Final answer:
To transform the graph of y = x^2 to y = -2(x - 3)^2 + 5, we apply a horizontal shift to the right by 3 units, a reflection over the x-axis combined with a vertical stretch by a factor of 2, and a vertical shift upwards by 5 units. This corresponds to option (a).
Step-by-step explanation:
To transform the graph of y = x^2 to y = -2(x - 3)^2 + 5, let's analyze the equation step by step:
- The (x - 3) inside the parentheses indicates a horizontal shift to the right side of the coordinate system by 3 units, because we are effectively saying where the graph used to be 'zeroed' at x=0, it is now 'zeroed' at x=3.
- The leading negative sign in front of the 2 indicates a reflection over the x-axis, which is a vertical flip that causes our parabola which opened upwards to now open vertically downward in the coordinate system.
- The coefficient of 2, which is the constant multiplier of the squared term, increases the steepness of the parabola, which is not just a stretch but also involves the reflection due to the negative sign. In absence of the negative, this would be a vertical stretch by a factor of 2; with the negative, it's a vertical stretch combined with a vertical flip.
- Finally, the +5 outside the parentheses indicates a vertical shift upwards by 5 units in the coordinate system.
So, the correct sequence to transform y = x^2 into y = -2(x - 3)^2 + 5 is a horizontal shift to the right by 3 units, reflection and vertical stretch by a factor of 2 (due to multiplication by -2), and a vertical shift upwards by 5 units.
This corresponds to option (a), as it fully describes the transformations applied.