To determine the factor \( k \) for the vertical compression from the function \( f(x) = x^3 + 2 \) to \( f(x) = 14x^3 + 12 \), we start by analyzing the two functions.
- The original function is \( f(x) = x^3 + 2 \).
- The transformed function is \( f(x) = 14x^3 + 12 \).
To express the transformed function in terms of the original function, we can look at how each term is modified:
- The \( x^3 \) term is multiplied by 14, suggesting a vertical stretch rather than compression.
- The constant term \( 2 \) in the original function translates to \( 12 \) in the transformed function, which shifts the graph vertically but does not affect the factor \( k \).
Regarding the vertical compression factor, it is generally defined when the function is of the form \( k \cdot g(x) \) where \( g(x) \) is the original function.
Here, since the transformation is a vertical stretch (the coefficient of \( x^3 \) increased from \( 1 \) to \( 14 \)), it indicates \( k \) is not a compression factor in this case.
Thus, the correct response for \( k \) identifying the vertical transformation would be:
k = 14
Therefore, the correct answer is:
- k = 14