To analyze the transformation of the function $f(x) = |x|$ as shown in the graph:

1. The original graph of $f(x) = |x|$ is a V-shaped graph with a vertex at the origin (0, 0) and an upward direction.
2. The transformed graph has a vertex shifted from the origin to $(0, -3)$, indicating a vertical translation downward by 3 units.
3. Additionally, the transformed graph opens downward instead of upward, which signifies a vertical reflection across the x-axis.

Given these transformations:

• Reflecting $f(x) = |x|$ vertically results in $-|x|$.
• Shifting it downward by 3 units modifies it to $-|x| - 3$.

Therefore, the equation for the transformed graph is: $y = -|x| - 3$

Would you like further explanation on this transformation process or have additional questions?

Here are some related questions to deepen your understanding:

1. What transformations would result in shifting $f(x) = |x|$ up by 4 units and reflecting it horizontally?
2. How would the equation change if the vertex of the graph was moved to $(2, -3)$?
3. What is the general formula for shifting an absolute value function horizontally and vertically?
4. How does the absolute value function behave when multiplied by a constant greater than 1 or between 0 and 1?
5. What impact does a horizontal reflection have on the absolute value function's equation?

Tip: Remember that vertical shifts affect the constant term, while reflections across the x-axis change the sign of the entire function.