Final answer:
The correct transformations to alter the graph of y=x² to y=-2(x-1)²+3 are a horizontal shift to the right by 1 unit, a vertical stretch by a factor of 2 combined with a reflection over the x-axis, and a vertical shift upward by 3 units, making the correct answer option D.
Step-by-step explanation:
The transformation of the graph of y=x² to y=-2(x-1)²+3 involves multiple steps that alter the position and shape of the original parabola. Firstly, there is a horizontal shift, which can be determined by observing the change in the x term within the parentheses. Since the equation is written as (x-1), this implies a horizontal shift to the right by 1 unit, not to the left.
The next transformation is a vertical stretch and reflection. The coefficient of -2 in front of the squared term indicates that the graph of the parabola is flipped over the x-axis due to the negative sign (reflection) and stretched vertically by a factor of 2 because the absolute value of the coefficient is greater than 1.
Finally, there is a vertical shift. The +3 at the end of the equation indicates that the whole graph is shifted up by 3 units in the vertical direction.
Therefore, the correct sequence of transformations to obtain the graph of y=-2(x-1)²+3 from y=x² is Shift to the right by 1 unit, reflect over the x-axis, and stretch vertically by a factor of 2, and then shift upward by 3 units. The correct answer is option D.