The graph you provided is a V-shaped graph, which resembles the graph of an absolute value function. To express this graph using transformations of the absolute value function $f(x) = |x|$, let's break down the changes:

Key observations:

1. Vertex: The vertex of the graph is at $(2, -2)$, indicating a horizontal shift to the right by 2 units and a vertical shift downward by 2 units.
2. Slope:
   • For $x \leq 2$, the slope of the left arm is $-1$, which indicates that this part of the graph has been reflected over the y-axis.
   • For $x \geq 2$, the slope of the right arm is $1$, indicating no vertical stretching or compression.

Thus, the equation for the graph is based on the general transformation of the absolute value function:

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

where:

• $a$ is the vertical stretch/compression factor (in this case, $a = 1$ because the slope is unchanged).
• $h = 2$ (horizontal shift).
• $k = -2$ (vertical shift).

Final equation:

$f(x) = |x - 2| - 2$

This represents the transformed absolute value function for the given graph.

Would you like more details or have any questions?

Relative questions:

1. How does the slope of each arm affect the transformations?
2. What other transformations could be applied to the absolute value function?
3. How would the equation change if the vertex was at $(2, 3)$?
4. How do we account for reflections in transformations of functions?
5. What would happen to the equation if the slope of the arms were steeper?

Tip:

When identifying transformations, always start by locating the vertex of the function and observe changes in slope to detect reflections and stretches.