Parametric equations

Curves defined by a parameter, plotting points from parameter values, eliminating the parameter to get the Cartesian equation, parametric parabolas and circles, and restrictions.

Curves with a parameter

Instead of linking $x$ and $y$ directly, a curve can give each coordinate as a function of a third variable, the parameter (often $t$ or $\theta$). Each value of the parameter gives one point:

$$x = 2t, \qquad y = t^2$$

For example $t = 3$ gives $(6, 9)$ and $t = -1$ gives $(-2, 1)$. A single number now labels each point on the curve.

The parabola x = 2t, y = t²

Drag P along the curve. The readout shows which value of the parameter labels each point.

Try this

Which point has parameter $t = 0$? What happens to $t$ as you move left?

Eliminating the parameter

Example 1: a parabola●●

Find the Cartesian equation of $x = 2t$, $y = t^2$.

$t = \dfrac{x}{2}$

$y = \left(\dfrac{x}{2}\right)^2 = \dfrac{x^2}{4}$, so $x^2 = 4y$.

The circle x = r cos θ, y = r sin θ

As θ runs from 0 to 2π the point travels once around the circle. Blue: the radius to the point.

Try this

Explain why $x^2 + y^2$ always equals $r^2$, whatever $\theta$ is.

Example 2: a circle●●

Find the Cartesian equation of $x = 3\cos\theta$, $y = 3\sin\theta$.

$x^2 + y^2 = 9\cos^2\theta + 9\sin^2\theta = 9(\cos^2\theta + \sin^2\theta)$

$x^2 + y^2 = 9$, a circle with centre the origin and radius $3$.

Restrictions

The parametric form can cover only part of the Cartesian curve, so always check which $x$- and $y$-values are actually reached.

Example 3●●●

Find the curve given by $x = t^2$, $y = 2t^2 + 1$.

$y = 2x + 1$

Substitute t² = x.

Only the part with $x \ge 0$: a ray starting at $(0, 1)$.

x = t² can never be negative.

Flashcards

Parameter

A third variable that gives both $x$ and $y$, labelling each point with one number.

$x = 2at$, $y = at^2$

The parabola $x^2 = 4ay$.

$x = rcos heta$, $y = rsin heta$

The circle $x^2 + y^2 = r^2$.

Eliminating the parameter

Make it the subject and substitute, or use a trig identity.

Practice

The Cartesian equation of $x = 2t$, $y = t^2$ is:
For $x = 2t$, $y = t^2$, find $y$ when $t = 3$.
On $x = 2t$, $y = t^2$, find the parameter of the point $(-4, 4)$.
The Cartesian equation of $x = 3\cos\theta$, $y = 3\sin\theta$ is:
Find the radius of the circle $x = 5\cos t$, $y = 5\sin t$.
The curve $x = t^2$, $y = 2t^2 + 1$ is:
A curve has $x = t + 1$ and $y = t^2$. Find $y$ when $x = 4$.
Eliminating the parameter means:
Eliminate $t$ from $x = 4t$, $y = 2t^2$ and write the result as $x^2 = ky$. Find $k$.
For $x = \cos\theta$, $y = \sin\theta$, the possible $x$-values are: