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Let's break down the problem and solve it step by step. The function given is:

\[
f(x) = |3x + 6| - 4
\]

### Step 1: Analyze the function

The expression involves an absolute value, so we need to break it into two cases based on when \(3x + 6\) is positive or negative.

#### Case 1: \(3x + 6 \geq 0\), or \(x \geq -2\)
In this case, the function becomes:

\[
f(x) = (3x + 6) - 4 = 3x + 2
\]

#### Case 2: \(3x + 6 < 0\), or \(x < -2\)
Here, the absolute value will flip the sign of \(3x + 6\), and the function becomes:

\[
f(x) = -(3x + 6) - 4 = -3x - 6 - 4 = -3x - 10
\]

### Step 2: Find where the function changes
The function changes at \(x = -2\), where \(3x + 6 = 0\). This is the critical point where the absolute value changes its behavior.

### Step 3: Evaluate the function at \(x = -2\)
At \(x = -2\), we substitute into the original function:

\[
f(-2) = |3(-2) + 6| - 4 = |0| - 4 = -4
\]

So, \(f(-2) = -4\).

### Step 4: Behavior of the function
- For \(x < -2\), the function is \(f(x) = -3x - 10\), which is decreasing because the slope is negative.
- For \(x \geq -2\), the function is \(f(x) = 3x + 2\), which is increasing because the slope is positive.

Thus, the function has a minimum at \(x = -2\).

### Step 5: Conclusion
The minimum value of the function is \(-4\), and it occurs at \(x = -2\).

### Answer:
- The point where an extreme value occurs is **\((-2, -4)\)**.
- It is a **minimum** because the function changes from **decreasing** to **increasing**.
- This occurs when \(x = -2\).
- The absolute **minimum** value of the function is **-4**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you graph absolute value functions?
2. What are the key features of piecewise functions?
3. How do you determine increasing and decreasing intervals of a function?
4. Can you explain how to find the critical points of a function?
5. What is the effect of shifting absolute value functions horizontally or vertically?

**Tip**: Always check the turning points of absolute value functions, as these often indicate key features like minimums or maximums.

What is the extreme value of f(x) = |3x + 6| - 4?

Learn how to find the extreme value of the absolute value function f(x) = |3x + 6| - 4 by breaking it into cases based on critical points. This problem explores key features like increasing, decreasing intervals, and minimum points for students in Grades 9-11.

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