Ximpledu

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

Example

(-4, -5)
Reflection in the x-axis
Solution

Example

y = x2 + 1
Reflection in the x-axis
Solution

Example

Point P moves on the x-axis.
Minimum value of AP + PB = ?
Solution

Reflection: y-axis

Formula


(x, y) → (-x, y)

Example

(3, 2)
Reflection in the y-axis
Solution

Example

(-4, -5)
Reflection in the y-axis
Solution

Example

y = 2x + 1
Reflection in the y-axis
Solution

Reflection: Origin

Formula


(x, y) → (-x, -y)

Example

(3, 2)
Reflection in the origin
Solution

Example

(-1, -4)
Reflection in the origin
Solution

Example

y = (x - 2)2 + 1
Reflection in the origin
Solution

Reflection: y = x

Formula


(x, y) → (y, x)

Example

(5, 2)
Reflection in y = x
Solution

Example

y = x2
Reflection in y = x
Solution

Translation: Point

Formula


(x, y) → (x + a, y + b)

Example

(3, 1)
Translation (x, y) → (x + 4, y + 5)
Solution

Example

(8, -1)
Translation (x, y) → (x - 7, y + 3)
Solution

Translation: Function

Formula


f(x, y) = 0 → f(x - a, y - b) = 0
Unlike the translation of a point,
change the signs of a and b.

Example

y = 2x + 4
Translation (x, y) → (x + 5, y + 3)
Solution

Example

y = -x + 1
Translation (x, y) → (x - 2, y + 6)
Solution

Example

y = x2
Translation (x, y) → (x + 3, y + 2)
Solution

Dilation

Formula


(x, y) → (kx, ky)

Example

(3, 2)
Dilation of 2
Solution

Example

(3, 2)
Dilation of 1/2
Solution

Example

(3, 2)
Dilation of -2
Solution

Rotation: 90 Degrees Counterclockwise

Formula


(x, y) → (-y, x)

Example

(3, 2)
Rotation of 90º counterclockwise about the origin
Solution

Example

(-4, -5)
Rotation of 90º counterclockwise about the origin
Solution

Rotation: 90 Degrees Clockwise

Formula


(x, y) → (y, -x)

Example

(3, 2)
Rotation of 90º clockwise about the origin
Solution

Example

(-4, -5)
Rotation of 90º clockwise about the origin
Solution