As we know, a curve in $2$-dim Euclidean space is defined as a map $\alpha: I \rightarrow \mathbb{R}^2$, where $I$ is an open interval. The set $\alpha(I) \subset \mathbb{R}^2$ represents the trace or image of the curve. The property of being smooth or analytic for a curve depends on whether the map $\alpha$ is smooth or analytic, respectively.
Interestingly, the smoothness of a curve does not necessarily correlate with the smoothness of its image. For instance, the function $y = \sqrt[3]{x^2}$ is not differentiable at $x=0$. However, we can identify an analytic map $x = t^3$, $y = t^2$ with same trace.
My question focuses on how to construct a smooth, or even an analytic, map that corresponds to the trace $y=|x|$.
For a smooth map, I have managed to construct $x = t e^{-1/t^2}$, $y = |t| e^{-1/t^2}$. It is verifiable that the $n$-th derivative at $t=0$ is $0$.
How would one go about constructing an analytic map for this trace? Alternatively, is it possible to prove that no such analytic map exists for this trace?