The given graph depicts a transformation of the absolute value function, which generally follows the form:

$y = a |x - h| + k$

We need to determine the values of $a$, $h$, and $k$, based on the graph.

Observations:

1. Vertex (h, k): The vertex of the transformed absolute value function appears to be located at the point (0, 3), so $h = 0$ and $k = 3$.
2. Slope (a): The slope of the arms of the absolute value function helps determine $a$. From the graph, as $x$ increases by 1, $y$ decreases by 1 unit. This indicates that the slope of the lines is -1 (on the left) and 1 (on the right). So, $a = -1$.

Thus, the equation for this transformed absolute value function is:

$y = -|x - 0| + 3 \quad \text{or} \quad y = -|x| + 3$

Final Values:

• $a = -1$
• $h = 0$
• $k = 3$

Would you like further details or clarifications on how these transformations were determined?

Here are some related questions to deepen understanding:

1. How do different values of $a$ affect the shape of the graph?
2. What happens to the graph if $h$ is non-zero?
3. How does $k$ influence the vertical shift of the graph?
4. Can you describe the role of each parameter in the transformation equation?
5. How would the graph change if $a$ were positive instead of negative?

Tip: When transforming absolute value functions, always look at the vertex for $h$ and $k$, and use the slopes to find $a$.