---
title: "Parametric Equations of Curves"
description: "Describe a curve by giving x and y separately in terms of a parameter, plot it from a table of parameter values, and convert to Cartesian form by eliminating the parameter."
canonical: https://lightmysky.com/learn/mathematics/parametric-equations-of-curves-mt_7NC7SCeU2P
source: https://lightmysky.com/learn/mathematics/parametric-equations-of-curves-mt_7NC7SCeU2P.md
retrieved: 2026-09-12
---

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# Parametric Equations of Curves

Describe a curve by giving x and y separately in terms of a parameter, plot it from a table of parameter values, and convert to Cartesian form by eliminating the parameter.

Subject: Mathematics · Area: Geometry · Ages 16 to 17
Page: https://lightmysky.com/learn/mathematics/parametric-equations-of-curves-mt_7NC7SCeU2P

## Ready when they can

- Plot the curve x = 2t, y = t² for a range of t values
- Eliminate t to get the Cartesian equation of that curve
- Use sin²t + cos²t = 1 to show x = 3 cos t, y = 3 sin t is a circle

## Lesson: Curves drawn by a moving dot

Parametric equations give x and y separately through a third variable. Think of t as time and the point as a moving dot. Pick a spread of t values, compute each pair, and join the dots. Arrows show the travel direction.

**Example.** Take x equals 2t and y equals t squared. Halve x to get t, then substitute: y equals x squared over 4. The table of values traces a parabola. Direction still matters: the same shape backwards is a different parametrisation.

Matching cosine and sine spell a circle. With x as 3 cos t and y as 3 sin t, squaring and adding gives x squared plus y squared equals 9. That is a circle of radius 3 at the origin. As t grows the dot travels anticlockwise from (3, 0).

**Tip.** Always state what the parameter forces on x or y. The Cartesian form can describe a bigger curve than the parameter traces. Elimination is the bridge, but the bridge can widen the road.

**Recap.** Table the parameter, follow the arrows, and eliminate t to reach the Cartesian form.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [The Pythagorean and Quotient Identities](https://lightmysky.com/learn/mathematics/the-pythagorean-and-quotient-identities-mt_NXBq37vxQh)

## Opens up

- [Conic Sections: Ellipse, Parabola and Hyperbola](https://lightmysky.com/learn/mathematics/conic-sections-ellipse-parabola-and-hyperbola-mt_V4YSI3a_Eu)
- [Vector-Valued Functions and Motion Along a Curve](https://lightmysky.com/learn/mathematics/vector-valued-functions-and-motion-along-a-curve-mt_Xcjt6exMy3)