Parametric Functions

Unit 4 · Functions Involving Parameters, Vectors, and Matrices

What AP Precalc asks here

A parametric function is a pair $(x(t), y(t))$ where both coordinates depend on a parameter $t$ . As $t$ varies, the point $(x(t), y(t))$ traces a curve in the plane. Eliminating $t$ — usually by solving for $t$ in one equation and substituting into the other — recovers a rectangular relation $y = f(x)$ or $F(x, y) = 0$ .

Parametric form

Pair of equations

x = x(t), \;\; y = y(t)

Eliminate t

\text{Solve one for } t, \text{ substitute into the other.}

AP Tip: Some parametrizations (e.g.,

x = \cos t, y = \sin t

) are easier to eliminate using identities (

\cos^2 t + \sin^2 t = 1

gives

x^2 + y^2 = 1

) than by solving for

t

Type 1

Evaluate at a value

Plug $t$ into both component functions and report the point.

Example 1

For

x = 2t + 1, y = t^2

, find the point at

t = 3

. (A)

(6, 6)

(B)

(7, 9)

(C)

(7, 6)

(D)

(3, 9)

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Type 2

Eliminate the parameter

Solve one equation for $t$ , substitute into the other.

Example 2

Eliminate the parameter from

x = t, y = t^2

. (A)

y = x^2

(B)

y = \sqrt x

(C)

y^2 = x

(D)

y = x

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Type 3

Match parametric equations to a curve

Use elimination or recognize standard parametrizations of circles, lines, parabolas.

Example 3

Identify the curve traced by

x = \cos t, y = \sin t

for

t \in [0, 2\pi]

. (A) Unit circle (B) Parabola (C) Line through the origin (D) Ellipse with semi-axes 2 and 1

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