1.4 – Parametric Equations

A parametric curve is represented by a pair of equations, $(f(t),g(t))$, defining the coordinates $(x,y)$ for each value of a parameter $t$ over a specified interval. These equations are known as parametric equations, where $f(t)$ and $g(t)$ determine the positions of $x$ and $y$, respectively.

Desmos and graphing calculators typically render a parametric curve over a closed interval. This means that for $a\le t\le b$, the curve is plotted starting from $(f(a),g(a))$ and ending at $(f(b),g(b))$.

Example

A parametric curve is defined by the parametric equations $x=t^2$ and $y=t^3$. When examining the curve over $-2\le t\le 2$:

$t$	$x=t^2$	$y=t^3$
-2	4	-8
-1	1	-1
0	0	0
1	1	1
2	4	8