(-x)=-f(x)$ (odd)
4. Asymptotes — vertical asymptotes occur where the denominator vanishes but the numerator doesn't; horizontal asymptotes come from $\lim_{x\to\pm\infty} f(x)$
5. Derivative analysis — use $f'(x)$ and $f''(x)$ to refine shape between asymptotes

Derivative Information

The first derivative $f'(x)$ reveals where $f$ increases ($f'>0$) or decreases ($f'<0$). Points where $f'(x)=0$ or is undefined are extrema candidates; confirm them by checking sign changes. The second derivative $f''(x)$ determines concavity: $f''>0$ means concave up, $f''<0$ means concave down, and sign changes at $f''(x)=0$ yield inflection points. Together, sign charts for $f'$ and $f''$ partition the domain into intervals where local behavior is fully determined.

Example: $f(x) = \dfrac{x^2}{x-1}$

• Domain: all reals except $x=1$
• Intercepts: origin at $(0,0)$
• Asymptotes: vertical at $x=1$; slant asymptote $y=x+1$
• First derivative: $f'(x) = \dfrac{x(x-2)}{(x-1)^2}$ gives local max at $(0,0)$ and local min at $(2,4)$
• Second derivative: $f''(x) = \dfrac{2}{(x-1)^3}$ shows concave down on $(-\infty,1)$ and concave up on $(1,\infty)$

Common Pitfall

Don't skip steps or reorder them. Later conclusions (extrema, concavity) must be consistent with earlier ones (domain, asymptotes). A local minimum cannot exist where the function approaches infinity, and the overall shape must respect all constraints simultaneously.