In the proof of the fact that after a projective transformation, the equation for any nonsingular cubic curve can be put in the form  y^2z=x(x-z)(x-\lambda z), where  \lambda \neq 0,1, my notes in a middle-step arrives at the cubic

(y-\frac{\alpha x+\beta z}{2})^2z=b_2(x,z),

and then the projective transformation

 [x,y,z]\mapsto [x,y-\frac{\alpha x+\beta z}{2},z]

is applied to transform it to


I can't really see how this works - shouldn't we use the inverse of this projective transformation? To me it seems that replacing y with  y-\frac{\alpha x+\beta z}{2} should give us the cubic  (y-\alpha x-\beta z)^2z=b_4(x,z), which is not really what we wanted.