To identify the factor \( k \) for the vertical compression from the original function \( y = |x + 6| \) to the dilated function \( y = 12|x + 6| \), we need to compare the two functions.
In this case, the original function \( y = |x + 6| \) is being transformed into \( y = 12|x + 6| \).
To determine the factor \( k \), we note that a vertical dilation is represented as:
\[ y = k \cdot f(x) \]
where \( f(x) \) is the original function.
Here, the original function is:
\[ f(x) = |x + 6| \]
and the dilated function is:
\[ y = 12|x + 6| = 12f(x). \]
This means that the vertical dilation factor \( k \) is \( 12 \).
Now, for vertical compression, a factor \( k < 1 \) would mean a compression. Since the transformation from \( |x + 6| \) to \( 12|x + 6| \) is an expansion (not a compression), it appears there might be a misunderstanding in the problem statement. However, if it were asked about a vertical compression factor for another example, it would require a fraction less than 1.
In summary, the factor \( k \) in this case, based on the provided information and comparison, is:
\[ k = 12. \]
If you're looking for a fraction representing compression specifically, it would be \( k = \frac{1}{12} \) in the context of using it for compression rather than dilation.