I'm a little confused with the definition of graph and trace. If I have a function (or a curve) $f:\mathbb R\to \mathbb R,\ f(t)=t^2$ and I draw the graph we have a parabola since the graph is the ordered pairs of this form $(t,f(t))$ but what's the trace? the $\mathbb R$?
On the other hand, we can draw the trace of this function (or curve) $\alpha:\mathbb R\to \mathbb R^2$, $\alpha (t)=(t,t^2)$ which has the same drawing the graph of $f$ and because of this fact is there some relation between $f$ and $\alpha$? Concerning the graph of $\alpha$, can we say it is $(t^2,t^3,\alpha(t))$?
I wrote anything wrong? I would be grateful if someone can help me with my doubts.
Thanks in advance.
Consider $\alpha(t)=(t,t^2) $with $t\in\boldsymbol{R}_+$ and $\beta(t)=\bigl(\exp(t),\exp(2t)\bigr)$ for $t\in\boldsymbol R$. The traces are the same, but $\alpha$ and $\beta$ are different paths: the velocity vectors are different, for example.
Edit (an even more simple example): the traces of $t\mapsto t(1,0)$ and $t\mapsto t(-1,0)$ for $t\in\boldsymbol R$ are the same, namely the first coordinate axis. But the paths are obviously different.