Power functions

The parabola $y = x^2$ is only one member of a large family. Raise the exponent to $3$, $4$ or more, and related curves with their own character appear. They are all called power functions and have the plain form $y = x^n$. Whoever knows their few rules can predict the shape of any of them without computing a single point.

The family y = x to the n

A power function has the form $y = x^n$, where $n$ stands for one of the numbers $1, 2, 3, \dots$. For $n = 1$ it is the line $y = x$, for $n = 2$ the parabola, for $n = 3$ the cubic curve. They all pass through the points $(0, 0)$ and $(1, 1)$, because $0^n = 0$ and $1^n = 1$ hold for every such $n$. The larger $n$ grows, the flatter the curve lies against the $x$-axis between $-1$ and $1$, and the more steeply it shoots upwards for $x > 1$.

Even and odd exponents

The decisive difference lies in whether $n$ is even or odd. For even $n$, such as $x^2$ or $x^4$, the value for negative $x$ is always positive, because minus times minus cancels in pairs. The curve therefore never dips below the $x$-axis and is mirrored across the $y$-axis. For odd $n$, such as $x^3$ or $x^5$, one negative sign is left over: negative $x$ give negative values, the curve runs from lower left to upper right.

Two kinds of symmetry

These two behaviours carry names. Even exponents lead to axis-symmetric curves: the left half is the mirror image of the right across the $y$-axis, just like the parabola. Odd exponents lead to point-symmetric curves: turn the curve by $180$ degrees around the origin and it lies on itself again. That is why for odd $n$ the relation $(-x)^n = -x^n$ holds.