Eliminate the parameter

Warm-up

Three quick questions from earlier lessons.

Discover

$x = t + 2$ and $y = t^2$ . Which equation links $x$ and $y$ directly?

$y = (x - 2)^2$
$y = x^2$

Key idea

Removing $t$ shows the shape of the curve.

To eliminate the parameter: solve one equation for

t

, then substitute into the other. For

x = r\cos t

y = r\sin t

, use

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

instead:

x^2 + y^2 = r^2

Worked example

Eliminate $t$ : $x = 3t$ , $y = 6t - 1$ .

t = \dfrac{x}{3}

Solve the simpler one for t.

y = 6 \cdot \dfrac{x}{3} - 1

Substitute into y.

y = 2x - 1

Simplify: a line.

Your turn, guided

Your turn: $x = t - 4$ , $y = t^2$ .

t = x + 4

Add 4.

y = (x + 4)^2

Substitute into y = t². Type y.

Practice ★

$x = t + 6$ and $y = t^2$ . Write $y$ in terms of $x$ .
$x = t + 8$ and $y = t^2$ . Write $y$ in terms of $x$ .
$x = t + 5$ and $y = t^2$ . Write $y$ in terms of $x$ .

Watch out!

Practice ★★

$x = 3t$ and $y = 4t + 9$ . Write $y$ in terms of $x$ .
$x = 2t$ and $y = 2t + 3$ . Write $y$ in terms of $x$ .
$x = 6t$ and $y = 8t + 7$ . Write $y$ in terms of $x$ .

What curve do $x = 5\cos t$ and $y = 5\sin t$ draw, for $0 \le t < 2\pi$ ?
1. A circle of radius 5 centered at the origin
2. A line
3. A circle of radius $25$

Exit ticket

Three questions to check you have got it.

$x = 3t$ and $y = 4t + 9$ . Write $y$ in terms of $x$ .
$x = 2t$ and $y = 2t + 3$ . Write $y$ in terms of $x$ .
$x = 6t$ and $y = 8t + 7$ . Write $y$ in terms of $x$ .

Summary card

Eliminating the parameter

Solve one equation for $t$ .
Substitute into the other.
With $\cos t$ and $\sin t$ : use $\cos^2 t + \sin^2 t = 1$ .

Watch out: solve for $t$ ; do not just replace $t$ by $x$ .

End of the lesson

Next lesson: Circles and ellipses in parametric form Back to the chapter

About this lesson