To graph the function $v(x) = \frac{1}{3} |x + 6| - 5$ using transformations, we will start with the basic absolute value function $f(x) = |x|$ and apply a series of transformations step-by-step. Here's how to do it:

1. Start with the basic function: The basic graph is $f(x) = |x|$, a "V"-shaped graph with its vertex at the origin (0,0) and symmetry along the y-axis.

2. Horizontal shift: The expression inside the absolute value, $|x + 6|$, represents a horizontal shift. The graph of $|x + 6|$ is the graph of $|x|$ shifted 6 units to the left. The vertex is now at (-6, 0) instead of (0, 0).

3. Vertical stretch/compression: The coefficient $\frac{1}{3}$ outside the absolute value affects the steepness of the graph. Specifically, it causes a vertical compression by a factor of $\frac{1}{3}$, making the V-shape wider. Each y-coordinate of the graph is multiplied by $\frac{1}{3}$.

4. Vertical shift: The constant term $-5$ at the end of the equation shifts the entire graph down by 5 units. The vertex, which was at (-6, 0), is now moved to (-6, -5).

Step-by-Step Transformation Summary:

• Start with the basic function $f(x) = |x|$.
• Shift the graph 6 units to the left.
• Compress the graph vertically by a factor of $\frac{1}{3}$, making it wider.
• Finally, shift the entire graph 5 units downward.

Vertex of the transformed function:

The new vertex of the graph is at the point (-6, -5).

Would you like more details or have any questions?

Here are some related questions to deepen understanding:

1. How would the graph change if we replaced $\frac{1}{3}$ with $2$?
2. What if the constant outside the absolute value was $+5$ instead of $-5$?
3. How do horizontal shifts affect the domain of the function?
4. How can you find the axis of symmetry from the function $v(x)$?
5. How would the graph look if the absolute value was negative, e.g., $-|x+6|$?

Tip: Always identify the vertex first when working with absolute value functions, as it's the key to understanding transformations.