Correlation between $x$ and $x^{2}$ Parts of my question have been answered on the website separately. I was trying to find a way to reconcile between the two. The two parts are:

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*The correlation between $x$ and $x^{2}$ is not zero if, let's say, $x$ is distributed uniformly between $0$ and $1$. However, if we shift the $x$ values by $-0.5$, so that $x$ is distributed symmetrically around the y-axis, the correlation becomes close to $0$. This is covered in the answers here: Why are $x$ and $x^2$ correlated?

*Correlation is translation invariant. This is covered here: Pearson correlation - can negative values in your data artificially increase the size of the correlation?
My question is, if correlation is indeed translation invariant, why does it change from non-zero to close to $0$ in the $x$ and $x^{2}$ case after the shift by $-0.5$?

The correlation between $x$ and $y$ is translation invariant:
$$ \text{cor}(x,y) = \text{cor}(x+a,y+b). $$
Thus,
$$ \text{cor}(x,x^2) = \text{cor}(x+a,x^2+b). $$
However, you are not comparing these two quantities. You are comparing $\text{cor}(x,x^2)$ and
$$ \text{cor}(x+a,(x+a)^2)=\text{cor}(x+a,x^2+2ax+a^2) \neq \text{cor}(x+a,x^2+b). $$
The important difference is in the $2ax$ term in the middle expression, which is not a constant $b$ but varies with $x$.
