A quotient map from $[0,1]$ to $S^1$

Maybe here is a general method:

Note that $[0,1]$ is compact and $S^1$ is Hausdorff. So, actually, your function is a closed map. Therefore, your function is a quotient map.

Edit:

Let $f:X\to Y$ be a continuous function where $X$ is compact and $Y$ is Hausdorff. Then, $f$ is a closed map.

Proof:

Pick a closed set $U$ in $X$. Since $X$ is compact, $U$ is compact. Hence, $f(U)$ is compact. Since $Y$ is Hausdorff, $f(U)$ is closed.

You can use the fact that a map $q:X\rightarrow Y $ is a quotient map, if it is continuous, and if it sends saturated open sets to saturated open sets; a subset is saturated if (def.) it containes every fiber {$f^{-1}(y)$} that it intersects.