https://file.aiquickdraw.com/mathbot/file/1728332263662p316ryw4.jpg

The solid black graph in the image is \( y = |x| \), which is a V-shaped graph centered at the origin. The dotted graph appears to be a transformation of \( y = |x| \).

Let's analyze the transformations to determine which equation represents the dotted graph:

1. The dotted graph is a reflection of \( y = |x| \) across the x-axis. Reflecting \( y = |x| \) across the x-axis gives \( y = -|x| \).

2. The dotted graph appears to be shifted downwards compared to \( y = |x| \). This suggests that we need to subtract a value from \( y = -|x| \).

3. From visual inspection, the dotted graph has been shifted 1 unit downwards. So, the equation of the dotted graph is \( y = -|x| - 1 \).

The correct answer is:

**\( y = -|x| - 1 \)**.

Would you like a more detailed explanation or have any questions about graph transformations? Here are five related questions:

1. What are the basic transformations that can be applied to the absolute value function?
2. How do you identify horizontal and vertical shifts in a graph?
3. What is the effect of reflecting a function across the x-axis or y-axis?
4. How do you determine the vertex of an absolute value function?
5. How can you find the equation of a transformed graph given its visual representation?

**Tip**: When analyzing transformations, always start by identifying shifts, reflections, and stretches/compressions separately for clarity.

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 graphs, including reflections and vertical shifts. This example explains the transformation of y = |x| to y = -|x| - 1 by reflecting across the x-axis and shifting the graph downwards. Suitable for students in Grades 9-12.

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