Curve sketching — systematic approach

WHY do we sketch curves systematically?

A graphing calculator hides understanding. The point of curve sketching is to read a formula and see its geometry. The systematic approach gives a fixed checklist so you never miss a feature (a hidden asymptote, a sneaky inflection point). This is the 80/20 of graphing: ~6 questions capture ~90% of any curve's behaviour.

WHAT to compute — the 7-step checklist

HOW each tool works — derived from first principles

Increasing / decreasing from $f'$

A critical point is where $f'(x)=0$ or $f'$ is undefined. By the First Derivative Test, if $f'$ changes $+\to-$ there is a local max; $-\to+$ a local min; no change → neither.

Concavity from $f''$

Asymptotes — derived as limiting behaviour

Worked Example 1 — a rational function

Sketch $f(x)=\dfrac{x^2}{x^2-1}$.

1. Domain. Denominator $x^2-1=0$ at $x=\pm 1$, so domain is $\mathbb{R}\setminus\{-1,1\}$. Why this step? Excludes points where $f$ blows up.

2. Intercepts. $f(0)=0$ → passes through origin. $f(x)=0\Rightarrow x^2=0\Rightarrow x=0$ only. Why? Origin anchors the middle of the picture.

3. Symmetry. $f(-x)=\frac{x^2}{x^2-1}=f(x)$ → even, mirror over $y$-axis. Why? Halves the work — sketch $x\ge 0$, reflect.

4. Asymptotes. Vertical at $x=\pm1$. Horizontal: $\lim_{x\to\infty}\frac{x^2}{x^2-1}=1$, so $y=1$. Why? These are the invisible walls/floors the curve approaches.

5. First derivative. Quotient rule: $f'(x)=\frac{2x(x^2-1)-x^2(2x)}{(x^2-1)^2}=\frac{-2x}{(x^2-1)^2}.$ Numerator $-2x=0$ at $x=0$; denominator $>0$ always. So $f'>0$ for $x<0$, $f'<0$ for $x>0$. → local max at $(0,0)$. Why? Sign of $f'$ gives the climb/fall; $x=0$ is a peak.

6. Second derivative. Differentiating $f'=\frac{-2x}{(x^2-1)^2}$ gives $f''(x)=\frac{6x^2+2}{(x^2-1)^3}.$ Numerator $6x^2+2>0$ always; denominator $>0$ when $|x|>1$, $<0$ when $|x|<1$. So concave up for $|x|>1$, concave down for $|x|<1$. No inflection (sign change only at undefined $x=\pm1$). Why? Tells us the bending in each region.

7. Assemble: peak at origin, dips down toward $y=1$ asymptote from below for $|x|<1$ side... actually for $|x|<1$, $x^2-1<0$ so $f<0$; for $|x|>1$, $f>1$. Curve sits below $y=1$ between the walls and above $y=1$ outside.

Worked Example 2 — a polynomial

Sketch $f(x)=x^3-3x$.

Domain: all $\mathbb{R}$. Intercepts: $f(0)=0$; $x^3-3x=x(x^2-3)=0\Rightarrow x=0,\pm\sqrt3$. Symmetry: $f(-x)=-x^3+3x=-f(x)$ → odd, symmetric about origin. Asymptotes: none (polynomial; ends $\to\pm\infty$). $f'=3x^2-3=3(x^2-1)=0$ at $x=\pm1$. Sign: $+$ for $|x|>1$, $-$ for $|x|<1$. → local max at $(-1,2)$, local min at $(1,-2)$. Why this step? Identifies the two turning points. $f''=6x=0$ at $x=0$, sign changes $-\to+$ → inflection at $(0,0)$. Assemble: rises, peaks at $(-1,2)$, falls through origin (bending switch), bottoms at $(1,-2)$, rises again.

Common Mistakes

Active Recall

Connections

• Limits and continuity — asymptotes are defined via limits.
• First derivative and monotonicity — increasing/decreasing intervals.
• Second derivative test — concavity and extrema classification.
• Mean Value Theorem — justifies why $f'$ sign controls trend.
• Rational functions and asymptotes — vertical/oblique cases.
• Optimization — finding extrema reuses the same critical-point machinery.

Concept Map

Hinglish (regional understanding)