https://file.aiquickdraw.com/mathbot/file/1726115205450hiquvmcu.jpg

The function shown in the image is:

\[
f(x) = -\frac{1}{2} |x + 3| + 4
\]

This represents a transformed absolute value function. Let’s break down the components:

1. **Absolute Value**: The term \( |x + 3| \) represents an absolute value function with a horizontal shift of 3 units to the left (since it's \( x + 3 \)).

2. **Vertical Compression & Reflection**: The coefficient \(-\frac{1}{2}\) in front of the absolute value compresses the graph vertically by a factor of \( \frac{1}{2} \) and reflects it across the x-axis (since the coefficient is negative).

3. **Vertical Shift**: The +4 at the end shifts the entire graph upwards by 4 units.

### Key Characteristics of the Function:
- **Vertex**: The vertex of the function occurs when the expression inside the absolute value equals 0, i.e., when \( x = -3 \). At this point, the function value is:
  \[
  f(-3) = -\frac{1}{2} \cdot 0 + 4 = 4
  \]
  So, the vertex is at \( (-3, 4) \).

- **Slope**: On the left of \( x = -3 \), the slope of the function is \( \frac{1}{2} \) (positive), and on the right, the slope is \( -\frac{1}{2} \) (negative). This creates the characteristic "V" shape of absolute value functions, though it is inverted due to the negative sign.

- **Range**: Since the function is reflected downwards, the maximum value is at the vertex (4), and the range of the function is \( (-\infty, 4] \).

Do you want to explore more details, such as the graph or specific points?

### Related Questions:
1. What is the domain of this absolute value function?
2. How does the graph change if the coefficient of \( |x+3| \) is altered?
3. How would you describe the symmetry of the function?
4. What happens if the absolute value term \( |x+3| \) is replaced by \( |x-3| \)?
5. Can you derive the function's equation from the graph of a basic absolute value function?

### Tip:
Absolute value functions are always V-shaped, and any transformations affect the direction, steepness, and position of the "V".

Understand the transformations applied to the absolute value function f(x) = -1/2 * |x + 3| + 4. Learn how to find the vertex, analyze the function's slope, and describe the range. Suitable for high school students (Grades 9-12).

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