y = x^3 - 3x

Graph of the Cubic Function $y = x^3 - 3x$

$y = x^3 - 3x$ is the most convenient cubic with both a maximum and a minimum¹. The bare $y = x^3$ merely increases, but subtracting $3x$ produces a crest and a trough. Because the coordinates of the extrema are integers, it is a favourite example for building sign charts.

Domain and symmetry

Both the domain and the range are all of the real numbers. Since $f(-x) = -x^3 + 3x = -f(x)$ , the function is odd and the graph has rotational symmetry about the origin.

Intercepts

Factoring as $x^3 - 3x = x(x^2 - 3)$ gives three $x$ -intercepts, at $x = 0$ and $x = \pm\sqrt{3} \approx \pm 1.732$ . This is the case of a cubic equation with three distinct real roots.

Monotonicity and extrema

The derivative is $y' = 3x^2 - 3 = 3(x-1)(x+1)$ .

$x$	$\cdots$	$-1$	$\cdots$	$1$	$\cdots$
$y'$	$+$	$0$	$-$	$0$	$+$
$y$	increasing	maximum $2$	decreasing	minimum $-2$	increasing

Both extrema have integer coordinates, and the crest sits $4$ above the trough.

Concavity

The second derivative is $y'' = 6x$ , which changes sign at $x = 0$ . The inflection point is the origin, with the curve concave down to its left and concave up to its right. The general fact that a cubic's inflection point is its centre of symmetry appears here as the origin itself.

Counting solutions

Intersecting with the horizontal line $y = k$ counts the real solutions of $x^3 - 3x = k$ . The minimum $-2$ and maximum $2$ are the thresholds.

Range of $k$	Real solutions
$\|k\| > 2$	$1$
$\|k\| = 2$	$2$ , one of them repeated
$\|k\| < 2$	$3$

At the thresholds a root is repeated, as the factorisations $x^3 - 3x - 2 = (x+1)^2(x-2)$ and $x^3 - 3x + 2 = (x-1)^2(x+2)$ confirm.

A trigonometric form

Substituting $x = 2\cos\theta$ and using the triple angle formula $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$ turns the function into $y = 2\cos 3\theta$ . The interval $-2 \leq x \leq 2$ corresponds to one full turn of $\theta$ , which is why the extrema are exactly $\pm 2$ .

In terms of Chebyshev polynomials the function is $y = 2T_3\!\left(\dfrac{x}{2}\right)$ ², and it is the equioscillation property of those polynomials, crests and troughs of equal height alternating across the interval, that makes the extreme values come out as neat integers.

Solving the cubic

That substitution is also a practical tool. For $|k| \leq 2$ , solving $2\cos 3\theta = k$ gives the three real roots.

x = 2\cos\left(\frac{1}{3}\arccos\frac{k}{2} + \frac{2n\pi}{3}\right)

This is the casus irreducibilis, where all three roots are real and yet Cardano's formula can only reach them through cube roots of complex numbers; trigonometry gets there directly.

Cubic function, Wikipedia
Chebyshev polynomials, Wikipedia

Graph of the Cubic Function y=x3−3xy = x^3 - 3xy=x3−3x