Check the notes of __single transformation__ before going through this note!!

When an object is transformed under two successive transformations i.e., the initial image is re-transformed to obtain final image is called combination of transformation.

__Combination of two translations:__

Let T_{1}=(a, b) be a translation vector and T_{2}=(c, d) be another translation vector. If T_{1} maps P→P’ and T_{2} maps P’→P”, the translation vector(T) which transforms P→P’’ is given by;

** ** **T=T _{2}oT_{1}**

**#Transform P(x, y) under combined translation of T _{1}oT_{2} where T_{1}=(1, 2) and T_{2}**

→Under combined transformation T

T

The image under combined translation of T

P(x, y)→P’(x-2, y+3), is the final image

__Combination of two Rotation__

Let R_{1} be a rotation and R_{2} be another rotation such that R_{1} transforms P→P’ and R_{2} transforms P’→P”, the combined rotation(R) which transforms P→P’’ is given by;

**R=R _{2}oR_{1}=R_{2}+R_{1}**

**#Transform P(4, 5) through +90 ^{o} with center O and again P’ is rotated through +90 with center O. Find the final image of P under combined transformation R_{2}oR_{1}.**→Under combined transformation R

R

So, the image under combined transformation of R

P(4,5 )→P’(-4,-5), is the final image.

__Combination of two Reflection__

i. When the axes are parallel.

The combination of two reflections when the axes are parallel is given by the translation and the distance of the translation is twice the distance between the axes of reflection.

**#If P(4, 5) is reflected through R1:x=2 and R2****:x=-2, find the image under combined transformation.**

→distance between the axes=-2-2=-4

Required translation vector=(2*-4, 0) ** {as it is reflection under x axes}**

Now, the combined transformation is given by translation vector

(-8, 0)

P(4,5 )→P’(-4,5), is the final image.

ii. When the axes of reflection intersect at a point.

The combined transformation when the axes of reflection intersect at a point is given by the rotation about the point of intersection of axes of reflection and twice the angle between them. (angle between them is determined with the help of graph.)

**#If P(4, -5) is reflected through x axis and then again through y axis, find the combined transformation.**

Here,

Reflection under x-axis

P(4, -5)→P’(4, 5)

Reflection under y-axis

P’(4, 5)→P”(-4,5)

Here, the axes intersects at (0,0) and the angle between them is 90.

So, this is combined transformation through (90*2=180)^{o} about origin

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