If $x$ and $y$ are continuous functions of $t$ on an interval, then the equations $x = x(t)$ and $y=y(t)$ are parametric equations, and $t$ is the parameter

We can graph parametric equations as follows

Parameterizing a Curve

Writing a function in terms of $y$ and $x$ into two functions in terms of $t$

• Set $x(t)$ to a value inside the domain (can be $t$), or anything else to get another parameterization
• Sub in $x(t)$ into $y(t)$

Parametric

$(\cos(t) \sin(2t))$

• Has a speed
• Has a direction
• Plug in values of $t$ to get a point on a curve

Implicit Function

$y^2 =4(1-x^2)x^2$

• Need both $x$ and $y$
• Hard to find points on the curve