What is the reflection of the point (-1, 3) in the vertical line x = -4 in the Cartesian plane?

Introduction / Context:
This is another reflection problem in coordinate geometry, again across a vertical line. It reinforces how reflections change coordinates depending on whether the line of reflection is vertical or horizontal.

Given Data / Assumptions:

• Original point is A(-1, 3).
• Line of reflection is the vertical line x = -4.
• We must determine the coordinates of the reflected image of A.

Concept / Approach:
Reflection in a vertical line x = k keeps the y coordinate the same and changes the x coordinate to 2k - x. This ensures that the line x = k is exactly midway between the original point and its reflected image. The reflected point will lie on the same horizontal line (same y value) as the original point.

Step-by-Step Solution:
Step 1: Identify k from x = -4, so k = -4.Step 2: The original point is (x, y) = (-1, 3).Step 3: Apply the reflection formula: image (x', y') = (2k - x, y).Step 4: Compute x': 2 * (-4) - (-1) = -8 + 1 = -7.Step 5: The y coordinate stays the same, so y' = 3.Step 6: Therefore, the reflected point is (-7, 3).

Verification / Alternative check:
Consider the distances from the line of reflection x = -4. The original point has x = -1, which is 3 units to the right of -4. The reflected point at x = -7 is 3 units to the left of -4. Both points lie on the horizontal line y = 3, confirming that the reflection is correct.

Why Other Options Are Wrong:
The points (-7, -3) and (-1, -3) change the y coordinate, which would correspond to reflection across some horizontal line, not x = -4. The points (7, -3) and (7, 3) appear on the opposite side of the y axis and do not maintain equal distance from x = -4. Only (-7, 3) preserves the vertical line symmetry with respect to x = -4.

Common Pitfalls:
Learners may confuse reflection across a vertical line with reflection across the y axis, or they might incorrectly change both coordinates. Forgetting that the y coordinate remains unchanged for vertical reflection is also a typical error. Relying on the formula (2k - x, y) helps avoid such mistakes.

Final Answer:
The reflection of the point (-1, 3) in the line x = -4 is (-7, 3).