Is regularity usually preserved by adding a constant to the polynomials cutting out an affine variety?

Question

More explicitly, my question is: if $Y$ is a regular affine variety cut out by $(P_1,\dots,P_r) \subseteq k[X_1,\dots,X_n]$, is the affine variety $Y_d$ cut out by $(P_1+d,\dots,P_r+d)$ regular for almost all $d \in k$ (whatever "almost all" means in this context)? Of course the Jacobian matrix has the same polynomial entries (because the partial derivatives don't care about constants), but they are evaluated at different points of $k^n$, so that I cannot see any obvious reason why this should be true. My motivation comes from easy examples like the circle $X^2+Y^2+d$, that is regular except for $d=0$.