Describe how the graph of \( g(x)=\frac{1}{6} x^{3}-2 \) can be obtained from the \( g \) raph of \( f(x)=x^{3} \). Then graph the function \( g(x) \). How can the graph of \( g(x)=\frac{1}{6} x^{3}-2 \) be obtained trom the graph of \( f(x)=x^{3} \) ? A. Shrink vertically and shift up 2 units. B. Stretch vertically and shift down 2 units. C. Stretch vertically and shift up 2 units. D. Shrink vertically and shift down 2 units.

To transform the graph of \( f(x) = x^3 \) into \( g(x) = \frac{1}{6} x^3 - 2 \), you first need to shrink the graph vertically by a factor of 6 because of the coefficient \( \frac{1}{6} \), which makes it less steep. After that, you shift the entire graph down by 2 units, which is achieved by the \(-2\) in the function. So, the correct choice here is **D. Shrink vertically and shift down 2 units**. Now, if you want to visualize this, first, start with the standard cubic curve of \( f(x) = x^3 \) which has its typical "S" shape. When you apply the vertical shrink, the arms of the graph will spread out more gently. After that, moving the graph down will lower the entire curve, making its new position more pronounced in the lower part of the coordinate plane. Get your graphing tools ready to see the transformation in action!