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".