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To find the vertex of the function \( f(x) = 4|x + 6| \), we analyze the form of an absolute value function.

1. The function is given by:
   \[
   f(x) = 4|x + 6|
   \]

2. The general form of an absolute value function is:
   \[
   f(x) = a |x - h| + k
   \]
   where \((h, k)\) represents the vertex of the function.

3. In \( f(x) = 4|x + 6| \), we can rewrite it as:
   \[
   f(x) = 4 |x - (-6)|
   \]
   This shows that \( h = -6 \) and \( k = 0 \).

4. Therefore, the vertex of this absolute value function is located at:
   \[
   (-6, 0)
   \]

**Answer:** The coordinates of the vertex are \((-6, 0)\).

Would you like further explanation on this topic or have any questions?

### Related Questions
1. How do you determine the vertex of a general absolute value function?
2. What effect does changing the coefficient of the absolute value term have on the graph?
3. How would the graph shift if we added a constant term to \( f(x) = 4|x + 6| \)?
4. What would the graph look like if the absolute value term had a negative coefficient?
5. How does the slope of the lines formed in an absolute value graph relate to the coefficient outside the absolute value?

### Tip:
Remember, the vertex of an absolute value function \( f(x) = a |x - h| + k \) is always at the point \((h, k)\), where \(h\) shifts the graph horizontally and \(k\) shifts it vertically.

What are the coordinates of the vertex of the graph of f(x) = 4|x + 6|?

Learn how to find the vertex of the absolute value function f(x) = 4|x + 6| with a clear, step-by-step solution. This problem covers algebraic concepts related to vertex form and absolute value functions, ideal for students in Grades 7-10.

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