I have an elliptic curve given of the form:
$$ E \hspace{2mm} : \hspace{2mm} y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$
The discriminant of $E$ is given by $$\Delta(E) = -b_2^2b_8-8b_4^3-27b_6^2+9b_2b_4b_6,$$where $ b_2=a_1^2+4a_4$, $ b_4=2a_4+a_1a_3 $, $b_6=a_3^2+4a_6$ and $b_8=a_1^2a_6+4a_2a_6-a_1a_3a_4+a_2a_3^2-a_4^2$. I now want to show that the discriminant is $\neq0$ if and only if the function $E$ is non-singular.
In order to show one direction I assume that there is a singular point in E, namely $(x_0,y_0)$ . Now a proof in a book says within one sentence: "Since the discriminant doesn't change by transformation $x=x+x_0$ and $y=y+y_0$ w.l.o.g. we can assume that the singular point is $(0,0)$."
But is there a simple way to see that the discriminant doesn't change within such a modification?
Thanks a lot for help