Curve Sketching

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.

Limits and asymptotes

Use one-sided limits near domain breaks to determine vertical-asymptote behaviour. Use limits as $x\to\pm\infty$ to describe end behaviour and horizontal or polynomial asymptotes.

For rational functions, factoring distinguishes holes from vertical asymptotes, while division can reveal an oblique asymptote.

First derivative analysis

Compute $f'$, find critical numbers, and include domain boundaries on the interval chart. Determine where $f'>0$ or $f'<0$.

This gives increasing/decreasing intervals and classifies local maxima and minima through sign changes.

Second derivative analysis

Compute $f''$ and find candidates where it is zero or undefined. Test its sign to determine concave-up and concave-down intervals.

An inflection point occurs only when concavity changes and the original function has a point there.

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.

Plot in a deliberate order

1. Draw axes and asymptotes.
2. Mark holes, intercepts, extrema, and inflection points.
3. Respect increasing/decreasing arrows.
4. Shape each interval with the correct concavity.
5. connect only across continuous parts of the domain.
6. match end behaviour.

A sketch need not be pixel-perfect, but it must satisfy all established facts.

Technology check

Compare with a graphing calculator using a window that shows relevant features. If the graph seems to disagree, examine window scale, missed roots, algebra errors, and domain restrictions.

Do not copy a screen without documenting the analysis; the goal is understanding why the curve has its shape.

Contextual graphs

A mathematical function may be defined more broadly than a real model. Restrict time, length, quantity, or other variables to meaningful values and label units.

Only extrema and intercepts within the contextual domain should be interpreted as real events.

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?

Related topics

• Critical Points
• First Derivative Test
• Concavity
• Limits