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.
The original function is defined as:
\[ y = |x + 6| \]
The dilated function is defined as:
\[ y = 12 |x + 6| \]
In a vertical dilation, the general form can be expressed as:
\[ y = k \cdot f(x) \]
where \( f(x) \) is the original function and \( k \) is the factor by which the function is compressed or stretched vertically.
In this case, we can compare the equations:
\[ y = 12 |x + 6| = k \cdot |x + 6| \]
From this comparison, we can see that \( k = 12 \).
However, the problem asks for the factor related to vertical compression. A vertical compression occurs when the absolute value of \( k \) is less than 1. In contrast, when \( k > 1 \), the transformation is a vertical stretch.
To determine the vertical compression factor \( k \) that leads to the dilated function, we would express it as:
\[ k = \frac{1}{12} \]
Thus, the vertical compression factor from \( y = |x + 6| \) to \( y = 12 |x + 6| \) is:
\[ \boxed{\frac{1}{12}} \]
This indicates a compression that makes the function less stretched compared to the original.