Parametric Equations

4 min read

A drone flies across a field, and at each instant its shadow has an x position and a y position. Both depend on time. To describe where the drone is, you do not need a single equation relating x and y. You need two equations, each giving one coordinate in terms of time. Parametric equations work exactly this way: a third variable, the parameter, drives both coordinates and traces out a path.

By the end of this page you can:

• Read a pair of parametric equations as a moving point and plot the curve it traces.
• Track the direction of motion and the range of the parameter.
• Eliminate the parameter to find a single rectangular equation for the curve when one exists.

The floor. A parametric curve is given by x = f(t) and y = g(t). As t increases the point (x, y) moves along a path. Substituting away t gives an ordinary equation in x and y for the same path.

Quick Reference

Before You Start

• Evaluate a function at a value, such as finding cos t and sin t for a given t.
• Read the unit-circle definition where x = cos t and y = sin t.
• Solve a simple equation for one variable so you can substitute it.

The Idea

A parametric curve describes a path one point at a time. Instead of pairing x directly with y, you let a parameter, usually called t, run through an interval. For each value of t you compute an x and a y, and the resulting point (x, y) lands somewhere in the plane. As t increases, the points connect into a curve, and the order in which they appear gives the curve a direction.

The pair x = f(t) and y = g(t) carries more information than a plain equation in x and y. It records where the path starts, which way it moves, and how fast. Often you can eliminate the parameter, meaning you solve one equation for t and substitute into the other, to recover a familiar rectangular equation. The rectangular equation shows the shape, but it forgets the direction and the start and stop, so the parameter form is the fuller description.

Key Formulas

A plane curve is described by parametric equations

\[ x = f(t), \qquad y = g(t), \]

where t ranges over an interval and is called the parameter. The point (x, y) = (f(t), g(t)) traces the curve as t increases.

Source: Stewart 7e, Section 8.4 Plane Curves and Parametric Equations.

The unit circle is parametrized by

\[ x = \cos t, \qquad y = \sin t, \qquad 0 \le t \le 2\pi, \]

since \( x^2 + y^2 = \cos^2 t + \sin^2 t = 1 \).

Source: Stewart 7e, Section 8.4 Plane Curves and Parametric Equations.

Worked Example

Sketch the curve given by x = t + 1 and y = t^2 for t in the interval from -2 to 2, then eliminate the parameter to find a rectangular equation.

Step 1. Build a small table. For t = -2: x = -1, y = 4. For t = -1: x = 0, y = 1. For t = 0: x = 1, y = 0. For t = 1: x = 2, y = 1. For t = 2: x = 3, y = 4. Plotting these points and joining them in order of increasing t traces a parabola opening upward, moving left to right.

Step 2. Eliminate the parameter. Solve the first equation for t: from x = t + 1, you get t = x - 1.

Step 3. Substitute into the second equation: y = t^2 = (x - 1)^2.

Connections

This builds on the unit-circle values of sine and cosine, since the most important parametric curve is the circle x = cos t, y = sin t, and on function notation, since each coordinate is its own function of t. It leads forward to graphing motion in calculus, where the parameter is usually time and the derivatives describe velocity, and it pairs naturally with polar coordinates, a different way to trade two rectangular coordinates for an angle and a radius. Polar graphs are a separate topic and are not covered here.

Resources

• Stewart, Redlin, Watson, Precalculus 7e, Section 8.4 Plane Curves and Parametric Equations (primary).
• OpenStax Precalculus 2e, Section 10.7 Parametric Equations (alternate).

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Last updated: 2026-06-24