ALGEBRAIC REPRESENTATIONS OF DILATIONS

About "Algebraic representations of dilations"

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.

Algebraic representations of dilations - Examples

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'.

After having gone through the stuff given above, we hope that the students would have understood "Algebraic representations of dilations". 

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