The general third degree curve in the Euclidean plane has the equation:

φ = ax^{3} + bx^{2}y + cxy^{2} + dy^{3}
+ ex^{2} + fxy + gy^{2} + hx + iy + j = 0.

If we multiply each coefficient by some non zero field element, we define the same curve.
Suppose we translate the origin: (x, y) = (x' + α, y' + β):

φ = a(x' + α)^{3} + b(x' + α)^{2}(y' + β) + c(x' + α)(y' + β)^{2} + d(y' + β)^{3}
+ e(x' + α)^{2} + f(x' + α)(y' + β) + g(y' + β)^{2}
+ h(x' + α) + i(y' + β) + j

= ax'^{3} + bx'^{2}y' + cx'y'^{2} + dy'^{3}
+ (3aα+e)x'^{2} + (2bα+2cβ+f)x'y' + (3dβ+g)y'^{2}
+ (3aα^{2}+2bαβ+cβ^{2}+h)x' + (bα^{2}+2cαβ+3dβ^{2}+i)y'

+ (aα^{3}+bα^{2}β+cαβ^{2}+dβ^{3}+eα^{2}+fαβ+gβ^{2}+hα+iβ+j)

= ax'^{3} + 3ax'^{2}α + 3ax'α^{2} + aα^{3}

our 3rd order terms remain unchanged, out 2nd order terms are

If we translate the origin to some point on the curve, j becomes 0.