Parametric equations: points and motion

Warm-up

Three quick questions from earlier lessons.

Discover

A drone flies so that after $t$ seconds it is at $x = 2t$ meters east and $y = t + 1$ meters up. Where is it after 3 seconds?

$(6, 4)$
$(4, 6)$

Key idea

Parametric equations give $x$ and $y$ separately, in terms of a third variable $t$ (often time).

The point at time

t

is

\big(x(t), y(t)\big)

. As

t

increases, the point moves along the curve; the parameter tells when it is where.

Worked example

$x(t) = t^2 - 1$ , $y(t) = 3t$ . Find the point at $t = -2$ .

x(-2) = (-2)^2 - 1 = 3

Replace t by −2 in x, with parentheses.

y(-2) = 3 \cdot (-2) = -6

Same t in y.

The point is

(3, -6)

Both coordinates.

Your turn, guided

Your turn: same curve, $t = 2$ .

x(2) = 3

2² − 1.

y(2) = 6

, so the point is

(3, 6)

3 · 2.

Practice ★

A point moves along $x(t) = t^2 - 4$ , $y(t) = 4t + 2$ . Where is it at $t = 4$ ?
A point moves along $x(t) = t^2 - 7$ , $y(t) = 5t + 2$ . Where is it at $t = 3$ ?
A point moves along $x(t) = t^2 - 5$ , $y(t) = 3t + 2$ . Where is it at $t = 4$ ?

Watch out!

Practice ★★

$x(t) = t + 1$ and $y(t) = t^2$ . Tap the point where the curve is at $t = 1$ .
$x(t) = t + 1$ and $y(t) = t^2$ . Tap the point where the curve is at $t = 0$ .

$x(t) = 4t + 4$ and $y(t) = t - 1$ . At what time $t$ does the point have $x = 8$ ?
$x(t) = 2t + 6$ and $y(t) = t - 1$ . At what time $t$ does the point have $x = 18$ ?
$x(t) = 2t + 1$ and $y(t) = t - 1$ . At what time $t$ does the point have $x = 3$ ?

Exit ticket

Three questions to check you have got it.

$x(t) = 4t + 4$ and $y(t) = t - 1$ . At what time $t$ does the point have $x = 8$ ?
$x(t) = 2t + 6$ and $y(t) = t - 1$ . At what time $t$ does the point have $x = 18$ ?
$x(t) = 2t + 1$ and $y(t) = t - 1$ . At what time $t$ does the point have $x = 3$ ?

Summary card

Parametric equations

The point at time $t$ is $\big(x(t), y(t)\big)$ .
"Where at time $t$ ": evaluate. "When at $x = c$ ": solve $x(t) = c$ .

Watch out: $x$ first, then $y$ .

About this lesson