Parametric Functions Modeling Planar Motion

Unit 4 · Functions Involving Parameters, Vectors, and Matrices

What AP Precalc asks here

Planar motion is naturally modeled by parametric equations. The parameter $t$ represents time; $(x(t), y(t))$ gives the object's position at time $t$ . AP Precalc tests position recovery, direction-of-motion reasoning (without invoking calculus derivatives), and identifying the shape of the trajectory.

Position function

General form

\vec r(t) = (x(t), y(t))

Direction (no calculus)

\text{compare } \vec r(t) \text{ to } \vec r(t + h) \text{ for small } h > 0

AP Tip: To find direction at time

t

without using calculus, compute the position at

t

and at

t + \epsilon

for some small

\epsilon

. The change in coordinates gives the rough direction.

Type 1

Position at a time

Substitute the time into both component functions.

Example 1

An object's position is

\vec r(t) = (3t, t^2 - 1)

. Where is it at

t = 2

? (A)

(2, 3)

(B)

(3, 4)

(C)

(6, 3)

(D)

(6, 4)

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Type 2

Direction of motion

Compare position at $t$ with position at $t + \epsilon$ to see in which direction each coordinate is changing.

Example 2

An object moves with

\vec r(t) = (\cos t, \sin t)

. At

t = 0

, in which direction is it moving? (A) Right (B) Up (C) Left (D) Down

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Type 3

Trajectory shape and key points

Eliminate $t$ to find the rectangular equation. Identify the curve family.

Example 3

An object follows

\vec r(t) = (2\cos t, 3 \sin t)

. What is the shape of its trajectory? (A) Circle of radius 2 (B) Circle of radius 3 (C) Ellipse with semi-axes 2 and 3 (D) Line

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