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The original function given is \( f(x) = |x| \), which is the absolute value function. The graph shows a transformation of this function. Let's analyze the transformation step by step.

1. **Reflection**: The graph is reflected over the x-axis compared to \( f(x) = |x| \), which means a vertical reflection. This indicates a negative sign in front of the function, making it \( -|x| \).

2. **Vertical Shift**: The graph is shifted downward by 3 units. This indicates a vertical shift, adding \( -3 \) to the function.

Thus, the equation for the transformed function \( g(x) \) is:

\[
g(x) = -|x| - 3
\]

Would you like further details or have any questions?

Here are 5 related questions for further practice:

1. What happens if we reflect the function \( |x| \) over the y-axis instead of the x-axis?
2. How would the graph change if the function was \( g(x) = 2|x| - 3 \)?
3. How can you describe a horizontal shift for the function \( f(x) = |x| \)?
4. What is the vertex of the function \( g(x) = -|x| - 3 \)?
5. How does adding or subtracting a constant outside the absolute value affect the graph?

**Tip**: Reflections over the x-axis involve multiplying the entire function by \(-1\), while reflections over the y-axis involve changing the sign inside the function.

The graph above is a transformation of the function f(x) = |x|. Write an equation for the function graphed above.

Analyze the transformation of the absolute value function f(x) = |x|. This includes a reflection over the x-axis and a vertical shift downward. Suitable for Grades 9-11, this problem helps students understand graph transformations.

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