Order of transformations for a quadratic curve

Express \(2x^2-12x+13\) in the form \(a(x-h)^2+k\).[2]

Hence describe a sequence of three transformations that maps the curve \(y=x^2\) onto the curve \(y=2x^2-12x+13\).[3]

A student claims that the sequence "translate \(3\) units right, translate \(5\) units down, then stretch vertically by factor \(2\)" also works. Show that the claim is false, and correct the second step so that the student’s order of transformations works.[3]