The graph of $y = |x|$ is shown as the solid black line, which is the standard V-shaped absolute value graph. The dotted graph appears to be a transformation of this absolute value graph, specifically reflected and/or shifted.

Observing the Dotted Graph:

1. Reflection: The dotted graph opens downwards, indicating that it is a reflection of $y = |x|$ across the x-axis. This transformation can be achieved by introducing a negative sign, so we are looking for a function of the form $y = -|x|$.

2. Vertical Shift: The vertex of the dotted graph is at $y = -2$, while the vertex of $y = |x|$ is at $y = 0$. This suggests a downward shift of 2 units.

Combining these transformations, the equation for the dotted graph would be: $y = -|x| - 2$

Matching with Given Options:

Among the choices:

• $y = -|x + 2|$
• $y = -|x - 2|$
• $y = | - x| - 2$
• $y = -|x| - 2$

The correct answer is: $\boxed{y = -|x| - 2}$

Would you like further clarification or a detailed breakdown of transformations?

Additional Related Questions:

1. What does a reflection across the x-axis do to the function $y = |x|$?
2. How does shifting a graph down by 2 units affect its equation?
3. What would the equation be if the dotted graph was shifted 2 units up instead?
4. How would you represent a horizontal shift in the graph of $y = |x|$?
5. How can you determine if an absolute value graph opens upwards or downwards?

Tip:

When analyzing transformations, start by looking for reflections, shifts, and changes in orientation of the vertex to quickly identify the new function.