Functions
AI Assistant

Power Functions

Explore y = xⁿ — from parabolas to hyperbolas, one exponent at a time

A power function has the form y = xⁿ, where the exponent n controls everything about the shape. When n = 2 you get the familiar parabola. When n = 3 you get an S-shaped curve. When n = ½ you get the square root. When n = −1 you get a hyperbola.

All of these are the same family of functions — they just differ in their exponent. In this lesson, you'll use a slider to sweep through different values of n and watch the curve transform in real time. You'll discover why even exponents make symmetric shapes, why odd exponents pass through the origin with an S-curve, and what happens when the exponent is negative or a fraction.

The gray line y = x stays on the graph as a reference, so you can always see how the power function compares to simple proportionality.

Graph

FAQ

What is a power function?
A power function is any function of the form y = x^n, where n is a constant exponent. Examples include y = x^2 (parabola), y = x^3 (cubic), y = x^{0.5} = \sqrt{x} (square root), and y = x^{-1} = \frac{1}{x} (reciprocal/hyperbola). The variable x is the base and the constant n is the exponent.
What is the difference between a power function and an exponential function?
In a power function y = x^n, the variable is the base and the exponent is constant. In an exponential function y = a^x, the base is constant and the variable is the exponent. For example, y = x^3 is a power function (x is the base), while y = 3^x is exponential (x is the exponent). They grow very differently — exponential functions eventually outpace any power function.
How does the exponent affect the shape of a power function?
The exponent n determines everything: Even integers (n = 2, 4, ...) give U-shaped curves symmetric about the y-axis. Odd integers (n = 3, 5, ...) give S-shaped curves that pass through the origin. Fractions like n = 0.5 give root functions (only defined for x ≥ 0). Negative exponents like n = −1 give hyperbolas with vertical and horizontal asymptotes. The larger |n| is, the steeper the curve grows away from the origin.
What happens when the exponent is 0 or 1?
When n = 0, y = x^0 = 1 for all x ≠ 0 — a horizontal line at y = 1. When n = 1, y = x^1 = x — a straight line through the origin with slope 1. These are the simplest power functions and serve as useful reference points when exploring other exponents.

Related Topics

Exponential Growth & DecayLogarithmic FunctionsQuadratic Equations