Comprehensive Curve Sketching

By the end of this lesson, you will be able to perform a complete curve sketch using calculus: domain, intercepts, symmetry, asymptotes, intervals of increase/decrease, local extrema, concavity, and inflection points.

The Complete Sketch Checklist

1. Domain — where is $f$ defined?
2. Intercepts — $x$-intercepts (set $f = 0$) and $y$-intercept ($f(0)$)
3. Symmetry — even ($f(-x) = f(x)$) or odd ($f(-x) = -f(x)$)?
4. Asymptotes — horizontal (limits at $\pm\infty$), vertical (denominator = 0)
5. Increasing/Decreasing — sign of $f'$; critical points where $f' = 0$
6. Local Extrema — via First or Second Derivative Test
7. Concavity — sign of $f''$; inflection points
8. Sketch — combine all information

Example 1 — Polynomial Sketch

$f(x) = x^4 - 8x^2 + 7$

1. Domain: all reals

2. Intercepts: $f(0) = 7$; $f(x) = 0$: $x^2 = \frac{8 \pm 6}{2} = 7$ or $1$; $x = \pm\sqrt{7} \approx \pm 2.65$ and $x = \pm 1$.

3. Symmetry: $f(-x) = f(x)$ → even (symmetric about $y$-axis).

4. Asymptotes: none (polynomial)

5. $f'(x) = 4x^3 - 16x = 4x(x^2-4) = 4x(x-2)(x+2)$

Critical points: $x = 0, \pm 2$

Sign: $(-\infty,-2)$: $f' < 0$; $(-2,0)$: $f' > 0$; $(0,2)$: $f' < 0$; $(2,\infty)$: $f' > 0$

6. Local max at $x = 0$: $f(0) = 7$; Local min at $x = \pm 2$: $f(\pm 2) = 16-32+7 = -9$

7. $f''(x) = 12x^2 - 16 = 0$: $x = \pm\frac{2}{\sqrt{3}} \approx \pm 1.15$

$f'' > 0$ for $|x| > 2/\sqrt{3}$ (concave up); $f'' < 0$ for $|x| < 2/\sqrt{3}$ (concave down)

Inflection points at $x = \pm 2/\sqrt{3}$: $f(\pm 2/\sqrt{3}) = \frac{16}{9} - \frac{32}{3} + 7 = \frac{16-96+63}{9} = -\frac{17}{9}$

Example 2 — Rational Function Sketch

$f(x) = \dfrac{x}{x^2-4}$

Domain: $x \neq \pm 2$

Intercepts: $(0, 0)$ (both $x$ and $y$-intercept)