You must start with y = x² and end up with y = -2(x - 1)² + 3

Start with this red equation: 

y = x²

which has this red curve as its graph:



Then you multiply the right side of the red
equation by 2 to get the green equation:

y = 2x²

That stretches the red curve vertically by a 
factor of 2, giving the green curve below.



Then you multiply the right side of the green
equation by -1 to get

y = -2x²

That reflects the green curve across the x-axis, 
giving the blue curve below:



Then you replace x by (x - 1) in the right side of
the blue equation to get

y = -2(x - 1)²

That shifts the blue graph 1 unit right, giving the
purple curve below:



Finally you add 3 to the right side of the purple equation
to get

y = -2(x - 1)² + 3 

That shifts the purple graph 3 units upward, giving the
light blue graph below:



That light blue graph is hard to see, so here it is in red:

 

But as you see the transformations in order were:

1. Stretch vertically by a factor of 2.  (Multiply right side by 2)
2. Reflect across the x-axis.  (Multiply right side by -1)
3. Shift right 1 unit. [Replace x by (x-1) ]
4. Shift up 3 units. {Add 3 to the right side.]

Edwin