Function Animation

How to use?

Write an expression using x and t as variables. Since the parameter t grows on its own, the animation plays out. Everything from Math is available: sin, cos, tan, sqrt, abs, PI, and so on.

Animating a function with a parameter

A function animation plots f(x) while a parameter t changes over time. Take f(x) = sin(x + t): let t grow from 0 to 2π and you animate the phase shift of a wave, so the sinusoid looks like it's sliding sideways. Watching it move is what makes affine transformations of graphs click. f(x − h) shifts right by h, a·f(x) stretches vertically by factor a, f(b·x) compresses horizontally by factor b, and f(x) + k shifts up by k. Tie the moving parameter to any one of these and a student sees oscillations, dilations and translations happening as motion, instead of trying to imagine the difference between two still pictures.

Applications: classroom, physics and computer graphics

You'll find it in math teaching (Brazilian BNCC for high-school functions, US Common Core), in physics (graphs of simple harmonic motion, waves, AC circuits) and in computer graphics, where keyframe interpolation and easing curves are parameterized by t. Tools like GeoGebra and Desmos animate sliders to get across exactly the same point.

FAQ

What does the parameter t represent? Think of it as a "time" knob. Turn it and the function changes shape or position. Mathematically, though, it's just another variable.

Which transformation does each placement of t produce? f(x − t) translates horizontally, t·f(x) dilates vertically, f(t·x) dilates horizontally, f(x) + t translates vertically.

Is this the same as a parametric curve? No. Here y depends on x while a separate t controls the shape. In a parametric curve, both x and y depend on t.