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?

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.