What are the coordinates of the reflection of the point (−0.5, 6) in the x-axis on the Cartesian plane?

Introduction: This coordinate geometry question tests your understanding of reflections in the Cartesian plane, specifically about reflecting a point across the x-axis. It is a common concept used in graphs, transformations, and analytic geometry.

Given Data / Assumptions:

• Original point: (−0.5, 6).
• We must reflect this point across the x-axis.
• The x-axis is the horizontal axis where y = 0.

Concept / Approach: Reflection in the x-axis changes the sign of the y-coordinate, while the x-coordinate remains unchanged. In other words, the reflection of a point (x, y) in the x-axis is (x, −y). This arises because the x-axis is the line of symmetry and vertical distances are mirrored across it.

Step-by-Step Solution: Step 1: Identify the coordinates of the given point: x = −0.5, y = 6. Step 2: Use the rule for reflection in the x-axis: (x, y) → (x, −y). Step 3: Apply this rule to the given point: (−0.5, 6) → (−0.5, −6). Step 4: Therefore, the reflected point is (−0.5, −6).

Verification / Alternative check: Graphically, if you draw the point (−0.5, 6) on the plane, it lies 6 units above the x-axis. Its reflection must lie 6 units below the x-axis directly underneath, which has the same x-coordinate but a y-coordinate of −6. This confirms that (−0.5, −6) is indeed the correct reflected point.

Why Other Options Are Wrong:

• (0.5, −6): Changes both the x and y signs; this reflects across the origin, not merely the x-axis.
• (−6, −0.5) and (6, −0.5): These swap coordinates and adjust signs; they do not follow the rule of reflection in the x-axis.
• (−0.5, 6): This is the original point and shows no reflection at all.

Common Pitfalls: Students sometimes confuse reflections in the x-axis and y-axis. For reflection in the x-axis, only the y-coordinate changes sign. For reflection in the y-axis, only the x-coordinate changes sign. Mixing these up leads to the wrong image point.

Final Answer: The reflection of the point (−0.5, 6) in the x-axis is (−0.5, −6).