Q. What does it mean to reflect over \(y = x\)?

What it means to reflect over the line y = x — step-by-step explanation

1. Basic description. Reflecting a point across the line \(y = x\) means producing its mirror image with respect to the line \(y = x\). In coordinates, the reflection map simply swaps the x- and y-coordinates. Concretely, the image of the point \((x, y)\) is the point \((y, x)\).
2. Algebraic rule (how to compute the reflection). Given an original point \((a, b)\), the reflected point across the line \(y = x\) is

   \[(a, b) \mapsto (b, a)\]

   This is the explicit procedure: exchange the coordinates.

3. Why swapping coordinates is the correct mirror image (geometric justification). Take the original point \((a, b)\) and its candidate reflected point \((b, a)\). Two facts show this is the correct reflection across the line \(y = x\).
   1. Midpoint lies on y = x. The midpoint of the segment joining \((a, b)\) and \((b, a)\) is

      \[\left(\frac{a+b}{2}, \frac{a+b}{2}\right)\]

      Because the coordinates are equal, the midpoint lies on the line \(y = x\), which is required for a mirror reflection.

   2. The joining segment is perpendicular to y = x. The slope of the segment from \((a, b)\) to \((b, a)\) equals

      \[\frac{a-b}{b-a} = -1\]

      The slope of the line \(y = x\) is 1. Since the product of slopes is -1, the segment is perpendicular to \(y = x\). A reflection requires that the segment connecting a point and its image be perpendicular to the mirror line.

4. Fixed points. Points that lie on the mirror line \(y = x\) do not move under this reflection. If a point satisfies \(y = x\), i.e. has coordinates \((t, t)\), then swapping coordinates yields \((t, t)\) again.
5. Effect on graphs of functions (relation to inverses). Reflecting the graph of a relation across \(y = x\) interchanges the roles of x and y. If the graph is \(y = f(x)\), the reflected graph satisfies

   \[x = f(y)\]

   If f is invertible, this can be written as

   \[y = f^{-1}(x)\]

   Thus geometric reflection across \(y = x\) corresponds to taking the inverse relation or inverse function (when it exists).

6. Linear-algebra viewpoint (matrix form). Reflection across \(y = x\) is the linear map that swaps coordinates. In matrix form, using column vectors, the map is

   \[\begin{pmatrix} x \\[4pt] y \end{pmatrix} \mapsto \begin{pmatrix} 0 & 1 \\[4pt] 1 & 0 \end{pmatrix} \begin{pmatrix} x \\[4pt] y \end{pmatrix} = \begin{pmatrix} y \\[4pt] x \end{pmatrix}\]

   This matrix is its own inverse, preserves distances, and has determinant -1 (indicating a reflection rather than a rotation).

7. Examples.
   1. Reflect the point \((2, 5)\). The image is \((5, 2)\).
   2. Reflect the line \(y = 2x + 3\). Swap x and y to get \(x = 2y + 3\). Solving for y gives \(y = \frac{x – 3}{2}\). The reflected line is \(y = \frac{x – 3}{2}\), which is the inverse relation of the original line.
8. Summary (compact statement). Reflection across \(y = x\) is the transformation that maps every point \((x, y)\) to \((y, x)\). Geometrically, it is the mirror reflection across the line \(y = x\): the midpoint of each original-image pair lies on \(y = x\) and the segment joining them is perpendicular to \(y = x\). For graphs of functions, reflecting across \(y = x\) corresponds to exchanging x and y, i.e. taking the inverse relation.