Final answer:
The function transformation from y = x² to y = -2x² reflects the graph across the x-axis due to the negative sign and vertically stretches it by a factor of 2, making the parabola narrower and upside down.
Step-by-step explanation:
The function transformation from y = x² to y = -2x² involves two main changes. First, the introduction of the negative sign reflects the graph across the x-axis. In other words, all points on the graph of y = x² are flipped to the opposite side of the x-axis, creating a visual symmetry where if (x, y) is on the original graph, then (x, -y) will be on the transformed graph.
Secondly, the coefficient 2 is a vertical stretch which multiplies all y-values by 2, further affecting the shape of the graph. The graph of y = -2x² will be narrower and steeper than the graph of y = x². Together, these transformations make the function y = -2x² an upside-down and vertically stretched version of the original function y = x².