REFLECTION OVER Y EQUALS X

Graph the image of the figure using the transformation given.

Problem 1 :

Reflection across the y = x.

Solution :

By observing the figure given above, the coordinate of T is

(-4, 3)

Rule :

(x, y) ==> (y, x)

T (-4, 3) ==> T' (3, -4)

Problem 2 :

Reflection across the y = x of the point U(-3, 1). Find the new coordinate of U'.

Solution :

U (-3, 1)  ==> U' (1, -3)

Problem 3 :

Reflection across the line y = x

T(2, 2), C(2, 5), Z(5, 4), F(5, 0)

Solution :

By observing the coordinates, to use the rule we have to remember

(x, y) ==> (y, x)

T (2, 2) ==>  T' (2, 2)

C (2, 5)  ==>  C' (5, 2)

Z (5, 4)  ==> Z' (4, 5)

, F (5, 0)  ==>  F' (0, 5)

Problem 4 :

Reflection across y = x for F(2, 2), W(2, 5), K(3, 2). Find the new coordinates of F', W' and K'.

Solution :

Rule for reflection of y = x :

(x, y) ==> (y, x)

F (2, 2) ==>F' (2, 2)

W (2, 5) ==> W' (5, 2)

K (3, 2)  ==>  K' (2, 3)

Graph the image of the figure using the transformation given.

Problem 5 :

Reflection about y = x.

Solution :

By observing the coordinates of the vertices given above,

J(-5, 0), M(-5, -1), L(-1, 3) and K(-3, 5)

Using the rule :

(x, y) ==> (y, x)

J(-5, 0) ==> J' (0, -5)

M(-5, -1) ==> M' (-1, -5)

L(-1, 3) ==> L' (3, -1)

K(-3, 5) ===> K' (5, -3)

Problem 6 :

Graph △ABC with vertices

A(1, 3), B(5, 2), and C(2, 1) and its image after the reflection described.

a. In the line n: x = 3

b. In the line m: y = 1

Solution :

Point A is 2 units left of line n, so its reflection A′ is 2 units right of line n at (5, 3). Also, B′ is 2 units left of line n at (1, 2), and C′ is 1 unit right of line n at (4, 1).

Problem 7 :

Graph FG with endpoints F(−1, 2) and G(1, 2) and its image after a reflection in the line y = x.

Solution :

Point A is 2 units above line m, so A′ is 2 units below line m at (1, −1). Also, B′ is 1 unit below line m at (5, 0). Because point C is on line m, you know that C = C′.

Problem 8 :

Graph FG with endpoints F(−1, 2) and G(1, 2) and its image after a reflection in the line y = x.

Solution :

Reflection about the line y = x

F(-1, 2) ==> F'(2, -1)

G(1, 2) ==> G'(2, 1)

Problem 9 :

The vertices of △JKL are J(1, 3), K(4, 4), and L(3, 1).

a) Graph △JKL and its image after a reflection in the x-axis.

b) Graph △JKL and its image after a reflection in the y-axis. 7

c) Graph △JKL and its image after a reflection in the line y = x.

d) Graph △JKL and its image after a reflection in the line y = −x

e) verify that FF′ is perpendicular to y = −x

Solution :

J(1, 3), K(4, 4), and L(3, 1)

a) Reflection across x-axis :

(x, y) ==> (x, -y)

J(1, 3) ==> J'(1, -3)

K(4, 4) ==> K'(4, -4)

L(3, 1) ==> L'(3, -1)

b) reflection in the y-axis

(x, y) ==> (-x, y)

J(1, 3) ==> J'(-1, 3)

K(4, 4) ==> K'(-4, 4)

L(3, 1) ==> L'(-3, 1)

c)  reflection in the line y = x :

(x, y) ==>(y, x)

J(1, 3) ==> J'(3, 1)

K(4, 4) ==> K'(4, 4)

L(3, 1) ==> L'(1, 3)

d) reflection in the line y = −x

(x, y) ==>(-y, -x)

J(1, 3) ==> J'(-3, -1)

K(4, 4) ==> K'(-4, -4)

L(3, 1) ==> L'(-1, -3)

e) verify that FF′ is perpendicular to y = −x

Problem 10 :

Graph triangle ABC with vertices A (-1, 4) B (2, -1) and C (4, 3) and its image after the glide reflection.

Translation (x, y) ==> (x + 2, y - 1)

Reflection : in the line y = x

Solution :

Given points are A (-1, 4) B (2, -1) and C (4, 3)

Reflection about y = x

A (-1, 4) ==> A' (4, -1)

B (2, -1) ==> B'(-1, 2)

C (4, 3) ==> C' (3, 4)

Translation (x, y) ==> (x + 2, y - 1)

A (4, -1) ==> A' (4 + 2,  -1 - 1) ==> A' (6, -2)

B (-1, 2) ==> B'(-1 + 2, 2 - 1) ==> B' (1, 1)

C (3, 4) ==> C' (3 + 2, 4 - 1) ==> C' (5, 3)