 Hello and welcome to this session. Now in level 1 under topic similarity and transformation we have discussed the various types of transformations, translation, reflection, rotation and dilation. In this session we will see that the transformation acts as a function that takes points in the plane as input and give other points as output. Now let us recall the three types of transformations. Now translation moves the object so that every point on the object moves in same direction at same distance. Here only the object size, shape and size remain same. Now this figure represents translation of image where every point has moved in same direction and at same distance. Now let us see what is rotation. Now rotation is a transformation in which the object is rotated about a fixed point. The direction of rotation can be clockwise or empty clockwise and here also shape and size remain same that is shape and size of the figure remains same. Now here you can see this figure represents rotation of image about center of rotation A that is the point A by some angle. Now let us see the third type of transformation which is reflection in reflection. An object is split across a line of reflection so shape and size remain same. Now this figure represents a reflection of image in vertical axis. The image is split in the line of reflection. Now let us discuss transformation as a function. Now transformation is an operation that moves, fits or changes a figure to create a new figure and here the new figure created by a transformation is called image and original figure is called the P image. Now if the P image is given by A then its image will be given by A dash and image of A dash will be given by A dash. Now a transformation is one to one correspondence between two sets of points to only one point in set A dash called its image and every point in set A dash is image of one and only one point in set A called its P image. Thus a transformation maps an original figure sample let image of point with coordinates x y in set A be a point with coordinates 2 minus x y which is a point in set A that is transforms to now and with this transformation a point with coordinates 2 minus 1 transforms to a point with coordinates 2 minus x y now where x is 2 y is minus 1 so coordinates of new point will be 2 minus 2 that is 0 and y that is minus 1 thus this point with coordinates 0 minus 1 in set A dash is the image point with coordinates 2 minus 1 in set A under this transformation here the point 2 minus 1 that is the point with coordinates 2 minus 1 is called input point and the point with coordinates 0 minus 1 is called so here we can see that whenever we are given the coordinates of any figure along with the transformation rule we can easily find the coordinates of the new image after the transformation and hence write the coordinates of the image of the given figure now let us discuss transformation rules for reflection first of all let us discuss reflection in x axis now under reflection in x axis the image of point A with coordinates x y the point A dash with coordinates x minus y so here we write x y transforms in y under reflection in x axis now let us discuss an example here when we reflect the point A with coordinates 3 4 in x axis we get its image point as the point A dash with coordinates 3 minus 4 now let us discuss reflection in y axis now under reflection in y axis the image of the point A with coordinates x y is A dash with coordinates minus x y and we write x y transforms to minus x y under reflection in y axis now here we can see under reflection in y axis the image of point A with coordinates 3 4 is the point A dash with coordinates minus 3 4 now let us see reflection in the line y is equal to x now under reflection in the line y is equal to x the image of the point A with coordinates x y is the point A dash with coordinates y x y transforms to y x now here we can see under reflection in the line y is equal to x the image of point A with coordinates 3 5 is the point A dash with coordinates 5 3 now let us see reflection in original now under reflection in original the image of the point A with coordinates x y is point A dash with coordinates minus x minus y and we write x y transforms to minus x minus y now we know that the point where the x axis and y axis intersect each other is called original now under reflection in original the image of point A with coordinates 3 3 is point A dash with coordinates minus 3 minus 3 now let us discuss transformation rules for translation now a translation of A units in horizontal direction and B units in vertical direction is the transformation of the plane such that the image of point A with coordinates x y is the point A dash with coordinates x plus A y plus B and we can write it as x y transforms to x plus A y plus B now let us discuss an example here and the translation where x y transforms to x minus 3 y plus 2 the image of the point 4 minus 3 will be 3 y plus 2 now where x is 4 y is minus 3 so image of this point will be 4 minus 3 that is 1 minus 3 plus 2 that is minus 1 also we must note that in a translation if a point is moved to the right then A is positive and if it is moved to the left then A is negative similarly when point is moved up then B is positive if it is moved down then B is negative now let us discuss transformation rules for rotation now let us discuss rotation at angle of 90 degrees about original now when image is rotated at 90 degrees angle about original in clockwise image of point A with coordinates A dash with coordinates y minus and where we write transforms to y minus and when image is rotated at 90 degrees angle about original in anti clockwise direction then the image on A with coordinates x y with coordinates minus y transforms to minus y x now on rotation of 90 degrees angle about original in clockwise direction the image of point A with coordinates minus 3 minus 2 will be the point A dash now where y is minus 3 so minus x will be minus of minus 3 which is 3 dash with coordinates minus 2 3 the image of point A with coordinates minus 3 minus 2 on rotation of 90 degrees about original in anti clockwise direction is the point A double dash with coordinates where y is minus 2 minus y will be minus of minus 2 which is 2 so image point is A double dash with coordinates 2 now here we can see when we have rotated the point A with coordinates minus 3 minus 2 at an angle of 90 degrees about original in clockwise direction then we get its image point as A dash with coordinates minus 2 3 and when we rotated it at an angle of 90 degrees about original in anti clockwise direction then its image point is the point A double dash with coordinates 2 minus 1 just discuss rotation at angle of 180 degrees about original now when image is rotated at 180 degrees angle the direction does not matter in this rotation then image of point A with coordinates x y is the point A dash with coordinates minus x minus y and we write x y transforms to minus x minus y for example image of point A with coordinates when rotated at 180 degrees angle about original is point A dash with coordinates minus 2 minus 1 now here we can see when we rotate the point A with coordinates of 180 degrees about original in anti clockwise direction we get the image point as A dash with coordinates minus 2 minus 1 and when we rotate at an angle of 180 degrees about original in clockwise direction then again we get the same point that is the image point A dash with coordinates minus 2 minus 1 so we can see in this rotation direction does not matter and now let us discuss rotation at angle of 270 degrees about original now then image is rotated at 270 degrees angle about original in clockwise direction then image of the point A with coordinates x y is the point A dash with coordinates minus y x and we write x y transforms to minus y x and when image is rotated at 270 degrees angle about original in anti clockwise direction then of the point A with coordinates x y is the point A dash with coordinates y minus x y transforms to y minus for example image of point A with coordinates 2 minus 3 when rotated at an angle of 270 degrees about original in clockwise direction then image will be A dash that is the point A dash with coordinates 3 2 and when it is rotated at an angle of 270 degrees about original in anti clockwise direction then its image point is the point A dash with coordinates minus 3 minus 2 so in this session we have learnt that transformation x as a function that takes points in the plane as input and gives other points as output and this completes our session hope you all have enjoyed the session