y = x^4 - x^2

Graph of the Quartic Function $y = x^4 - x^2$

$y = x^4 - x^2$ is a W-shaped curve with two valleys and a hill between them. In physics it is the double-well potential, the standard shape used to explain symmetry breaking¹. Where the bare $y = x^4$ has a single valley, subtracting $x^2$ splits its floor into two.

Domain and symmetry

The domain is all real numbers. Only even powers appear, so the function is even and the graph is symmetric about the $y$ -axis.

Intercepts

Factoring as $x^4 - x^2 = x^2(x-1)(x+1)$ gives a double root at $x = 0$ and simple roots at $x = \pm 1$ . The curve touches the $x$ -axis at the origin and crosses it at $\pm 1$ .

Minimum by completing the square

Treating $x^2$ as a single quantity and completing the square gives the following.

x^4 - x^2 = \left(x^2 - \frac{1}{2}\right)^2 - \frac{1}{4}

The first term on the right is non-negative, so the minimum value is $-\dfrac{1}{4}$ , attained when $x^2 = \dfrac{1}{2}$ , that is at $x = \pm\dfrac{1}{\sqrt{2}}$ . Getting both the depth and the position of the valleys without calculus is the appeal of this form.

Extrema and inflection

The derivative is $y' = 4x^3 - 2x = 2x(2x^2 - 1)$ and the second derivative is $y'' = 12x^2 - 2$ .

$x$	$y$	Role
$\pm\dfrac{1}{\sqrt{2}} \approx \pm 0.707$	$-\dfrac{1}{4}$	minimum
$\pm\dfrac{1}{\sqrt{6}} \approx \pm 0.408$	$-\dfrac{5}{36}$	inflection
$0$	$0$	local maximum

The range is $y \geq -\dfrac{1}{4}$ . The central hill rises $\dfrac{1}{4}$ above the valleys, and that height is the barrier separating them. The curve is concave down between the inflection points and concave up outside.

Viewing it as a quadratic in $x^2$

Setting $t = x^2$ turns the function into $y = t^2 - t$ , a quadratic in $t$ whose vertex is at $t = \dfrac{1}{2}$ with value $-\dfrac{1}{4}$ . That parabola has just one valley. But the map $t = x^2$ sends each positive $t$ back to two values of $x$ , and it is this folding that splits the single valley in two.

Sign of the $x^2$ coefficient	Vertex in $t$	Shape in $x$
negative	$t > 0$ , reachable	two valleys (W shape)
positive	$t < 0$ , unreachable	a single valley

Every even quartic that takes a W shape can be understood this way.

Symmetry breaking

The reason this shape matters in physics is that the formula is symmetric while the most stable state is not: something must choose one side or the other². In Landau theory the free energy is written $F = a(T - T_c)\varphi^2 + b\varphi^4$ ; above the transition temperature the coefficient of $\varphi^2$ is positive and there is one valley, while below it the coefficient turns negative and the double well appears.

A ferromagnet settling spontaneously into one direction of magnetisation as it cools is precisely this splitting of the valley, and the Higgs mechanism in particle physics rests on a potential of the same form.

Quantum mechanics

With two valleys, a classical particle that has fallen into one can never surmount the barrier. In quantum mechanics, however, tunnelling lets it move between them, and as a result the lowest energy level splits slightly into two. The inversion of the ammonia molecule is the classic example.

Double-well potential, Wikipedia
Spontaneous symmetry breaking, Wikipedia

Graph of the Quartic Function y=x4−x2y = x^4 - x^2y=x4−x2