To get f'(x) = 3^(2x) + 8 from f(x) = 3^x + 6, a horizontal compression (dilation) by a factor of 1/2 and a vertical translation up by 2 units are applied.
Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of the function. Given the original function f(x) = 3^x + 6 and the transformed function f'(x) = 3^(2x) + 8, we can analyze the changes to determine the transformations applied.
To obtain f'(x) from f(x), the exponent in 3^x is multiplied by 2, which indicates a horizontal compression by a factor of 1/2. This type of transformation is known as a dilation. Furthermore, the constant term in f(x) is increased from 6 to 8, which is a vertical translation up by 2 units.
Therefore, the correct transformations applied are a dilation and a translation, which matches option C: a dilation and a translation.