To identify the factor \( k \) for the vertical compression from the function \( f(x) = x^3 + 2 \) to the function \( f(x) = 14x^3 + 12 \), we need to analyze how the function is transformed.
The original function \( f(x) = x^3 + 2 \) can be compared to the new function \( f(x) = 14x^3 + 12 \). To see the vertical scaling factor, we can focus on the leading coefficients of the \( x^3 \) term in each function.
The leading coefficient of the original function \( f(x) = x^3 + 2 \) is 1 (since \( x^3 \) has a coefficient of 1).
The leading coefficient of the new function \( f(x) = 14x^3 + 12 \) is 14.
The relationship between these coefficients can be expressed as: \[ k = \frac{\text{new coefficient}}{\text{original coefficient}} = \frac{14}{1} = 14 \]
Thus, \( k = 14 \) indicates a vertical stretch rather than a compression.
Since the question asks specifically for vertical compression:
- A vertical compression would mean that \( k < 1 \).
Based on the comparison, the correct identification of \( k \) for the transformation (although it is a vertical stretch, not a compression) is: \( k = 14 \).
Therefore, the correct response is: k = 14.