Let $M=\{(t,f(t))\mid t\in (-1,1)\}$ a sub-manifold of $\mathbb R^2$ of dimension $1$ (so $f$ is at least $\mathcal C^1$). A theorem of my course says that for all $a\in M$, there is a chart $(\varphi ,W)$ where $W$ is an open of $\mathbb R^2$ s.t. $\varphi (a)=0$ and $$\varphi (M\cap W)=\{(x,y)\in W\mid y=0\}.$$
What could be such a Chart in my case ? I was thinking about something as using the derivative, but I'm not sure it really make sense... I'm quite confuse with those charts... can someone explain ?