Curve Sketching

Curve sketching combines algebra, limits, first derivatives, and second derivatives into a coherent graph.

Critical points and concavity changes organize the shape of a differentiable graph. A smooth curve has a local maximum, local minimum, and inflection point marked. Sign bands below the axis show intervals of increase and decrease, while curvature visibly changes near the inflection point.

A strong sketch is an evidence-based summary of a function's domain, key points, end behaviour, slopes, and curvature.

Start with the domain

Identify inputs for which the formula is defined. Denominator zeros, even-root restrictions, and logarithm arguments can split the graph into separate pieces.

Mark excluded values before derivative work. They may create holes, vertical asymptotes, or boundaries but are not automatically critical points.

Intercepts and symmetry

Find the $y$-intercept by evaluating $f(0)$ when allowed. Find $x$-intercepts by solving $f(x)=0$.

Check $f(-x)$. If $f(-x)=f(x)$, the graph is even and symmetric about the $y$-axis. If $f(-x)=-f(x)$, it is odd and has rotational symmetry about the origin.

Worked example: a polynomial sketch

These facts determine the sketch's shape and scale landmarks.

Sign-chart organization

Use separate but aligned charts for $f'$ and $f''$. Mixing critical numbers with inflection candidates in one unlabeled row can confuse their meanings.

Label intervals, signs, and conclusions. A derivative zero may influence monotonicity without changing concavity, or vice versa.

Common mistakes

Beginning with derivative calculations before the domain. Excluded values organize everything else.

Connecting across an asymptote or hole. Respect discontinuities.

Calling every $f'=0$ point an extremum. Check sign changes.

Calling every $f''=0$ point an inflection point. Concavity must change.

Ignoring end behaviour after plotting local features. The whole curve must remain consistent.

Quick self-check

Are domain, intercepts, symmetry, and asymptotes known?
Where is the function increasing and decreasing?
Which critical points are actually extrema?
Where is the graph concave up or down, and where does concavity change?
Do endpoints and end behaviour match the algebra?
Does technology confirm rather than replace the reasoning?