In coordinate geometry, what is the image of the point (−3, 6) when it is reflected in the x-axis?

Introduction / Context: This problem checks your understanding of reflections in coordinate geometry, specifically reflection in the x axis. Such transformations are fundamental when working with graphs, symmetry, and geometric interpretations on the Cartesian plane.

Given Data / Assumptions:

• Original point P has coordinates (−3, 6).
• The point is reflected in the x axis.
• We must find the coordinates of the reflected point.

Concept / Approach: Key ideas:

• Reflection in the x axis keeps the x coordinate unchanged.
• The y coordinate changes sign: (x, y) becomes (x, −y).
• This comes from the fact that points symmetric about the x axis are equidistant above and below that axis.

Step-by-Step Solution: Start with the original point P(−3, 6). Under reflection in the x axis, x remains the same, so x = −3. The y coordinate changes sign: y becomes −6. Therefore the image point P′ is (−3, −6).

Verification / Alternative check: You can visualise the point on the coordinate plane. P(−3, 6) lies in the second quadrant. Reflecting in the x axis moves it straight down vertically to the point with the same x coordinate and the opposite y coordinate, which is (−3, −6) in the third quadrant. The vertical distance from the x axis is 6 units in both cases, confirming the reflection.

Why Other Options Are Wrong: Option a: (3, 6) represents reflection in the y axis, not in the x axis. Option b: (6, −3) changes both coordinates and does not preserve the correct symmetric distance. Option d: (−6, 3) is reflection about the line y = x combined with sign changes and is not appropriate here. Option e: (3, −6) corresponds to reflection in the origin, changing both signs, rather than only flipping across the x axis.

Common Pitfalls: A common mistake is to change the wrong coordinate or both coordinates. Always remember: reflection in the x axis affects only the y coordinate, and reflection in the y axis affects only the x coordinate. Keeping a mental picture of the coordinate plane helps avoid confusion.

Final Answer: The image of the point after reflection in the x axis is (−3, −6).