Final answer:
The graph of f(x) has undergone three transformations to become g(x): it has been reflected over the x-axis, stretched vertically by a factor of 4 away from the x-axis, and translated vertically downward by 9√3 units.
The correct option is G.
Step-by-step explanation:
To determine which transformations have been performed on the graph of f(x) = x√3 to obtain the graph of g(x) = -12x - 9√3, we need to look at the changes in the coefficients and understand how these affect the graph's position and shape.
Reflection Over the X-Axis
The original function f(x) has a positive coefficient in front of x. The new function g(x) has a negative coefficient in front of x, indicating that the graph has been reflected over the x-axis since the sign change results in all the y-values being negated.
Vertical Stretch
The original coefficient of x in f(x) was √3, and it has become -12 in g(x). Since the new coefficient is four times larger in magnitude (-12 = -4√3), this indicates that the graph has been stretched vertically by a factor of 4 away from the x-axis.
Vertical Translation
The presence of the -9√3 term in g(x) indicates a vertical translation downward in the coordinate system by 9√3 units.
The correct option is G.