We are given the function $f(x) = |x|$ and asked to identify the transformations that result in the function $g(x) = \frac{1}{3}|x + 5| - 6$.

To analyze this, we will identify the transformations in terms of shifts, stretches, and reflections:

Step-by-Step Transformations:

1. Horizontal Shift:

   • In $g(x)$, the term inside the absolute value is $x + 5$. This corresponds to a horizontal shift to the left by 5 units.
     (If the form is $|x - h|$, the shift is $h$, so here it's $-5$).

2. Vertical Stretch:

   • The coefficient $\frac{1}{3}$ outside the absolute value scales the function vertically. Since $\frac{1}{3} < 1$, this is a vertical compression by a factor of 3 (the graph is "squished" vertically).

3. Vertical Shift:

   • The $-6$ at the end represents a vertical shift downward by 6 units.

Summary of Transformations:

• Horizontal shift left by 5 units.
• Vertical compression by a factor of 3.
• Vertical shift downward by 6 units.

Would you like more details or have any further questions?

Here are five related questions:

1. How would the graph change if the coefficient outside the absolute value was negative?
2. What transformation would occur if the inside of the absolute value was $x - 5$ instead?
3. How would the graph change if the $-6$ was positive instead of negative?
4. Can you describe the effect of multiplying $f(x)$ by 2 instead of $\frac{1}{3}$?
5. What happens to the graph if no horizontal shift is applied?

Tip: When dealing with absolute value transformations, always focus on the inside expression for horizontal changes and outside terms for vertical transformations.