Question

what is the equation of y=x^3 with the transformation: vertical compression by a factor of 1/2, horizontal shift 7 units to the left, reflection across the x-axis?

Answer 1

Assuming the transformations are in order,
because S(x,y) = (x, 1/2y), reverse what is done to y and simplify,
y = x^3 > y = 1/2*x^3 to compress vertically,
because S(x,y) = (x-7,y), reverse what is done to x,
y = 1/2*(x+7)^3 to shift to the left,
reflection across the x-axis changes positives to negatives,
y = -1/2(x+7)^3 to reflect across the x-axis.