**Algebraic representations of dilations :**

When a dilation in the coordinate plane has the origin as the center of dilation, we can find points on the dilated image by multiplying the x and y coordinates of the original figure by the scale factor.

For example, if the scale factor is "k", the algebraic representation of the dilation is

(x, y) → (kx, ky)

For enlargements, k > 1 and for reductions, k < 1.

**Example 1 :**

The triangle PQR shown on the grid is the pre-image. If the center of dilation is the origin and the scale factor is 3, graph the dilated image P'Q'R'.

**Solution : **

**Step 1 :**

List the coordinates of the vertices of the pre image.

P(1, 3), Q(3, 1) and R(1, 1)

**Step 2 :**

Since the scale factor is 3, the rule to get the coordinates of the vertices of the image is

(x, y) → (3x, 3y)

**Step 3 :**

List the coordinates of the vertices of the image.

P(1, 3) ---> P'(3, 9)

Q(3, 1) ---> Q'(9, 3)

R(1, 1) ---> R'(3, 3)

**Step 4 : **

Graph the image P'Q'R'.

**Example 2 :**

The triangle ABC shown on the grid is the pre-image. If the center of dilation is the origin and the scale factor is 1/3, graph the dilated image A'B'C'.

**Solution : **

**Step 1 : **

List the coordinates of the vertices of the pre image.

A(3, 9), B(9, 9) and C(3, 3)

**Step 2 : **

Since the scale factor is 1/3, the rule to get the coordinates of the vertices of the image is

(x, y) → [(1/3)x, (1/3)y]

**Step 3 : **

List the coordinates of the vertices of the image.

A(3, 9) ---> A'(1, 3)

B(9, 9) ---> B'(3, 3)

C(3, 3) ---> C'(1, 1)

**Step 4 : **

Graph the image A'B'C'.

**Example 3 :**

The arrow ABCDE shown on the grid is the pre-image. If the center of dilation is the origin and the scale factor is 0.5, graph the dilated image A'B'C'D'E'.

**Solution : **

**Step 1 :**

List the coordinates of the vertices of the pre image.

A(1, 1), B(3, 1), C(3, 4), D(1, 4) and E(2, 3)

**Step 2 :**

Since the scale factor is 1/2, the rule to get the coordinates of the vertices of the image is

(x, y) → [0.5x, 0.5y]

**Step 3 :**

List the coordinates of the vertices of the image.

A(-4, 2) ---> A'(-2, 1)

B(0, 5) ---> B'(0, 2.5)

C(4, 2) ---> C'(2, 1)

D(2, -4) ---> D'(1, -2)

E(-2, -4) ---> E'(-1, -2)

**Step 4 : **

Graph the image A'B'C'D'E'.

**Example 4 :**

The rectangle JKLM shown on the grid is the pre-image. If the center of dilation is the origin and the scale factor is 2, graph the dilated image J'K'L'M'.

**Solution : **

**Step 1 :**

List the coordinates of the vertices of the pre image.

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

**Step 2 :**

Since the scale factor is 2, the rule to get the coordinates of the vertices of the image is

(x, y) → (2x, 2y)

**Step 3 :**

List the coordinates of the vertices of the image.

J(1, 1) ---> J'(2, 2)

K(3, 1) ---> K'(6, 2)

L(3, 4) ---> L'(6, 8)

M(1, 4) ---> M'(2, 8)

**Step 4 : **

Graph the image J'K'L'M'.

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