Recipe

Sketch a curve using calculus

Use when: you need an accurate sketch of $y=f(x)$ without a graphing tool, or need to justify a shape a tool already drew

Goal: a sketch whose every feature -- where it's defined, where it turns, which way it bends -- is proved, not guessed

Taught in Curve sketching -- putting it all together

Quick recipe

Find the domain
Find the asymptotes
Find critical numbers and classify them
Find concavity and inflection points
Assemble the sketch

1
Determine every $x$ where $f$ is actually defined.

Example

$f(x)=x^3-3x^2$ is a polynomial, defined for every real $x$ .

Why: every later stage only makes claims about $x$ -values inside the domain, so getting this wrong invalidates everything that follows.
2
Check for vertical asymptotes (a denominator vanishing where the numerator doesn't) and horizontal asymptotes (the limits as $x\to\pm\infty$ ).

Example

$f(x)=x^3-3x^2$ has no denominator and is a polynomial, so it has no asymptotes of either kind.

Why: asymptotes are the frame the rest of the sketch has to respect. Finding them before the fine detail avoids drawing a shape that contradicts them later.
3
Solve $f'(x)=0$ (or find where $f'$ is undefined), then run the First Derivative Test on each one.

Example

$f'(x)=3x^2-6x=3x(x-2)$ , zero at $x=0$ and $x=2$ . Testing signs: $f'(-1)=9>0$ , $f'(1)=-3<0$ , $f'(3)=9>0$ , so $x=0$ is a local max ( $f(0)=0$ ) and $x=2$ is a local min ( $f(2)=-4$ ).

Why: this stage finds every place the curve turns around, and in which direction it's climbing or falling between those turns.
4
Solve $f''(x)=0$ (or find where $f''$ is undefined), then check for an actual sign change on each side.

Example

$f''(x)=6x-6$ , zero at $x=1$ . $f''(0)=-6<0$ (concave down), $f''(2)=6>0$ (concave up), a genuine sign change, so $(1,-2)$ is an inflection point.

Why: concavity is the texture Step 3's turns alone can't capture. The same "climbing" stretch can be curving two different ways.
5
Plot every landmark from Steps 1–4, then connect them respecting each fact: which stretches climb or fall, which way each stretch bends, and where any asymptote constrains the picture.

Why: by this stage every fact needed to fix the curve's shape has already been proved — there's really only one curve left consistent with all of it.

Common mistakes

Skipping straight to Step 5 without running the sign tests, and guessing the shape instead.
Finding a critical number but never actually classifying it with the First Derivative Test.
Confusing "f is increasing" (Step 3) with "f is concave up" (Step 4) -- they answer different questions.
Sketching a curve that visually crosses an asymptote found in Step 2.