Suppose I have an affine variety $X$ in $\mathbb{A}^n_{\mathbb{C}}$. Say I shift it by $(x_1, ..., x_n) \rightarrow (x_1 + a_1, ..., x_n + a_n)$ so that it becomes another affine variety $Y$ in $\mathbb{A}^n_{\mathbb{C}}$.
How come the dimension and degree of $Y$ remain the same as those of $X$? It's intuitively clear but I wasn't really sure why... Any explanation would be appreciated. Thanks you.