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

$y = x^4$ is the fourth-degree monomial obtained by multiplying $x$ by itself four times¹. It resembles the parabola $y = x^2$, but it is flatter near the origin and climbs far more steeply away from it.

Domain and range

• The domain is all real numbers
• The range is $y \geq 0$
• The minimum value $0$ is attained at $x = 0$
• Even function

The exponent is even, so the values are never negative.

Symmetry

Since $(-x)^4 = x^4$, the function is even: $x$ and $-x$ give the same value, and the graph is symmetric about the $y$-axis.

Monotonicity and minimum

The derivative is $y' = 4x^3$, negative for $x < 0$ and positive for $x > 0$. The function therefore decreases, turns at $x = 0$ where it attains its minimum, and increases thereafter.

The second derivative $y'' = 12x^2$ is never negative, so the curve is convex everywhere. At $x = 0$ we have $y'' = 0$, but the sign does not change there, so the origin is not an inflection point.

Compared with the parabola

For $|x| < 1$ the quartic is smaller, so the curve is flatter near the origin; for $|x| > 1$ it is larger and grows far faster. The two graphs meet at $x = 0$ and $x = \pm 1$.

Relationships with other functions

It belongs to the family of even power functions $x^n$: the larger the even exponent, the flatter the curve inside $|x| < 1$ and the more nearly vertical its rise outside. Restricted to $x \geq 0$ the function is one-to-one, and its inverse is the fourth root $y = \sqrt[4]{x}$.

Applications

• The deflection of a uniformly loaded beam is a quartic polynomial in position
• A double-well potential such as $x^4 - x^2$ relies on the quartic term to create two stable minima
• Kurtosis, which measures how heavy the tails of a distribution are, is defined from the fourth moment²

1. Quartic function, Wikipedia
2. Kurtosis, Wikipedia