reflection over y = x — Definition, Formula & Examples
Reflection over y = x is a transformation that flips every point across the diagonal line y = x by swapping its x- and y-coordinates. If a point starts at (a, b), its image lands at (b, a).
A reflection over the line y = x is an isometric mapping in the coordinate plane defined by the rule . The line y = x serves as the perpendicular bisector of the segment connecting each point to its image.
Key Formula
Where:
- = the original x-coordinate of the point
- = the original y-coordinate of the point
How It Works
To reflect any point over y = x, simply swap the two coordinates: the x-value becomes the new y-value, and the y-value becomes the new x-value. For an entire shape, apply this swap to every vertex, then connect the new vertices. Because the transformation preserves distances and angles, the reflected figure is congruent to the original. This reflection is also how you obtain the graph of an inverse function — reflecting f(x) over y = x produces f⁻¹(x).
Worked Example
Problem: Reflect the triangle with vertices A(1, 4), B(3, 2), and C(5, 6) over the line y = x.
Step 1: Apply the rule (x, y) → (y, x) to each vertex.
Step 2: Swap coordinates for B.
Step 3: Swap coordinates for C.
Answer: The reflected triangle has vertices A'(4, 1), B'(2, 3), and C'(6, 5).
Why It Matters
Reflecting over y = x is the geometric basis for finding inverse functions. In precalculus and calculus, sketching f⁻¹(x) by reflecting f(x) across y = x is a standard technique. It also appears in linear algebra when analyzing matrix transformations.
Common Mistakes
Mistake: Negating a coordinate instead of swapping, such as mapping (3, 5) to (−3, 5) or (3, −5).
Correction: Negating coordinates reflects over an axis (x-axis or y-axis). Reflection over y = x requires swapping: (3, 5) → (5, 3).