# Transformation (Math)

See how to find a point/function

under a reflection/transition/dilation/rotation.

24 examples and their solutions.

## Reflection: x-axis

### Formula

(x, y) → (x, -y)

### Example

(3, 2)

Reflection in the x-axis

Solution Reflection in the x-axis

(3, 2) → (3, -2)

Graph

Close

### Example

(-4, -5)

Reflection in the x-axis

Solution Reflection in the x-axis

(-4, -5) → (-4, 5)

Graph

Close

### Example

y = x

Reflection in the x-axis

Solution ^{2}+ 1Reflection in the x-axis

y = x

→ -y = x

y = -x

^{2}+ 1→ -y = x

^{2}+ 1y = -x

^{2}- 1Graph

Close

### Example

Point P moves on the x-axis.

Minimum value of AP + PB = ?

Solution Minimum value of AP + PB = ?

AB' = √(5 - 2)

^{2}+ (-3 - 1)

^{2}- [2]

= √3

^{2}+ 4

^{2}

= √9 + 16

= √25

= √5

^{2}

= 5

[1]

B(5, 3) → B'(5, -3)

PB = PB'

AP + PB = AP + PB'

So AP + PB becomes minimum

when A, P, B' are in a straight line.

So AB' is the minimum value of AP + PB.

PB = PB'

AP + PB = AP + PB'

So AP + PB becomes minimum

when A, P, B' are in a straight line.

So AB' is the minimum value of AP + PB.

[2]

A(2, 1), B'(5, -3)

Distance Formula

Distance Formula

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## Reflection: y-axis

### Formula

(x, y) → (-x, y)

### Example

(3, 2)

Reflection in the y-axis

Solution Reflection in the y-axis

(3, 2) → (-3, 2)

Graph

Close

### Example

(-4, -5)

Reflection in the y-axis

Solution Reflection in the y-axis

(-4, -5) → (4, -5)

Graph

Close

### Example

y = 2x + 1

Reflection in the y-axis

Solution Reflection in the y-axis

y = 2x + 1

→ y = 2(-x) + 1

y = -2x + 1

→ y = 2(-x) + 1

y = -2x + 1

Graph

Close

## Reflection: Origin

### Formula

(x, y) → (-x, -y)

### Example

(3, 2)

Reflection in the origin

Solution Reflection in the origin

(3, 2) → (-3, -2)

Graph

Close

### Example

(-1, -4)

Reflection in the origin

Solution Reflection in the origin

(-1, 4) → (1, -4)

Graph

Close

### Example

y = (x - 2)

Reflection in the origin

Solution ^{2}+ 1Reflection in the origin

y = (x - 2)

→ -y = (-x - 2)

-y = (x + 2)

y = -(x + 2)

^{2}+ 1→ -y = (-x - 2)

^{2}+ 1-y = (x + 2)

^{2}+ 1 - [1]y = -(x + 2)

^{2}- 1Graph

[1]

Close

## Reflection: y = x

### Formula

(x, y) → (y, x)

### Example

(5, 2)

Reflection in y = x

Solution Reflection in y = x

(5, 2) → (2, 5)

Graph

Close

### Example

## Translation: Point

### Formula

(x, y) → (x + a, y + b)

### Example

(3, 1)

Translation (x, y) → (x + 4, y + 5)

Solution Translation (x, y) → (x + 4, y + 5)

(x, y) → (x + 4, y + 5)

(3, 1) → (3 + 4, 1 + 5)

= (7, 6)

(3, 1) → (3 + 4, 1 + 5)

= (7, 6)

Graph

Close

### Example

(8, -1)

Translation (x, y) → (x - 7, y + 3)

Solution Translation (x, y) → (x - 7, y + 3)

(x, y) → (x - 7, y + 3)

(8, -1) → (8 - 7, -1 + 3)

= (1, 2)

(8, -1) → (8 - 7, -1 + 3)

= (1, 2)

Graph

Close

## Translation: Function

### Formula

### Example

y = 2x + 4

Translation (x, y) → (x + 5, y + 3)

Solution Translation (x, y) → (x + 5, y + 3)

(x, y) → (x + 5, y + 3)

y = 2x + 4

→ (y - 3) = 2(x - 5) + 4

y - 3 = 2x - 10 + 4

y - 3 = 2x - 6

y = 2x - 3

y = 2x + 4

→ (y - 3) = 2(x - 5) + 4

y - 3 = 2x - 10 + 4

y - 3 = 2x - 6

y = 2x - 3

Graph

Close

### Example

y = -x + 1

Translation (x, y) → (x - 2, y + 6)

Solution Translation (x, y) → (x - 2, y + 6)

(x, y) → (x - 2, y + 6)

y = -x + 1

→ (y - 6) = -(x + 2) + 1

y - 6 = -x - 2 + 1

y - 6 = -x - 1

y = -x + 5

y = -x + 1

→ (y - 6) = -(x + 2) + 1

y - 6 = -x - 2 + 1

y - 6 = -x - 1

y = -x + 5

Graph

Close

### Example

y = x

Translation (x, y) → (x + 3, y + 2)

Solution ^{2}Translation (x, y) → (x + 3, y + 2)

(x, y) → (x + 3, y + 2)

y = x

→ (y - 2) = (x - 3)

y = (x - 3)

y = x

^{2}→ (y - 2) = (x - 3)

^{2}y = (x - 3)

^{2}+ 2 - [1]Graph

[1]

Close

## Dilation

### Formula

(x, y) → (kx, ky)

### Example

(3, 2)

Dilation of 2

Solution Dilation of 2

(3, 2) → (2⋅3, 2⋅2)

= (6, 4)

= (6, 4)

Graph

Close

### Example

(3, 2)

Dilation of 1/2

Solution Dilation of 1/2

(3, 2) → (12⋅3, 12⋅2)

= (32, 1)

= (32, 1)

Graph

Close

### Example

(3, 2)

Dilation of -2

Solution Dilation of -2

(3, 2) → (-2⋅3, -2⋅2)

= (-6, -4)

= (-6, -4)

Graph

Close

## Rotation: 90 Degrees Counterclockwise

### Formula

(x, y) → (-y, x)

### Example

(3, 2)

Rotation of 90º counterclockwise about the origin

Solution Rotation of 90º counterclockwise about the origin

(3, 2) → (-2, 3)

Graph

Close

### Example

(-4, -5)

Rotation of 90º counterclockwise about the origin

Solution Rotation of 90º counterclockwise about the origin

(-4, -5) → (5, -4)

Graph

Close

## Rotation: 90 Degrees Clockwise

### Formula

(x, y) → (y, -x)

### Example

(3, 2)

Rotation of 90º clockwise about the origin

Solution Rotation of 90º clockwise about the origin

(3, 2) → (2, -3)

Graph

Close

### Example

(-4, -5)

Rotation of 90º clockwise about the origin

Solution Rotation of 90º clockwise about the origin

(-4, -5) → (-5, 4)

Graph

Close