The graph in the image represents a transformation of the function $f(x) = |x|$, which is the absolute value function.

To determine the function rule for $g(x)$, let's analyze the transformation:

1. The vertex of the parent function $f(x) = |x|$ is at the origin $(0, 0)$. In the graph of $g(x)$, the vertex has shifted to $(-3, 2)$.

2. The slope of the lines on both sides of the vertex appears unchanged, meaning that the function has not been vertically stretched or compressed.

This suggests the following transformations:

• Horizontal shift to the left by 3 units.
• Vertical shift up by 2 units.

Thus, the function rule for $g(x)$ is:

$g(x) = |x + 3| + 2$

This is the transformed absolute value function where:

• $+3$ inside the absolute value shifts the graph to the left.
• $+2$ outside shifts the graph upwards.

Would you like more details on the transformation steps?

Here are 5 related questions:

1. How do vertical and horizontal shifts affect the graph of a function?
2. What would happen if we added a coefficient in front of the absolute value, such as $2|x + 3| + 2$?
3. How can we use transformations to reflect a graph across the x-axis or y-axis?
4. What is the general form of a transformed absolute value function?
5. Can transformations like stretching or compressing occur together with shifts?

Tip: When analyzing graph transformations, always start by identifying the vertex and check for any changes in slope or direction.