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 specifically at the coefficients of the \( x^3 \) terms in both functions.
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Original function: \( f(x) = x^3 + 2 \)
- The coefficient of \( x^3 \) is 1.
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Transformed function: \( g(x) = \frac{1}{4}x^3 + \frac{1}{2} \)
- The coefficient of \( x^3 \) is \( \frac{1}{4} \).
The factor \( k \) represents how the \( y \)-values of the original function are scaled when transitioning to the new function. Specifically, this factor can be calculated by comparing the coefficients of \( x^3 \):
\[ k = \text{new coefficient} / \text{original coefficient} = \frac{1/4}{1} = \frac{1}{4} \]
This indicates a vertical compression by a factor of \( \frac{1}{4} \).
Therefore, the correct response is:
k=1/4