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the grpah of the parent function y=|x| is shifted right 3 units stretched by a factor of 3 shifted down 2 units and finally reflected across the x-axis

The parent function is \( y = |x| \). Let's apply each transformation step-by-step:

1. **Shift right by 3 units**: Replace \( x \) with \( x - 3 \), giving \( y = |x - 3| \).
2. **Stretch by a factor of 3**: Multiply by 3, yielding \( y = 3|x - 3| \).
3. **Shift down by 2 units**: Subtract 2, resulting in \( y = 3|x - 3| - 2 \).
4. **Reflect across the x-axis**: Multiply by -1, giving the final equation \( y = -3|x - 3| - 2 \).

So, the correct answer is the first option: \( y = -3|x - 3| - 2 \).

Would you like further clarification on any step? Here are 5 related questions to deepen your understanding:

1. What would happen if we shifted left instead of right?
2. How does changing the stretch factor affect the graph?
3. What if we reflected across the y-axis instead?
4. How do vertical and horizontal shifts interact with reflections?
5. How does each transformation affect the vertex of the absolute value function?

**Tip**: Always apply transformations in the given order to avoid errors.

The graph of the parent function y = |x| is shifted right 3 units, stretched by a factor of 3, shifted down 2 units, and finally reflected across the x-axis.

Explore the transformations of the absolute value function y = |x| with a right shift of 3 units, a vertical stretch by a factor of 3, a downward shift of 2 units, and a reflection across the x-axis. Learn how these transformations affect the graph step-by-step.

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