What are the coordinates of the reflection of the point (−3, 1) in the vertical line x = −2 in the coordinate plane?

Introduction / Context:
This coordinate geometry problem asks you to reflect a point across a vertical line. Reflection questions are common in analytic geometry and help you understand symmetry with respect to lines like x = a or y = b in the Cartesian plane.

Given Data / Assumptions:

• Original point P has coordinates (−3, 1).
• The line of reflection is the vertical line x = −2.
• We are asked to find the coordinates of the reflected point P'.
• Reflection across a vertical line changes only the x coordinate; the y coordinate remains the same.

Concept / Approach:
For reflection in a vertical line x = a, the x coordinate of a point and its reflection are equidistant from a but on opposite sides. If the original point has x coordinate x1, then the reflected point has x coordinate x2 such that: a − x1 = x2 − a or equivalently: x2 = 2a − x1 The y coordinate is unchanged because the line of reflection is vertical and symmetry is only left right across this line.

Step-by-Step Solution:
Step 1: Identify the line of reflection x = −2, so a = −2. Step 2: The original point P is (x1, y1) = (−3, 1). Step 3: Use the reflection formula for the x coordinate: x2 = 2a − x1. Step 4: Substitute a = −2 and x1 = −3: x2 = 2 * (−2) − (−3) = −4 + 3 = −1. Step 5: The y coordinate stays the same, so y2 = y1 = 1. Step 6: Therefore, the reflected point P' has coordinates (−1, 1).

Verification / Alternative check:
We can check distances to the line x = −2. For the original point P(−3, 1), the horizontal distance to the line is |−3 − (−2)| = |−1| = 1 unit to the left. For the reflected point (−1, 1), the horizontal distance is |−1 − (−2)| = 1 unit to the right. Both points have the same y coordinate and are symmetric with respect to x = −2, confirming that (−1, 1) is correct.

Why Other Options Are Wrong:
(1, 1) reflects the point across x = −1, not x = −2, so it is too far to the right. (−3, −5) and (−3, 5) keep the x coordinate unchanged and instead move the point vertically, which corresponds to reflection across a horizontal line, not a vertical one. (−5, 1) moves the point further left by 2 units rather than mirroring it across x = −2, so the distances from the line are not equal on both sides.

Common Pitfalls:
Students sometimes reflect in the wrong direction (up down instead of left right) or forget that reflection across a vertical line only changes the x coordinate. Another common mistake is to add the distance instead of using the formula x2 = 2a − x1, leading to asymmetric points. Drawing a quick sketch of the line and the point often helps visualize the correct reflection.

Final Answer:
The reflection of (−3, 1) in the line x = −2 is (−1, 1).