In parametric equation $x = x(t)$, $y = y(t)$, what does $y(x)$ describe?

Question

When I eliminate a parameter from the parametric equation and receive $y = y(x)$, what does $y$ actually describe? I know that $y$ describes the shape of the parametric equation ONLY, not speed or direction.
In case when $x = x(t)$, $y = y(t)$ has a complicated shape can we actually get the same shape for $y(x)$?
For example:
let$$x = 2\sin(1+3t)$$ and $$y = 2t^3$$ eliminating the parameter t I receive: $$y = 2\left(\frac{\arcsin(\frac{x}{2}) - 1}{3}\right)^3$$ and its graph is just a little part of the graph of the parametric equation. Am I missing something?

Answer 1

Too long for a comment.

The principal value of $\arcsin$ has range $[-\pi/2,\pi/2]\,.$ When $t=1$ your $1+3t$ is outside of that range. If you take another branch of $\arcsin$ (by flipping its sign and adding $\pi$ to the principal value) whose range contains $1+3t=4$ you should be able to find $dy/dx$ with that method:

\begin{align}t&=\frac{-\arcsin(x/2)+\pi-1}{3}\,,&y&=2t^3\,,&\frac{dy}{dx}=\frac{dy}{dt}\frac{dt}{dx}=\frac{-6t^2}{3\sqrt{4-x^2}}\,.\end{align}