Graph of the Quintic Function $y = x^5$

$y = x^5$ is the fifth-degree monomial obtained by multiplying $x$ by itself five times¹.

Domain and range

• Both the domain and the range are all real numbers
• The function increases monotonically
• Odd function
• The origin is an inflection point

Every real $x$ yields exactly one real value, very large and positive for large positive $x$, and large in magnitude but negative for large negative $x$.

Symmetry

Because the exponent is odd, $(-x)^5 = -x^5$, making it an odd function with point symmetry about the origin. For example, $x = 2$ gives $32$ while $x = -2$ gives $-32$: only the sign flips. The graph lies in the first and third quadrants.

Monotonicity

The derivative is $y' = 5x^4$, which is positive everywhere except at $x = 0$, so the function is monotonically increasing over the whole line, like the odd functions $y = x$ and $y = x^3$.

At the origin $y' = 0$, so the tangent line coincides with the $x$-axis. The second derivative $y'' = 20x^3$ changes sign there, so the origin is an inflection point. The graph is therefore extremely flat near the origin, even flatter than $x^3$, lying down before rising again.

Compared with the cube

For $|x| > 1$ the quintic grows and falls more steeply; for $|x| < 1$ it stays closer to $0$.

Relationships with other functions

It belongs to the family of power functions $x^n$ and is the archetype for odd $n$. Every odd power passes through $(-1, -1)$, $(0, 0)$ and $(1, 1)$.

Why degree five matters

General polynomials of degree five and higher have no solution formula in radicals, a fact known as the Abel-Ruffini theorem². The monomial $x^5$ is the simplest case where this begins: the equation $x^5 = a$ is still solvable by a fifth root, but the general quintic is not. It also appears as a higher-order term in Taylor series and in models of sharply varying phenomena.

1. Quintic function, Wikipedia
2. Abel-Ruffini theorem, Wikipedia