TRANSFORMATIONS ON THE COORDINATE PLANE

About the topic "Transformations on the coordinate plane"

Transformations on the coordinate plane :

We can perform transformations on a coordinate plane by changing the coordinates of the points on a figure. The points on the translated figure are indicated by the prime "symbol" to distinguish them from the original points

Transformations on the Coordinate Plane


Reflection

A point on a coordinate plane can be reflected across an axis. The reflection is located on the opposite side of the axis, at the same distance from the axis.


Translation

A translation "slides" an object a fixed distance in a given direction.  The original object and its translation have the same shape and size, and they face in the same direction.  It is a direct isometry.


Dilation

A dilation is a transformation that produces an image that is the same shape as the original, but is a different size.  NOT an isometry.  Forms similar figures. 


Rotation

To rotate a figure 90 degree clockwise about the origin, switch the coordinates of each point and then multiply the new first coordinate by -1.

To rotate 180 degree about origin, multiply both coordinates of each point by -1.

Reflection on coordinate plane

Example 1 :

Graph (3, −2). Then fold your coordinate plane along the y-axis and find the reflection of (3, −2). Record the coordinates of the new point in the table.

Solution :

Let us see the next example on "Transformations on the coordinate plane".

Example 2 :

Graph (3, −2). Then fold your coordinate plane along the x-axis and find the reflection of (3, −2). Record the coordinates of the new point in the table.

Solution :

Reflection about the x-axis

Reflection about the y-axis

Reflection about the line y=x

Once students understand the rules which they have to apply for reflection transformation, they can easily make reflection -transformation of a figure.

For example, if we are going to make reflection transformation of the point (2,3) about x-axis, after transformation, the point would be (2,-3). Here the rule we have applied is (x, y) ------> (x, -y). 

So we get (2,3) -------> (2,-3).

Translation on coordinate plane

Translation of h, k : 

 ( x , y ) -----------> ( x + h , y + k )

Translation for (h ,  k) = (-1 , 2) 

Let us see the next example on "Transformations on the coordinate plane".

Example 3 :

Let A ( -2, 1), B (2, 4) and (4, 2) be the three vertices of a triangle. If this triangle is translated for ( h, k ) = ( 2, 3) what will be the new vertices A' , B' and C' ?

Solution :

Step 1 :

First we have to know the correct rule that we have to apply in this problem.

Step 2 :

Here triangle is translated for (h , k ) = ( 2 , 3 ). So the rule that we have to apply here is (x , y) -------> ( x + h , y + k )

Step 3 :

Based on the rule given in step 1, we have to find the vertices of the translated triangle A'B'C'

Step 4 :

(x , y) ----------> ( x + h , y + k )

A ( -2, 1 ) ------------ A' ( 0, 4 )

B ( 2, 4 ) ------------ B' ( 4, 7 )

C ( 4, 2 ) ------------ C' ( 6, 5 )

Step 5 :

Vertices of the translated triangle are 

                       A' ( 0, 4) , B ( 4, 7 ) and C' ( 6, 5)  

Dilation on the coordinate plane

Dilation rule 

Dilation of scale factor "k" : 

 ( x , y ) -----------> (kx , ky )

Dilation for " k = 2 "

Example 4 :

Let A ( -2, -2), B ( -1, 2) and (2 , 1) be the three vertices of a triangle. If this triangle is dilated for the scale factor " k  = 2 ", what will be the new vertices A' , B' and C' ?

Solution :

Step 1 :

First we have to know the correct rule that we have to apply in this problem.

Step 2 :

Here triangle is dilated for the scale factor "k  = 2". So the rule that we have to apply here is (x , y) -------> ( kx , ky )

Step 3 :

Based on the rule given in step 1, we have to find the vertices of the dilated triangle A'B'C'

Step 4 :

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

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

B ( -1, 2 ) ------------ B' ( -2, 4 )

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

Step 5 :

Vertices of the dilated triangle are                      

                       A' ( -4, -4) , B ( -2, 4 ) and C' ( 4, 2)  

Rules on finding rotated image

90° rotation (clock wise)

90° rotation (counter clock wise)

180° rotation (clock wise and counter clock wise)

Example 5 :

Let A ( -2, 1), B (2, 4) and C (4, 2) be the three vertices of a triangle. If this triangle is rotated about 90° clockwise, what will be the new vertices A' , B' and C' ?

Solution :

Step 1 :

First we have to know the correct rule that we have to apply in this problem.

Step 2 :

Here triangle is rotated about 90° clock wise. So the rule that we have to apply here is (x , y) -------> (y , -x)

Step 3 :

Based on the rule given in step 1, we have to find the vertices of the reflected triangle A'B'C'

Step 4 :

(x , y) ----------> (y , -x)

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

B ( 2, 4 ) ------------ B' ( 4, -2 )

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

Step 5 :

Vertices of the reflected triangle are                      

                       A' ( 1, 2) , B ( 4, -2 ) and C' ( 2, -4)  

After having gone through the stuff given above, we hope that the students would have understood "Transformations on the coordinate plane". 

Apart from the stuff given above, if you want to know more about "Transformations on the coordinate plane", please click here

If you need any other stuff in math, please use our google custom search here. 

HTML Comment Box is loading comments...