To transform the graph of $y = |x|$ into $y = -\frac{1}{2}| \frac{1}{3}x + 4 | - 5$, we need to apply a series of transformations, each corresponding to a different manipulation of the function. Let's break it down step by step:

1. Horizontal Stretch/Compression (inside the absolute value)

• From $y = |x|$ to $y = | \frac{1}{3}x |$:
  • The term $\frac{1}{3}x$ inside the absolute value represents a horizontal stretch by a factor of 3. This means that the graph of $y = | \frac{1}{3}x |$ will be wider than the graph of $y = |x|$, with the graph spread out 3 times farther.

2. Horizontal Shift (inside the absolute value)

• From $y = | \frac{1}{3}x |$ to $y = | \frac{1}{3}x + 4 |$:
  • The term $+4$ inside the absolute value causes a horizontal shift to the left by 4 units. This moves the graph of $y = | \frac{1}{3}x |$ to the left so that the vertex (which was originally at the origin) is now at $x = -4$.

3. Vertical Stretch/Compression (outside the absolute value)

• From $y = | \frac{1}{3}x + 4 |$ to $y = -\frac{1}{2}| \frac{1}{3}x + 4 |$:
  • The coefficient $-\frac{1}{2}$ outside the absolute value causes two transformations:
    1. A vertical compression by a factor of $\frac{1}{2}$, meaning the graph will appear "shorter" or "squashed" vertically.
    2. The negative sign reflects the graph across the x-axis, flipping the graph upside down. So, the "V" shape of the absolute value graph will now open downward instead of upward.

4. Vertical Shift (outside the absolute value)

• From $y = -\frac{1}{2}| \frac{1}{3}x + 4 |$ to $y = -\frac{1}{2}| \frac{1}{3}x + 4 | - 5$:
  • The term $-5$ at the end causes a vertical shift downward by 5 units. This means that the entire graph of the function moves 5 units downward, and the vertex shifts from its previous position down by 5 units.

Final Summary of Transformations:

1. Horizontal stretch by a factor of 3 (from $y = |x|$ to $y = | \frac{1}{3}x |$).
2. Horizontal shift left by 4 units (from $y = | \frac{1}{3}x |$ to $y = | \frac{1}{3}x + 4 |$).
3. Vertical compression by a factor of $\frac{1}{2}$, and reflection over the x-axis (from $y = | \frac{1}{3}x + 4 |$ to $y = -\frac{1}{2}| \frac{1}{3}x + 4 |$).
4. Vertical shift down by 5 units (from $y = -\frac{1}{2}| \frac{1}{3}x + 4 |$ to $y = -\frac{1}{2}| \frac{1}{3}x + 4 | - 5$).

Would you like any additional details or clarification on any of these transformations?

Here are 5 related questions for further exploration:

1. How would the graph change if the term $-\frac{1}{2}$ were replaced with $2$?
2. What would happen if the shift $+4$ inside the absolute value was replaced with $-4$?
3. How would the graph look if we reflected it across the y-axis instead of the x-axis?
4. What effect would a vertical stretch by a factor of 3 have on this function?
5. How does the order of transformations affect the final graph?

Tip: Always follow the order of transformations: first horizontal changes (stretch/compression, shift), then vertical changes (stretch/compression, shift, reflection).