https://file.aiquickdraw.com/mathbot/file/17302034308756v4xxjnr.jpg

Let's solve this step-by-step.

1. **Initial Point**: The initial point is \((5, -1)\).

2. **Translation**:
   - Moving **5 units to the right** adds 5 to the \( x \)-coordinate: \(5 + 5 = 10\).
   - Moving **3 units down** subtracts 3 from the \( y \)-coordinate: \(-1 - 3 = -4\).
   - The new point after translation is \((10, -4)\).

3. **Reflection**:
   - Reflecting across the line \( x = 0 \) (the \( y \)-axis) changes the sign of the \( x \)-coordinate but leaves the \( y \)-coordinate the same.
   - So, the reflection of \((10, -4)\) across \( x = 0 \) is \((-10, -4)\).

4. **Quadrant Identification**:
   - The point \((-10, -4)\) lies in **Quadrant III** (where both \( x \) and \( y \) are negative).

**Answer**: The correct answer is **C. Quadrant III**.

Would you like more details on any of the steps?

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Here are 5 related questions to expand your understanding:
1. What would happen if the reflection was applied across the \( y \)-axis instead?
2. How would you translate a point by a different combination of units?
3. How can we reflect a point across the line \( y = x \)?
4. What is the effect of reflecting a point across both the \( x \)-axis and \( y \)-axis?
5. How can we determine the quadrant of a point based on its coordinates?

**Tip**: Remember that reflecting across the line \( x = 0 \) simply changes the sign of the \( x \)-coordinate, while reflecting across \( y = 0 \) changes the sign of the \( y \)-coordinate.

A translation 5 units right and 3 units down, then a reflection across x=0 is applied to the point (5, -1). Where is the image in the coordinate system?

Learn how to determine the quadrant of a point after a translation and reflection. This problem involves translating a point 5 units right and 3 units down, then reflecting it across x = 0. Suitable for Grades 8-10.

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