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

$y = x^5$ is the fifth-degree monomial obtained by multiplying $x$ by itself five times. Its domain is all real numbers and its range is all real numbers as well: 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 $x$.

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.

The derivative is $y' = 5x^4$, which is positive everywhere except at $x = 0$ (where it is zero), 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 ($y = 0$). The second derivative $y'' = 20x^3$ changes sign at $x = 0$, 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.

For $|x| > 1$, $x^5$ grows and falls more steeply than $x^3$, while for $|x| < 1$ it stays closer to $0$. Comparing values at the same $x$:

• $x = 2$: $x^3 = 8$, $x^5 = 32$
• $x = 0.5$: $x^3 = 0.125$, $x^5 = 0.03125$
• $x = -2$: $x^3 = -8$, $x^5 = -32$

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 $(1, 1)$. General polynomials of degree five and higher have no solution formula in radicals (the Abel–Ruffini theorem), and $x^5$ is the simplest case where this begins. It also appears as a higher-order term in Taylor series and in models of sharply varying phenomena.