Foundations — Curve sketching — systematic approach

Before you can sketch a curve, you must be fluent in the little marks the parent note uses without pausing: $f(x)$, $\lim$, $f^{\prime }$, $f^{\prime \prime }$, $\to$, $\infty$, and the words domain, concave, asymptote. This page builds each one from absolute zero — plain words first, then a picture, then why the topic can't live without it. Nothing here assumes you've met calculus before.

0. Two tiny bits of set language we'll reuse

Picture the number line drawn as an unbroken road ($\mathbb{R}$); the braces $\{-1,1\}$ point at two specific mileposts; the $\setminus$ punches those two mileposts out, leaving the road with two pin-prick holes. We need this language the moment we describe where a function is allowed to live.

1. The function machine: what $f(x)$ means

Think of $f$ as a machine. You drop a number $x$ into the top; a single number falls out of the bottom. That output is called $y$, so we write $y=f(x)$ — "$y$ is whatever $f$ does to $x$".

The picture: on the flat floor (the ==$x$-axis, the horizontal number line) we mark the input; we walk straight up to the curve; the height we reach is the output, read off the $y$-axis== (the vertical number line). Every dot on a curve is one pair "(input, its output)". A whole curve is just all those pairs at once.

2. Domain: where the machine is allowed to run

Picture a strip of the $x$-axis painted green where inputs are legal and red where they're forbidden. Above a red point there is simply no curve — a gap. Now, using the set language from Section 0:

Not every gap behaves the same way, and it's worth naming the difference now:

3. Intercepts and solving $f(x)=0$

Picture two pins: one pin on the vertical axis at height $f(0)$, and pins on the horizontal axis wherever the curve dips down to touch it. These pins anchor the middle of your sketch.

The mark "$=0$" is a question, not a statement: "for which inputs is the output zero?" Solving it is answering that question.

4. Symmetry: $f(-x)$ vs $f(x)$

Here $-x$ just means "the mirror-image input on the other side of zero". If $x=3$ then $-x=-3$. Checking symmetry is checking whether the machine treats $x$ and $-x$ as twins (even) or as opposites (odd).

5. The arrow $\to$, the symbol $\infty$, and limits

The picture: watch the height of the curve as you slide your finger along the $x$-axis toward $a$. If the height homes in on a level $L$, that level is the limit — even if there's a hole exactly at $a$. (This is exactly the "hole" from Section 2: a removable discontinuity is precisely the case where the limit exists but $f(a)$ is undefined.)

The little superscripts matter: $x\to a^{+}$ means "creep in from the right (bigger side)", $x\to a^{-}$ means "creep in from the left (smaller side)". A vertical wall can throw the curve to $+\infty$ on one side and $-\infty$ on the other, so we must track each side separately.

6. Asymptotes: the invisible walls, floors, and slants

7. The tangent line, the slope, and the derivative $f^{\prime }$

Putting these together: the steepness of the curve at a point is defined to be the slope of its tangent line there. That number is the derivative.

Reading that formula piece by piece — this is why a limit had to come first:

• $h$ is a tiny step sideways from $x$.
• $f(x+h)-f(x)$ is the rise: how much the height changed over that step.
• Dividing by $h$ makes it rise over run — the slope of the little line joining two nearby points (a secant).
• ${\lim }_{h\to 0}$ shrinks the step to nothing, so the two points merge and the secant becomes the tangent. We couldn't just set $h=0$ (that would be $\frac{0}{0}$, meaningless) — the limit is the only tool that lets $h$ vanish gracefully.

A critical point is where $f^{\prime }(x)=0$ or $f^{\prime }$ doesn't exist — the only places the trend can flip. See First derivative and monotonicity.

8. Concavity and the second derivative $f^{\prime \prime }$

9. How every symbol feeds the topic

Every foundation on the left flows into the seven-step checklist on the right. Notice the limit feeds two branches — asymptotes and the derivative — which is why it earns its place early.

Equipment checklist

Cover the right side and test yourself.

Connections

• Curve sketching — systematic approach — the parent this page equips you for.
• Limits and continuity — the $\lim$, $\to$, $\infty$ machinery in full.
• First derivative and monotonicity — sign of $f^{\prime }$ and critical points.
• Second derivative test — concavity and inflection detail.
• Rational functions and asymptotes — the invisible walls.
• Mean Value Theorem — why the sign of $f^{\prime }$ controls the trend.
• Optimization — using critical points to find best values.