What is the reflection of the point (-2, 5) in the vertical line x = -1 in the Cartesian plane?

Introduction / Context:
This question involves reflecting a point across a vertical line in the coordinate plane. Reflection problems help build intuition about symmetry and how x and y coordinates change under geometric transformations.

Given Data / Assumptions:

• Original point is A(-2, 5).
• Line of reflection is the vertical line x = -1.
• We want the coordinates of the reflected point across this line.

Concept / Approach:
Reflection across a vertical line x = k keeps the y coordinate unchanged and adjusts the x coordinate so that the line is the perpendicular bisector of the segment joining the point and its image. If a point (x, y) is reflected in x = k, its image has coordinates (2k - x, y). This formula guarantees that the horizontal distances from the line of reflection are equal and on opposite sides.

Step-by-Step Solution:
Step 1: Identify k from x = -1, so k = -1.Step 2: Original point has coordinates (x, y) = (-2, 5).Step 3: Use the reflection formula in a vertical line: image (x', y') = (2k - x, y).Step 4: Substitute values: x' = 2 * (-1) - (-2) = -2 + 2 = 0.Step 5: The y coordinate remains the same: y' = 5.Step 6: Hence the reflected point is (0, 5).

Verification / Alternative check:
You can check by considering distances. The original point has x = -2, which is 1 unit to the left of x = -1. The reflected point at x = 0 is 1 unit to the right of x = -1. Both points share the same y coordinate 5, which confirms a symmetrical reflection across the vertical line x = -1.

Why Other Options Are Wrong:
The points (-2, -7) and (-2, 7) change the y coordinate but keep x the same, which would represent reflection across a horizontal line, not a vertical one. The point (2, 5) is two units to the right of the line x = -1, giving unequal distances from the line. The point (-4, 5) lies two units to the left and also fails the equal distance requirement. Only (0, 5) satisfies the correct symmetry.

Common Pitfalls:
Students may accidentally reflect across the y axis instead of the given line, or forget that the y coordinate remains unchanged for vertical reflections. Another common oversight is miscalculating the horizontal distance and not applying the formula 2k - x correctly. Keeping the rule for vertical and horizontal reflections in mind avoids these errors.

Final Answer:
The reflection of the point (-2, 5) in the line x = -1 is (0, 5).