HOW TO DO HORIZONTAL AND VERTICAL DILATION WITH THE GIVEN POINTS

Dilations are enlargements or reductions.

The transformation equation for a dilation with center (0, 0) and factor k are x' = kx and y' = ky

• If k > 1, the image figure is an enlargement of the object
• If 0 < k < 1, the image figure is an reduction of the object.

Vertical dilation with fixed y-axis :

Suppose P(x, y) moves P'(x', y') such that P' lies on the line through N(0, y) and P, and

we call this a horizontal dilation with the factor k.

For a horizontal dilation with factor k, the transformation equation are 

x' = kx and y' = y

Vertical dilation with fixed x-axis :

Suppose P(x, y) moves P'(x', y') such that P' lies on the line through N(x, 0) and P, and

we call this a vertical dilation with the factor k.

For a vertical dilation with factor k, the transformation equation are 

x' = x and y' = ky

Problem 1 :

Find the image of (2, 3) under a dilation with center O and factor 3.

Solution :

Here x = 2 and y = 3

dilation factor = 3, so k = 3

(x, y) ==> (kx, ky)

(2, 3) ==> ( 3(2), 3(3) )

(6, 9)

So, image of the point (2, 3) after the dilation is (6, 9).

Problem 2 :

Find the image of (-1, 4) under a dilation with center O and factor 1/3.

Solution :

Here x = -1 and y = 4

dilation factor = 1/3, so k = 1/3

(x, y) ==> (kx, ky)

(-1, 4) ==> ( 1/3(-1), 1/3(4) )

(-1/3, 4/3)

So, image of the point (-1, 4) after the dilation is (-1/3, 4/3).

Problem 3 :

Find the image of (3, -1) under a vertical dilation with factor 4

Solution :

Here x = 3 and y = -1

dilation factor = 4, so k = 4

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

(3, -1) ==> (3, 4(-1))

(3, -4)

So, image of the point (3, -1) after the vertical dilation is (3, -4).

Problem 4 :

Find the image of (4, 5) under a vertical dilation with factor 2

Solution :

Here x = 4 and y = 5

dilation factor = 2, so k = 2

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

(4, 5) ==> (4, 2(5)

(4, 10)

So, image of the point (4, 5) after the vertical dilation is (4, 10).

Problem 5 :

Find the image of (-2, 1) under a horizontal dilation with factor 1/2

Solution :

Here x = -2 and y = 1

dilation factor = 1/2, so k = 1/2

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

(-2, 1) ==> (-2(1/2), 1)

(-1, 1)

So, image of the point (-2, 1) after the vertical dilation is (-1, 1).

Problem 6 :

Find the image of (3, -4) under a horizontal dilation with factor 3/2

Solution :

Here x = 3 and y = -4

dilation factor = 3/2, so k = 3/2

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

(3, -4) ==> ((3/2)(3), -4)

(9/2, -4)

So, image of the point (3, -4) after the vertical dilation is (9/2, -4).