Curve Sketching
A detective walks into a room and reads the whole story from a few clues: the open window, the
muddy footprint, the stopped clock. Curve sketching is exactly that kind of detective
work. You are handed nothing but a formula — no picture, no table, no plot — and from
it alone you deduce the entire life story of the graph: where it comes from, where it crosses
the axes, where it climbs and where it falls, where it turns, how it bends, and where it is
heading as x runs off to infinity. Every fact is proved
before a single point is drawn.
This is not a nostalgic hand-drawing exercise. Engineers sketch the response of a circuit
before simulating it, because a sketch tells them whether the answer can possibly be
right. Economists sketch cost and revenue curves before running the numbers, because the
shape — one break-even point or two? a maximum or a runaway? — is the decision, and
the decimals are just detail. A computer can plot a million points, but only you can say what
the plot means, and only a sketch done from the calculus can guarantee that nothing
interesting has been missed between the plotted points.
You already own every tool this page needs. Curve sketching adds nothing new — it is the
moment all your derivative detective gear gets used on one case at once.
The case-file checklist
Professionals work a curve in a fixed order, because each clue narrows down the next. Here is
the pipeline — commit it to memory:
-
Domain. Where does the formula even make sense? A polynomial lives on all of
\mathbb{R}, but 1/x is banned from
x=0 and \ln x from
x \le 0. Everything you deduce afterwards only holds
inside the domain.
-
Intercepts. The y-intercept is the free gift
f(0); the x-intercepts are the roots of
f(x)=0 — factor if you can.
-
f' — the motion report. Its
sign
says where the curve rises and falls; its zeros are the critical points, which the sign
changes sort into
peaks and valleys.
-
f'' — the bending report.
Concavity says
which way the curve cups, and a sign change in f'' pins an
inflection point — the exact spot where the bend flips.
-
End behaviour, then sketch. Ask what dominates as
x \to \pm\infty, plot the handful of special points, and join
them — respecting every rise/fall and every bend you proved.
Five steps, and the sketch practically draws itself. Let's run the full pipeline on a real
case.
Case file #1: f(x) = x^3 - 3x^2
Step 1 — domain and end behaviour. A polynomial: the domain is all of
\mathbb{R}, no traps. For large |x| the
x^3 term dwarfs everything, so the curve plunges down on the far
left and climbs away on the far right — the classic cubic signature.
Step 2 — intercepts. The y-intercept is
f(0) = 0. For the x-intercepts, factor:
x^3 - 3x^2 = x^2(x - 3) = 0 \quad\Rightarrow\quad x = 0 \ (\text{double}),\ \ x = 3.
The double root at x=0 is a clue in itself: the
x^2 factor doesn't change sign there, so the curve
touches the axis at the origin instead of cutting through it.
Step 3 — the first derivative.
f'(x) = 3x^2 - 6x = 3x(x - 2) = 0 \quad\Rightarrow\quad x = 0,\ x = 2.
Now the sign chart: 3x(x-2) is positive for
x < 0, negative for 0 < x < 2,
positive for x > 2. So the curve rises, falls, rises —
a local maximum at (0, 0) and a
local minimum at (2, -4) (since
f(2) = 8 - 12 = -4). Notice how the clues corroborate: the max at
(0,0) and the "touching" double root are the same event
seen by two different witnesses.
Step 4 — the second derivative.
f''(x) = 6x - 6 = 0 \quad\Rightarrow\quad x = 1, \qquad f(1) = 1 - 3 = -2.
f'' is negative before x=1 and positive
after — a genuine sign change, so (1, -2) is an
inflection point: concave down (frowning) on the left, concave up (smiling)
on the right. And look where it sits — exactly halfway between the two turning points. For any
cubic, it always does.
Step 5 — sketch. Five special points, a rise–fall–rise itinerary, and a
down-then-up bend. Watch the whole case assemble itself, one clue at a time:
Case file #2: a cubic with symmetry
Same pipeline, new suspect: f(x) = x^3 - 3x. This time run the
steps yourself before reading the summary — the whole investigation fits in three lines:
\begin{aligned}
&\textbf{Intercepts:}\ f(0)=0;\quad x^3-3x = x(x^2-3)=0 \Rightarrow x = 0,\ \pm\sqrt3 \\
&\textbf{Slope:}\ f'(x)=3x^2-3=0 \Rightarrow x=\pm1,\quad \text{max at } (-1,2),\ \text{min at } (1,-2) \\
&\textbf{Bend:}\ f''(x)=6x=0 \Rightarrow \text{inflection at } (0,0)
\end{aligned}
A bonus clue you get for free here: every power of x is odd, so
f(-x) = -f(x) — the graph has point symmetry through the origin.
The max at (-1, 2) had to be mirrored by a min at
(1, -2), and the inflection had to sit at the centre of
symmetry. Spotting symmetry early can halve the work. Step through the build:
The twist: a curve with an empty clue drawer
Now try g(x) = x^3 + 3x. It looks like the twin of case #2 — one
sign changed — but run step 3 and something strange happens:
g'(x) = 3x^2 + 3 \;\ge\; 3 \;>\; 0 \quad \text{for every } x.
The equation g'(x) = 0 has no solutions at all. No
critical points, no peaks, no valleys: the curve climbs relentlessly from
-\infty to +\infty without ever pausing.
A beginner might panic — "my recipe found nothing!" — but a detective knows that
no fingerprints is also evidence. An empty answer to step 3 is a positive fact about
the shape: this cubic is strictly increasing, so it crosses the
x-axis exactly once (at x=0) and never
turns back.
Step 4 still earns its keep: g''(x) = 6x changes sign at
x = 0, so there is an inflection at the origin — the climb
eases off as it approaches, then steepens again after. Turning points and inflection points
are independent clues; a curve can have either without the other.
Read the dashed derivative: it dips down towards the axis at x=0
(that's the inflection — the slope's minimum) but never actually reaches zero. The bold curve
obeys: gentlest at the origin, never flat.
The classic trap: pick a few x values, compute
f, join the dots. Try it on case #2,
f(x) = x^3 - 3x, with the innocent-looking sample
x = -2, 0, 2: you get the points
(-2,-2), (0,0),
(2,2) — three dots in a perfect line. Join them and you'd swear the
graph is the straight line y = x, missing both turning
points, both outer intercepts, and the inflection. Everything interesting happened
between your samples.
Points can only ever tell you about the points; it is f' and
f'' that tell you where to look. Solve
f'=0 and f''=0 first, and the
interesting x values hand themselves in — then a handful of
plotted points is genuinely enough.
Second trap: check the domain before trusting the recipe. Run the pipeline
blindly on f(x) = 1/x and you'll find no critical points and no
inflections, shrug, and draw one smooth increasing-or-decreasing curve — sailing straight
across the forbidden point x = 0, where the graph actually tears
into two separate branches with a vertical asymptote between them. Step 1 exists precisely to
catch this: the derivative story only holds inside the domain, one connected piece at
a time.
The three graphs at once
The whole method rests on one alignment: f (bold),
f' (dashed) and f'' (dotted) are three
views of the same object, and the features of the bold curve happen exactly where the other
two cross the axis. Where f'=0, the bold curve has a peak or a
valley; where f''=0 (with a sign change), it has an inflection.
Slide the marker across f(x) = x^3 - 3x and watch the three
witnesses agree. Park it at x = -1: the tangent lies flat, and sure
enough the dashed f' curve is crossing zero on its way down
— a maximum. Park it at x = 0: the tangent is at its steepest
descent, the dashed curve bottoms out, and the dotted f'' crosses
zero — the inflection. Every claim in the checklist is visible here as a crossing.
Because the plotter is doing something surprisingly close to dot-joining — just faster. A
graphing calculator or Desmos samples your function at a grid of x
values, then adaptively subdivides wherever neighbouring samples suggest the curve is
bending, and joins what it finds with tiny straight segments. It is brilliant engineering, but
it is still sampling: a feature narrower than the gap between samples — a needle-thin spike, a
removable hole, a wild oscillation like \sin(1/x) near zero — can
slip straight through the net, and the plotter will serenely draw a wrong picture with total
confidence. The calculus never misses: f' = 0 is an equation, and
an equation finds every solution, however narrow the spike.
This is why the hand-run pipeline survives in the age of software — nowhere more proudly than
in France, where l'étude de fonction is a beloved exam ritual with its own artefact:
the tableau de variations, a tidy table with the domain along the top, the sign of
f' in the middle row, and arrows (↗, ↘) showing the function's
rises and falls beneath, extremes perched at the turning points. Generations of French
students have summarised entire curves in one such table before drawing a single axis. It is
the same detective work you did above, filed in a very elegant folder.
Watch it on Khan Academy
Here's the full pipeline run live on another example — watch for the same rhythm: derivative,
sign chart, second derivative, sketch.