To identify the factor \( k \) for the vertical compression from the original function \( y = |x + 6| \) to the dilated function \( y = \frac{1}{2}|x + 6| \), we can compare the two functions.
In the context of vertical dilations, the general form of a function undergoing vertical compression can be expressed in terms of a factor \( k \):
\[ y = k f(x) \]
where \( f(x) \) is the original function. In this case, \( f(x) = |x + 6| \) and the dilated function is given as \( y = \frac{1}{2}|x + 6| \).
Here, we see that:
\[ k = \frac{1}{2} \]
Since \( k \) is less than 1, this indicates a vertical compression by a factor of \( \frac{1}{2} \) from the original function \( y = |x + 6| \) to the dilated function \( y = \frac{1}{2}|x + 6| \).
So, the factor \( k \) for the vertical compression is:
\[ \boxed{\frac{1}{2}} \]
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