Thank you for the clarification! Let's analyze the graph more carefully, now knowing the correct transformation is $g(x) = -\frac{1}{4}|x + 2| - 1$.

Step-by-Step Analysis:

1. Vertex Location:

   • From the graph, the vertex is at $(-2, -1)$. This indicates a horizontal shift 2 units to the left (which is represented by $x + 2$) and a vertical shift 1 unit down (which is represented by the $-1$ outside the absolute value).

2. Reflection:

   • The graph opens downwards, so it must be reflected across the x-axis. This is why there's a negative sign in front of the absolute value term.

3. Vertical Compression:

   • The slope of the "arms" of the V-shape is shallower than the standard $f(x) = |x|$. Normally, the slope of the absolute value graph is $1$, but here the slope is much smaller, indicating a vertical compression. The factor $\frac{1}{4}$ tells us that the graph has been vertically compressed by a factor of 4.

Correct Equation:

Now that we have identified all the transformations:

• Reflection: $-$
• Horizontal shift: $x + 2$ (shift 2 units to the left)
• Vertical compression: $\frac{1}{4}$
• Vertical shift: $-1$ (shift 1 unit down)

The correct equation is indeed: $g(x) = -\frac{1}{4}|x + 2| - 1$

This matches the correct answer you provided.

Summary of Transformations:

• Reflection: The graph is reflected across the x-axis.
• Vertical Compression: The graph is compressed by a factor of $\frac{1}{4}$.
• Horizontal Shift: The graph is shifted 2 units to the left.
• Vertical Shift: The graph is shifted 1 unit down.

Let me know if you'd like a deeper explanation of any of these transformations, or if you have further questions!

Related Questions:

1. How would the equation change if the graph were stretched instead of compressed?
2. What is the effect of changing the coefficient $\frac{1}{4}$ to another value?
3. How do horizontal shifts affect the domain of an absolute value function?
4. What is the domain and range of $g(x) = -\frac{1}{4}|x + 2| - 1$?
5. How can you graph an absolute value function with a different slope and vertex without plotting many points?

Tip:

When analyzing transformations, always break down the effects one by one (reflection, stretch/compression, and shifts) to build the final equation!