Let $y=x^2/2$. Its parametric form is $r(t)=t\,\hat i+t^2/2\,\hat j$, and its evolute is
$ c(t)=-t^3\,\hat i+\frac{3t^2+2}{2}\,\hat j.\tag{1} $
Visually,
When I rewrite $(1)$ as a normal function, by letting $x=-t^3$, I get
$ y=\frac{3x^{2/3}+2}{2}, $
but the graph of this evolute is nothing like the one above. What am I doing wrong?