Parametric equations

With $x=f(t)$ and $y=g(t)$, one parameter selects both coordinates. Eliminating t describes the curve, but can lose the direction of travel or parameter restrictions. Where dx/dt is nonzero, $dy/dx=(dy/dt)/(dx/dt)$. A second derivative requires differentiating this ratio with respect to t and dividing by dx/dt again.

Eliminate the parameter, preserve the path

For $x=t^2-2t$ and $y=t+1$:

t	-2	-1	0	1	2	3	4
x	8	3	0	-1	0	3	8
y	-1	0	1	2	3	4	5

Eliminate t using $t=y-1$: $x=(y-1)^2-2(y-1)=y^2-4y+3$. Unrestricted real t traces the parabola. Restricting $0\le t\le4$ gives $1\le y\le5$, starting at $(0,1)$ and ending at $(8,5)$, in increasing-t order.

Equal t steps need not mean equal distances; this particle slows towards t=1 and speeds up afterwards. In general $x=f(t),y=g(t)$ on $a\le t\le b$ starts at $(f(a),g(a))$ and ends at $(f(b),g(b))$. Keep direction and restrictions after eliminating t.