Final answer:
The rule for g after a horizontal shrink by a factor of 1/3, a translation 2 units up, and a reflection in the x-axis from the graph of A f(x) = x³ + 2x² - 9, is g(x) = -27x³ - 18x² + 11.
Step-by-step explanation:
To write a rule for g that represents the indicated transformations of the graph of A f(x) = x³ + 2x² - 9, we need to apply each transformation step by step. First, a horizontal shrink by a factor of 1/3 is achieved by multiplying the input variable by the reciprocal of 1/3, which is 3. Therefore, we replace every x in A f(x) with 3x to account for this shrink. Next, a translation 2 units up is applied by adding 2 to our function. Finally, a reflection in the x-axis is applying a negative sign to the whole function, reflecting it across the x-axis.
The combined transformation gives us the following rule for g:
g(x) = -(3x)³ - 2(3x)² + 9 + 2, which simplifies to g(x) = -27x³ - 18x² + 11.