Unit 4: Functions Involving Parameters, Vectors & Matrices

Parametric Curves

Graphing and interpreting parametrically defined curves.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Controlling x and y with a Third Variable

Polar coordinates just used one parameter, θ, to control a point's distance and direction at once. Before reading on: what could a curve gain from using a parameter to control x and y completely independently, instead of writing y directly in terms of x?

Definition — Parametric Curve

A parametric curve defines both coordinates as functions of a third variable, the parameter t: x = x(t), y = y(t). As t varies, the point (x(t), y(t)) traces a path — one that can encode direction and even speed of travel, not just the shape a rectangular equation alone shows.

Worked Example — Eliminating the Parameter

For x(t) = t + 1, y(t) = t² − 2, solve the first equation for the parameter: t = x − 1. Substitute into the second: y = (x − 1)² − 2, a rectangular equation for a parabola. But the parametric form carries more: at t = 0, (x, y) = (1, −2); at t = 1, (2, −1); at t = 2, (3, 2) — check against the rectangular equation: at x = 2, y = (1)² − 2 = −1, matching; at x = 3, y = (2)² − 2 = 2, matching. Since x = t + 1 increases steadily as t increases, the curve is traced left to right — a directional fact the rectangular equation y = (x − 1)² − 2 alone never states.

Worked Example — Parametrizing a Circle

For x(t) = 3cos(t), y(t) = 3sin(t), eliminate the parameter using the Pythagorean identity: (x/3)² + (y/3)² = cos²(t) + sin²(t) = 1, so x² + y² = 9 — a circle of radius 3 centered at the origin. At t = 0, the point is (3, 0); at t = π/2, it's (0, 3); at t = π, it's (−3, 0) — moving counterclockwise as t increases. A single rectangular equation, x² + y² = 9, describes the same circle but says nothing about direction or where the trace begins.

Tip

To eliminate a parameter, solve one equation for t and substitute — unless a trig identity offers a cleaner path, the way cos²(t) + sin²(t) = 1 turned x(t) = 3cos(t), y(t) = 3sin(t) directly into a circle's equation without ever isolating t explicitly.

Common Mistakes

Assuming the rectangular equation obtained by eliminating t captures everything the parametric form describes.
Eliminating t discards direction and timing information — two different parametrizations can trace the exact same rectangular curve while moving through it in opposite directions or at different rates.
Substituting the wrong variable's expression when eliminating the parameter, mixing up which equation was solved for t.
Solve one equation for t explicitly first, then substitute that whole expression into the other equation — working with both equations simultaneously without isolating t first invites sign and algebra errors.

Key Takeaways

A parametric curve defines x and y as separate functions of a shared parameter t, tracing a path as t varies.
Eliminating the parameter (by substitution or a matching identity) produces a rectangular equation for the same curve.
Unlike its rectangular equation, a parametric curve encodes direction of travel, and the trig-identity elimination for a circle mirrors how sin/cos identities appear throughout this course.

Summary

Parametric curves reveal how a curve is traced, not just its shape. The next lesson studies relations that resist even a parametric description as cleanly as a circle — equations left implicit, including the conic sections, that aren't solved for y at all.

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