To identify the factor \( k \) for the vertical compression from the function \( f(x) = x^3 + 2 \) to the function \( f(x) = \frac{1}{4} x^3 + \frac{1}{2} \), we compare the coefficients of \( x^3 \) in both functions.
The original function is: \[ f(x) = x^3 + 2 \]
The new function is: \[ g(x) = \frac{1}{4} x^3 + \frac{1}{2} \]
The term involving \( x^3 \) in the original function has a coefficient of \( 1 \), while in the new function, the coefficient is \( \frac{1}{4} \). The vertical compression factor \( k \) is the ratio of the new coefficient to the original coefficient:
\[ k = \frac{\text{new coefficient}}{\text{original coefficient}} = \frac{\frac{1}{4}}{1} = \frac{1}{4} \]
Therefore, the correct identification of the factor \( k \) for the vertical compression is:
\( k = \frac{1}{4} \)