Mathfolis

Parametric Equations

Unit 9 · Parametric, Polar & Vector

Parametric Equations and Derivatives

A parametric curve is defined by x = x(t) and y = y(t). The slope of the tangent is dy/dx = (dy/dt)/(dx/dt). Horizontal tangents occur where dy/dt = 0 (and dx/dt ≠ 0); vertical tangents occur where dx/dt = 0 (and dy/dt ≠ 0).

Parametric Derivative

dydx=dy/dtdx/dt,dx/dt≠0\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, \quad dx/dt \neq 0
Horizontal tangent: dy/dt = 0, dx/dt ≠ 0
Vertical tangent: dx/dt = 0, dy/dt ≠ 0
Caution: When both dy/dt = 0 and dx/dt = 0, the tangent is indeterminate — do not classify it as horizontal or vertical without further analysis.
Type 1

Slope of the Tangent Line

Compute dx/dt and dy/dt, form the ratio, and evaluate at the given t.

Example 1

For x = t² − t, y = t³ − 3t, find dy/dx at t = 1.

Example 2

For x = eᵗ, y = teᵗ, find the tangent line at t = 0.

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Type 2

Horizontal and Vertical Tangents

Set dy/dt = 0 for horizontal tangents and dx/dt = 0 for vertical tangents, then verify the other derivative is nonzero.

Example 3

For x = t³ − 3t, y = t², find all horizontal and vertical tangents.

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Type 3

Eliminating the Parameter

Solve one equation for t and substitute into the other to get y as a function of x.

Example 4

Eliminate the parameter from x = 2t − 1, y = t² + 1.

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← Arc LengthSecond Derivative (Parametric) →