Final answer:
The transformation from f(x) = x^2 to g(x) = 0.5x^2 represents a vertical compression of the original parabola by a factor of 0.5. This means the graph of g(x) will be wider and not rise as quickly, with all points being at half the height of their counterparts on the graph of f(x).
Step-by-step explanation:
The transformation described in the question involves the function f(x) = x^2 being changed to g(x) = 0.5x^2. This transformation affects the magnitude of the output of the function without altering the input values of x. Specifically, it is a vertical scaling transformation. Every y-value of the original function is multiplied by 0.5 in the new function. As a result, the graph of g(x) compared to f(x) is a vertically compressed version of the parabola f(x), where all the points on the graph of g(x) are halfway between the x-axis and the corresponding points on the graph of f(x).
For example, a point on the original function at y = 4 when x = 2 would be at y = 2 on the new function when x = 2. Consequently, the parabola representing g(x) will be wider and not rise as quickly as f(x)'s parabola. This kind of transformation does not involve any horizontal or vertical translation, nor does it reflect the graph across any axis; it only involves a change in the steepness of the graph.
The vertical scaling factor is 0.5, which indicates that the original function's y-values have been halved. Understanding this transformation can be very useful in various mathematical applications, like adjusting the amplitude of a wave in physics.