Question 1179841
**Understanding Absolute Value Transformations**

The general form of an absolute value function is:

f(x) = a |b(x - h)| + k

where:

* **a:** Controls vertical reflection and stretch/shrink.
    * If a is negative, there's a vertical reflection.
    * |a| > 1: Vertical stretch
    * 0 < |a| < 1: Vertical shrink
* **b:** Controls horizontal reflection and stretch/shrink.
    * If b is negative, there's a horizontal reflection.
    * |b| > 1: Horizontal shrink
    * 0 < |b| < 1: Horizontal stretch
* **h:** Horizontal shift (left or right).
    * (x - h): Shifts the graph h units to the right.
    * (x + h): Shifts the graph h units to the left.
* **k:** Vertical shift (up or down).
    * +k: Shifts the graph k units up.
    * -k: Shifts the graph k units down.

**Applying the Given Transformations**

1.  **Vertical Reflection:** a = -1
2.  **Horizontal Shrink by 1/2:** b = 2
3.  **Vertical Translation 3 Units Up:** k = 3
4.  **Horizontal Translation 2 Units Left:** h = -2

**Substituting the Values**

f(x) = -1 |2(x - (-2))| + 3

**Simplified Equation**

f(x) = - |2(x + 2)| + 3