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.

Related Topics

Exponential Growth & DecayLogarithmic FunctionsQuadratic Equations

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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.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.