Parametric Equations

Draw circles, ellipses, hearts, and curves that regular functions can't make

Most functions you've seen have the form y = f(x) — one y-value for each x. But what about circles, loops, and curves that double back on themselves? Those need a different approach: parametric equations.

Instead of y depending on x, both x and y depend on a third variable t (think of it as time). As t increases, the point (x(t), y(t)) moves and traces out a curve. The classic example is x = \cos(t),\; y = \sin(t) — as t goes from 0 to 2π, the point traces a perfect circle.

In this lesson, you'll start with that circle, stretch it into an ellipse, create wild Lissajous figures, and even draw a heart — all by changing the parametric formulas.

Related Topics

The Unit CircleSine and CosinePythagorean Theorem

Suggested Lessons

Slope Quadratic Equations Unit Circle Word Problems
What are parametric equations?
Parametric equations define a curve by expressing both x and y as functions of a third variable, usually called t (for time): x = f(t), y = g(t). As t varies over some interval, the point (x, y) traces out a curve. This allows you to draw shapes that aren't possible with a single y = f(x) equation, like circles, loops, and spirals.
How is the parameter t like time?
Think of t as time: at each moment t, a point sits at position (x(t), y(t)). As time moves forward, the point moves and draws a trail. At t = 0, the point starts somewhere; as t increases, it traces the curve. This is why parametric equations are natural for describing motion — projectiles, orbits, and animations all use them.
What is the difference between parametric and regular equations?
A regular equation y = f(x) gives one y for each x — the graph passes the vertical line test. Parametric equations can produce curves that fail the vertical line test (like circles, figure-eights, and spirals) because x and y are independent functions of t. Parametric is more general — any y = f(x) curve can be written parametrically as x = t, y = f(t).
What are some famous parametric curves?
Famous parametric curves include: Circle: (cos t, sin t). Ellipse: (a·cos t, b·sin t). Lissajous figures: (cos(at), sin(bt)) with different frequencies. Cycloid: (t - sin t, 1 - cos t) — the curve a point on a rolling wheel traces. Cardioid: heart-shaped curve. Spirograph patterns: nested circular motion.
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.