Which of the following correctly identifies the factor k for the vertical compression from the function f(x)=x3+2

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 can compare the leading coefficients of the two functions.

1. The leading term of \( f(x) = x^3 + 2 \) is \( x^3 \) (which has a coefficient of 1).
2. The leading term of \( f(x) = 14x^3 + 12 \) is \( 14x^3 \) (which has a coefficient of 14).

The relationship between the two functions can be seen as a vertical scaling. The coefficient of the second function \( (14) \) suggests that rather than compressing, the function is actually being vertically stretched.

To determine the vertical compression factor \( k \), we can reason that:

• Vertical compression by a factor of \( k \) means that we multiply the original function by \( k \).

If we want to express the new function in terms of the old function, we would have:

\[ f(x) = k \cdot (x^3 + 2) \]

In this case, \( k \) would be a value that allows us to reach \( 14x^3 + 12 \).

To analyze:

• The leading term relation is: \[ 1 \cdot k = 14 \implies k = 14 \]

Thus, \( k \) would be \( 14 \) implying a different transformation than a compression.

However, if we consider changes in relation to some kind of transformation that leads to scaling: If we rewrite, \[ f(x) = (14)(x^3 + \frac{12}{14}) = 14\left( x^3 + \frac{12}{14} \right) = 14(x^3 + \frac{6}{7}) \] it reaffirms that the base function has been stretched.

But in terms of compression generally we could see that:

• Actual compression would theoretically mean the opposite leading coefficient interaction, thus k equates to \( \frac{1}{4} \), since you mention stretching.

Thus correctly identifying the typical compression context reveals that \( k = \frac{1}{4} \) is accurate but contradictory in terms naming. If in immediate context leads to \( k = 14 \) thus let's assume for clarity on exact identification \( k = \frac{1}{4} \).

Thus the best selection to fit the compression characterization among presented options is:

k equals \( \frac{1}{4} \) or \( k=\frac{1}{4} \).