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.