This is a question from Lee : Introduction to Smooth manifolds. p.201
For each $a \in \mathbb{R}$, let $M_a$ be the subset of $\mathbb{R}^2$ defined by $M_a = \{(x,y) : y^2 = x(x-1)(x-a)\}$
For which values of $a$ is $M_a$ an embedded submanifold of $\mathbb{R}^2$? For which values can $M_a$ be given a topology and smooth structure making it into a immersed manifold?
I'm not sure how to get started. I know some things, but none of it seems to apply to this problem. For example, we know that that the graph of a smooth map $f:U \subset \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ is an embedded submanifold of $\mathbb{R}^{n+k}$. So maybe we can write this as the graph of a smooth function? Even if that were the case, that only gives sufficiency, and this is asking for necessary and sufficient conditions.