Parametric Equations

Starting point: what you already know

• A Cartesian equation links $x$ and $y$ directly, such as $y=2x+1$ or $x^2+y^2=9$.
• The circle centred at the origin with radius $r$ is $x^2+y^2=r^2$.
• Substitution: if $x=3$ and $y=x+2$, then $y=5$.
• The identity ${\sin }^{2}t+{\cos }^{2}t=1$ is true for every angle $t$.

What parametric form means

In parametric form a curve is given by two equations:

$$x=f(t),y=g(t)$$

The letter $t$ is the parameter. Each value of $t$ gives one point $(x,y)$. As $t$ changes, the point moves and traces the curve.

Getting back to the Cartesian equation

To find the Cartesian equation, eliminate the parameter: get rid of $t$ so only $x$ and $y$ are left.

Method 1: make $t$ the subject of one equation and substitute.

Method 2: use an identity. This is the trick for circles.

Writing a curve in parametric form

You can go the other way too. For the parabola $y=x^2$, let $x=t$. Then $y=t^2$. A curve has many possible parametric forms; any choice that traces the same points works.

Quick summary

• Parametric form gives $x$ and $y$ each in terms of a parameter $t$.
• To get the Cartesian equation, eliminate $t$: make it the subject and substitute, or use ${\sin }^{2}t+{\cos }^{2}t=1$.
• $x=h+r\cos t$, $y=k+r\sin t$ is a circle with centre $(h,k)$ and radius $r$.
• Check the possible values of $x$ and $y$, because the parameter may restrict the curve.

Quiz