Calculus · step 13 of 13

Curve sketching: asymptotes, concavity & inflection

A full sketch is a checklist, not an art: domain, intercepts, asymptotes, first derivative for increase and turning points, second derivative for concavity and inflection, then behaviour at the extremes. Vertical asymptotes come from zeros of a denominator that do not cancel; horizontal ones from the limit as x → ±∞; and an inflection needs the second derivative to actually CHANGE sign, not merely vanish.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Find the domain and interceptsExclude denominator zeros. Set x = 0 for the y-intercept and y = 0 for the x-intercepts.
Locate the asymptotesVertical where the denominator vanishes and the numerator does not; horizontal from lim_{x→±∞} f; oblique when the numerator degree exceeds the denominator by exactly one.
Use f′ for shapeSolve f′ = 0 for turning points and build a sign chart for increasing and decreasing intervals.
Use f″ for concavityf″ > 0 is concave up, f″ < 0 concave down. An inflection point requires a sign change.

Worked example

Find the asymptotes and inflection points of f(x) = x/(x² + 1).

Domain is all of ℝ since x² + 1 never vanishes — no vertical asymptotes.
As x → ±∞, f ≈ 1/x → 0, so y = 0 is a horizontal asymptote.
f′ = (1 − x²)/(x²+1)², zero at x = ±1 — turning points.
f″ = 2x(x² − 3)/(x²+1)³, zero at x = 0 and x = ±√3, and it changes sign at each.

Answer. Horizontal asymptote y = 0, no vertical asymptotes, turning points at x = ±1, and three inflection points at x = 0, ±√3.

Where marks get dropped

These are the specific errors that cost credit on curve sketching: asymptotes, concavity & inflection questions — QED's rubric penalises each of them separately.

Declaring an inflection point wherever f″ = 0. x⁴ has f″(0) = 0 but stays concave up — the sign must change.
Assuming a vertical asymptote at every denominator zero. If the numerator vanishes there too, it may be a removable hole instead.
Missing an oblique asymptote. When the numerator’s degree is one more than the denominator’s, polynomial division reveals a slanted asymptote.

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Curve sketching: asymptotes, concavity & inflection — frequently asked questions

How do I find an oblique asymptote?

Divide the numerator by the denominator. The polynomial quotient is the asymptote, since the remainder term tends to 0.

Can a graph cross its horizontal asymptote?

Yes. An asymptote describes behaviour at infinity only — x/(x²+1) crosses y = 0 at the origin.

What order should I do the checks in?

Domain, intercepts, asymptotes, f′, f″, then extremes. Doing derivatives before finding the domain wastes effort on points that are not in it.

The rest of Calculus

Limits, derivatives, integrals and their applications. Each subtopic below has its own method, worked example and mark-losing traps.

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