Y=X Reflection Rule (With Examples)

The y=x reflection rule allows us to find the reflection of a point about the diagonal line y=x. The reflection rule simply states that the reflection of the point (A,B) about the line y=x is (B,A).

Reflection about the Line Y=X:

In order to find the reflection of a point about the line y=x one must simply interchange the positions of the x and y co-ordinates. That is,

(x,y) ⟿ (y,x)

Examples:

1) A point in the first quadrant stays in the first quadrant on reflection about the line y=x. For example, the point (3,5) goes to the point (5,3).

(3,5) ⟿ (5,3)

2) A point in the second quadrant goes to the fourth quadrant on reflection about the line. For example, (-1,4) goes to (4,-1).

(-1,4) ⟿ (4,-1)

Conversely, a point lying in the fourth quadrant goes back to the second quadrant upon reflection. For example, (2,-5) goes to (-5,2).

(2,-5) ⟿ (-5,2)

3) A point in the third quadrant stays in the third quadrant on reflection about the line y=x. For example, the point (-1,-2) goes to the point (-2,-1).

(-1,-2) ⟿ (-2,-1)

Reflecting a Triangle about the Line Y=x:

In order to reflect a triangle about the line y=x we:

1. First label and find the co-ordinates of the vertices of the original triangle.
2. Reflect each of the vertices of the triangle via the reflection rule.
3. Join the new points obtained in order to find the new triangle.

For example, upon reflecting a triangle with vertices A(-1,1), B(-3,3) and C(-2,5) we obtain the new triangle as shown below.

The vertices of the reflected triangle are, D(1,-1), E(3,-3) and F(2,-5).

Reflecting a Function About the Line Y=X:

Suppose that we are given a function y = f(x). We perform the following procedure, in order to find the reflection of the function about the line.

1. Interchange the roles of x and y in the equation.
2. We obtain the new equation x = f(y).
3. Solve for y in the new equation to get the reflected function g(x)

For example, let y = x³ +4 be the given function.

Interchanging the roles of x and y we get,

x = y³ +4

y³ = x – 4

y = (x – 4)^1/3

Therefore g(x) = (x – 4)^1/3 is the new function obtained upon reflection.