To identify the factor \( k \) for the vertical compression from the function \( f(x) = x^3 + 2 \) to the function \( g(x) = \frac{1}{4}x^3 + \frac{1}{2} \), we can look at how the leading coefficients of the two functions relate to each other.
- The original function is \( f(x) = x^3 + 2 \), where the leading term (the coefficient of \( x^3 \)) is \( 1 \).
- The new function is \( g(x) = \frac{1}{4}x^3 + \frac{1}{2} \), where the leading term is \( \frac{1}{4} \).
The vertical compression factor \( k \) can be determined by comparing the leading coefficients:
\[ k = \frac{\text{leading coefficient of } g(x)}{\text{leading coefficient of } f(x)} = \frac{\frac{1}{4}}{1} = \frac{1}{4}. \]
Thus, the vertical compression factor \( k \) is \( \frac{1}{4} \).
Therefore, the correct identification of the factor \( k \) is \( \frac{1}{4} \).
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