I am trying to find a general way of finding parametric equations for a rectangular equation. The problem I am working on is $y=x^3$, and I have to find two examples of parametric equations. Obviously, for a first example, $x=t$ and $y=t^3$. But, for a second example, I want $x=t^2+1$ (just for the heck of it), how would I find the corresponding equation for y, in terms of t, so that when I substitute either equation into the other, I get $y=x^3$?
You have a little problem if you choose to take $x = t^2+1$ (assuming that $t\in\Bbb R$). The problem is that you parametrisation does not give the whole of the curve $y=x^3$, but only that part of it for which $x \geq 1$, since for any real $t$, we have $t^2+1 \geq 1$.
If you want a parametrisation for the whole of the curve $y=x^3$, you need to make sure that the function $f(t)$ that you choose for $x$ is injective and surjective.
How about $x=t+1, y=(t+1)^3$, or $x = t^3, y = t^9$?