The given function is:

$m(x) = \frac{1}{3} |x - 3| - 2$

This is an absolute value function, and we can use transformations to graph it. Let's break down the transformation process:

1. Starting with the base function: The base function is $|x|$, which forms a "V" shape with a vertex at the origin (0,0).

2. Horizontal shift: The expression $|x - 3|$ indicates a horizontal shift to the right by 3 units. This moves the vertex from (0,0) to (3,0).

3. Vertical stretch/compression: The coefficient $\frac{1}{3}$ in front of the absolute value compresses the graph vertically by a factor of $\frac{1}{3}$, making the arms of the "V" less steep compared to the base function $|x|$.

4. Vertical shift: The "-2" at the end of the function moves the entire graph down by 2 units. This means the new vertex is at $(3, -2)$.

5. Final graph: The function results in a "V" shape with a vertex at (3, -2), opening upwards, with the slopes of the arms being $\frac{1}{3}$ and $-\frac{1}{3}$.

Comparing to the provided graph:

• The graph you provided has a "V" shape.
• The vertex appears to be correctly located at $(3, -2)$, matching the transformations.
• The slopes of the lines appear to match the $\frac{1}{3}$ slope, indicating the compression is correct.

Would you like further details or have any questions about this explanation?

Here are 5 related questions for practice:

1. How would the graph change if the vertical stretch factor was $\frac{1}{2}$ instead of $\frac{1}{3}$?
2. What would happen if the expression was $|x + 3|$ instead of $|x - 3|$?
3. How would the graph shift if the equation had $+2$ instead of $-2$?
4. What would the graph look like if the absolute value was replaced by a square root $\sqrt{x-3}$?
5. How can you find the axis of symmetry of an absolute value function?

Tip: Absolute value functions are symmetric around the vertex, making them an excellent tool to explore reflections and transformations in graphs.