REFLECTION ACROSS THE Y AXIS

Reflection over y-axis

The rule of reflection about y-axis is

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

What is preimage ?

Preimage In a transformation, the original figure is called the preimage.

What is image ?

Image In a transformation, the final figure is called the image.

Graph the image of the figure using the transformation given.

Problem 1 :

Reflection across the y-axis

Solution :

Marking the point Q (3, 3). Reflection of Q across y-axis is

Q' (-3, 3)

Problem 2 :

Reflection across the x-axis U (-3, 1)

Solution :

Rule :

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

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

Problem 3 :

reflection across the y-axis

A (-1, -5), B (-2, -2), C (-1, 0), D (3, -2)

Solution :

Rule : 

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

A (-1, -5)  ==> A' (1, -5)

B (-2, -2)  ==> B' (2, -2)

C (-1, 0)  ==> C' (1, 0)

D (3, -2)  ==> D' (-3, -2)

Graph the image of the figure using the transformation given.

Problem 4 :

Reflection across the y-axis.

Solution :

By observing the points from the given figure,

S (-3, 2), B (-3, -3) and Z (1, 0)

Rule : 

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

S (-3, 2)  ==> S' (3, 2)

B (-3, -3) ==> B' (3, -3)

Z (1, 0)  ==> Z' (-1, 0)

Problem 5 :

Reflection across the y-axis D(-2, -3), E(2, -2), F(3, -4)

Solution :

Rule : 

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

D(-2, -3) ==> D' (2, -3)

E(2, -2)  ==> D' (-2, -2)

F(3, -4)  ==> F' (-3, -4)

Find the coordinates of the vertices of each figure after the given transformation.

Problem 6 :

Reflection across the y-axis K(1, −1), N(4, 0), Q(4, −4)

Solution :

Rule : 

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

K (1, −1) ==> K' (-1, -1)

N (4, 0) ==> N' (-4, 0)

Q (4, −4) ==> Q' (-4, -4)

Problem 7 :

Write a rule to describe each transformation. 

Solution :

By observing the coordinates of the triangle XGF.

X (3, 3), G (2, 2) and F (4, 1)

Observing the coordinates of the triangle X'G'F'

X' (-3, 3), G' (-2, 2) and F' (-4, 1)

X (3, 3) ==>  X' (-3, 3)

G (2, 2)  ==> G' (-2, 2)

F (4, 1)  ==>  F' (-4, 1)

Clearly the x-coordinate alone changes its sign. So, it is reflection about y-axis.

Problem 8 :

The coordinates of a point and its image are given. Is the reflection in the x-axis or y-axis?

a) (2, −2) ==> (2, 2)

b) (−4, 1) ==> (4, 1)

c) (−2, −5) ==> (2, −5)

d) (−3, −4) ==> (−3, 4)

Solution :

a) (2, −2) ==> (2, 2)

Here y is changed as -y, so the reflection is across x-axis.

b) (−4, 1) ==> (4, 1)

Here x is changed as -x. So, the reflection across y-axis.

c) (−2, −5) ==> (2, −5)

Here x is changed as -x. So, the reflection across y-axis.

d) (−3, −4) ==> (−3, 4)

Here y is changed as -y, so the reflection is across x-axis.

Problem 9 :

Graph the polygon and its image after a reflection in the given line.

x - axis 

Solution :

The coordinates of the original figures are

L(-3, 6), K(-1, 4), N(1, 0) and M(4, 2)

To follow the reflection across x-axis, we have to follow the rule

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

L(-3, 6) ==> L'(-3, -6)

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

N(1, 0) ==> N'(1, 0)

M(4, 2) ==> M'(4, -2)

Problem 10 :

Graph the polygon and its image after a reflection in the given line.

y = −1

Solution :

The coordinates of the original figures are

L(-2, -2), K(-4, 6), N(8, 7) and M(6, -1)

To follow the reflection across y = -1

The vertical distance between each point to each coordinates, 

• The vertical distance between y = -1 and L is, moving down 1 unit from the line of reflection. So, we have to move up of 1 unit from the line of reflection.

-1 + 1 ==> 0

L(-2, -2) ==> L'(-2, 0)

• The vertical distance between y = -1 and K is, moving up 7 units from the line of reflection. So, we have to move down of 7 units.

-1 - 7 ==> -8

K(-4, 6) ==> K'(-4, -8)

• The vertical distance between y = -1 and N is, moving up 8 units from the line of reflection. So, we have to move down of 8 units.

-1 - 8 ==> -9

N(8, 7) ==> N'(8, -9)

Problem 11 :

Translate the triangle 1 unit right and 5 units down. Then reflect the image in the y-axis 

Solution :

By observing the figure, the coordinates are 

R (-4, 1), S (-4, 4) and T (-2, 1)

Translating right of 1 unit and moving down of 5 units,

(x, y) ==> (x + 1, y - 5)

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

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

T(-2, 1) ==> T' (-2 + 1, 1 - 5) ==> T'(-1, -4)

Reflection across y-axis :

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

R'(-3, -4) ==> R'(3, -4)

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

T'(-1, -4) ==> T'(1, -4)