The graph provided depicts a piecewise linear function where the postman walks a fixed distance from the post office, and the transformation is based on the absolute value function $y = |x|$.

Step-by-step analysis:

1. Parent function: The parent function $y = |x|$ is V-shaped and symmetrical about the y-axis.
2. Transformations:
   • The graph has been reflected about the x-axis because the graph starts decreasing after $x = 0$.
   • There seems to be a vertical stretch by a factor of 2, as the slope is steeper than the standard $y = |x|$ graph.
   • The vertex of the graph has shifted to the right (indicating a horizontal shift of 4 units to the right) and upwards (vertical shift of 8 units up).

Thus, the correct series of transformations would include a reflection about the x-axis, vertical stretch by a factor of 2, horizontal shift 4 units to the right, and vertical shift 8 units up.

Answer:

The correct option is A.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you determine the vertical stretch or compression of a graph?
2. What impact does reflecting a graph across the x-axis have on the function?
3. How does a horizontal shift affect the equation of a function?
4. How can you differentiate between a horizontal and vertical shift in a graph?
5. What is the effect of a vertical shift on the vertex of an absolute value function?

Tip: When analyzing transformations, always consider the vertex's position, the slope, and the direction in which the graph opens.