https://file.aiquickdraw.com/mathbot/file/1730289647695zl2rm6kt.jpg

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.

The graph of \( y = |x| \) is the solid black graph below. Which function represents the dotted graph?

Learn how to identify transformations in absolute value functions with this example problem involving the graph of \( y = |x| \) and its transformations. This question covers reflections and vertical shifts, suitable for students in Grades 8-10.

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