What are the coordinates of the reflection of the point (1, 4) in the x-axis?

Introduction / Context:This coordinate geometry question checks your understanding of reflections in the Cartesian plane. When a point is reflected in the x axis, its horizontal position remains unchanged, while its vertical coordinate changes sign. Knowing this rule allows you to find the new coordinates quickly without complex calculations.

Given Data / Assumptions:

• Original point: (1, 4).
• The reflection is taken in the x axis.
• Coordinates are in the standard (x, y) format.

Concept / Approach:The rule for reflection in the x axis is simple: a point (x, y) becomes (x, -y). The x coordinate stays the same because reflection occurs vertically across the horizontal axis. The y coordinate changes sign because the point is mirrored to the opposite side of the axis. Applying this rule directly gives the image of the original point.

Step-by-Step Solution:

Verification / Alternative check:You can visualize the Cartesian plane: the x axis is horizontal. The point (1, 4) is one unit to the right of the origin and four units above the x axis. Reflecting it in the x axis puts it one unit to the right of the origin but four units below the x axis, which corresponds to (1, -4). This simple mental picture confirms the calculation.

Why Other Options Are Wrong:

Common Pitfalls:Students sometimes mistakenly change the sign of the x coordinate instead of the y coordinate for reflection in the x axis, which actually describes a reflection in the y axis. Another common mistake is swapping coordinates, which corresponds to reflection in the line y = x. Remember that the x axis affects only the vertical coordinate y when reflecting.

Final Answer:(1, -4)