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Curve Sketching

Curve sketching combines the tools of differential calculus to draw the graph of a function faithfully. This page follows Walz, Foundations of Mathematics, the second part of Section 6.2: it studies convexity and concavity, inflection points, zeros, symmetry, and behaviour for large $x$, and then assembles these into a complete analysis.

It builds directly on Applications of Derivatives and Extrema, where extrema and higher derivatives were introduced.

Convexity and Concavity

A region of the graph of $f$ is convex if, for every pair of points $x < y$ in it, the curve between $(x \mid f(x))$ and $(y \mid f(y))$ lies below the secant through those points; it is concave if the curve lies above the secant. Reflecting the graph in the $x$-axis (replacing $f$ by $-f$) turns a concave region into a convex one and vice versa.

For twice-differentiable functions these geometric notions are captured by the sign of the second derivative.

Convexity and the sign of $f''$. The graph of a twice-differentiable function $f$ is convex exactly where $f''(x) \ge 0$, and concave exactly where $f''(x) \le 0$.

The reason is that the convex regions are precisely where the first derivative $f'$ is monotonically increasing, i.e. where $(f')' = f'' \ge 0$. For example $f(x) = \tfrac{1}{3}x^3 - x^2 + \tfrac{2}{3}$ has $f''(x) = 2x - 2$, so its graph is concave for $x \le 1$ and convex for $x \ge 1$.

Inflection Points

An inflection point is a location $x_0$ where the graph changes between convex and concave. Equivalently:

Inflection points. A point $x_0$ is an inflection point of $f$ if $f'$ has a local extremum at $x_0$. The point $(x_0 \mid f(x_0))$ is then the turning point of the graph.

Because inflection points of $f$ are local extrema of $f'$, the extremum criteria from the previous page apply to $f'$:

Criteria for inflection points.

  • Necessary: if a twice-differentiable $f$ has an inflection point at $x_0$, then $f''(x_0) = 0$.
  • Sufficient: if $f$ is three times differentiable with $f''(x_0) = 0$ and $f'''(x_0) \neq 0$, then $x_0$ is an inflection point.

For $f(x) = \tfrac{1}{3}x^3 - x^2 + \tfrac{2}{3}$ we have $f''(x) = 2x - 2$, so $f''(1) = 0$, and $f'''(x) = 2 \neq 0$; hence $x_0 = 1$ is an inflection point, where the graph passes from concave to convex.

An inflection point at which the first derivative also vanishes ($f'(x_0) = 0$) is called a saddle point. The standard example is $f(x) = \tfrac{1}{3}x^3$ at $x_0 = 0$, where $f'(0) = f''(0) = 0$ but $f'''(0) = 2 \neq 0$.

Additional Sources of Information

Beyond derivatives, several other features help pin down the shape of a graph.

Zeros

The zeros of $f$ are the locations $x_0$ with $f(x_0) = 0$. They are found with the equation-solving methods of algebra: factoring out common terms, splitting off known linear factors, and applying the quadratic formula. For example $g(x) = x^2 - 2x = x(x - 2)$ has zeros $x_0 = 0$ and $x_1 = 2$.

Symmetry

  • The graph is axisymmetric about the vertical line $x = a$ if $f(a + x) = f(a - x)$ for all $x$. For example $f(x) = x^2$ is symmetric about the $y$-axis, since $f(-x) = f(x)$.
  • The graph is point-symmetric about a point $(a \mid b)$ if $f(a + x) = -f(a - x) + 2b$ for all $x$. For example $f(x) = \tfrac{1}{3}x^3$ and $f(x) = \dfrac{x}{x^2 + 1}$ are point-symmetric about the origin, since $f(-x) = -f(x)$.

Behaviour for Large $x$

Investigating $\lim_{x \to \infty} f(x)$ reveals the long-range behaviour. If the limit is a real number $S$, the constant function $a_f(x) = S$ is a horizontal asymptote; otherwise there is none. For a rational function, factor out the highest power in numerator and denominator, then compare degrees:

  • If the numerator has smaller degree, e.g. $\dfrac{x}{x^2 + 1} \to 0$, the asymptote is the $x$-axis.
  • If the degrees are equal, e.g. $\dfrac{x^2}{x^2 + 1} \to 1$, the asymptote is $y = 1$.
  • If the numerator has larger degree, e.g. $\dfrac{x^3}{x^2 + 1} \to \infty$, there is no horizontal asymptote.

A Complete Curve Discussion

A full curve sketch collects: domain, zeros, symmetry, monotonicity regions, local and global extrema, convexity and inflection points, and behaviour for large $x$. Two representative examples from Walz illustrate the process.

  • A family of cubics. For $f_t(x) = \tfrac{1}{t^2}x^3 - x$ with parameter $t > 0$, factoring gives zeros $x = 0, \pm t$; the graph is point-symmetric about the origin; the extrema are $x_{3,4} = \pm\tfrac{\sqrt{3}}{3}\,t$ (a minimum and a maximum, by the second-derivative test); and the single inflection point is $x_0 = 0$. As $t$ varies, the extreme points trace the line $g(x) = -\tfrac{2}{3}x$.
  • An optimisation problem. To minimise the surface area of a cylindrical can of fixed volume $V = 1$, express the area as $O(r) = 2\bigl(\pi r^2 + \tfrac{1}{r}\bigr)$ for $r > 0$. Then $O'(r) = 0$ gives $r_0 = \sqrt[3]{1/(2\pi)} \approx 0.54$, and $O''(r_0) = 12\pi > 0$ confirms a minimum. Checking $O(r) \to \infty$ as $r \to 0^+$ and $r \to \infty$ shows this local minimum is the unique global minimum.

Where This Leads

Differential calculus answers "how steep?" and "where largest?". The companion topic, integral calculus, answers "how much area?". The page The Definite Integral begins that development, which the Main Theorem will connect back to differentiation.