Parametric Equations

We will give some basic methods to eliminate the parameter and do the calculus in parametric equations.

Elimination

The core idea is substitution. Just like we handle with those integration before, we can use simple substitution and complex substitution to eliminate parameters.

e.g.

For ${\displaystyle x=t^2+2t}$, ${\displaystyle y=t-3}$, identify the corresponding curve.

We have ${\displaystyle t=y+3}$, substitute ${\displaystyle t}$ with ${\displaystyle y+3}$, then we have the equation for the parametric equation:

${\displaystyle x+1=(y+4{)}^2}$

When we go further, methods like trig-substitutions can be applied.

Calculus of Parametric Equations

A core formula is:

${\displaystyle \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}}$