 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that sketch the transformation of given function y is equal to f of x to the function y is equal to 1 by 2 into f of minus of 1 by 2 into x minus 1 the whole plus 1. Before starting the solution of this question we should know some results that is if we are given any function y is equal to f of x then y is equal to a into f of b into x minus c the whole plus d affects the graph in following ways. Now a causes the graph of the function to stretch when a is greater than 1 or compress when 0 is less than a is less than 1 vertically by factor a and if the value of a is less than 0 that is there is a negative sign before the given function then the graph is reflected over x axis. Similarly b causes the graph of the function to stretch when 0 is less than b is less than 1 or compress when b is greater than 1 horizontally by factor 1 upon b and if the value of b is less than 0 then the graph is reflected over y axis. Similarly c causes the graph to translate horizontally by c units. Also if the value of c is greater than 0 the curve moves to right and if the value of c is less than 0 the curve moves to left. d causes the graph of the function to translate vertically by d units. If the value of d is greater than 0 then the curve moves upwards and if the value of b is less than 0 then the curve moves downwards. With this key idea we proceed to the solution. Here in the question we are given graph of a function y is equal to f of s and we have to draw the graph of the function y is equal to 1 by 2 into f of minus 1 by 2 into x minus 1 the whole plus 1. Now this function is of the type y is equal to a into f of b into x minus c the whole plus d. So on comparing these two functions we can find out the values of a, b, c and d. So here a is equal to 1 by 2 b is equal to minus of 1 by 2 c is equal to 1 and d is equal to 1. So here we have a as 1 by 2 b as minus 1 by 2 c is equal to 1 and d is equal to 1. So from this key idea we can identify the transformations associated with a, b, c and d. So we have the following transformations a is equal to 1 by 2 so vertical compression by factor 1 by 2 b is equal to minus 1 by 2 implies that modulus of b will be equal to modulus of minus 1 by 2 which is equal to 1 by 2. So modulus of b is equal to 1 by 2 means there is a horizontal stretch by factor 1 upon modulus of b which is equal to 1 upon 1 by 2 which is equal to 2. Also b is equal to minus of 1 by 2 which is less than 0 so there is reflection in y-axis c is equal to 1 so curve moves one unit to the right d is equal to 1 so curve will move one unit up using these transformations we will sketch the curve since we are not given any order of transformation so we stretch or compress the graph and reflected first and later translate the curve. So first we will vertically compress the curve by factor 1 by 2 we know that in vertical stretch or compression by factor a the point with coordinates x y is transformed to the point with coordinates x a y that is we multiply the y coordinate by factor a here we see that this curve passes through the points that is minus 2 0 minus 1 minus 1 0 0 1 1 and 2 0 so to find coordinates after the stretch we multiply the y coordinate by 1 by 2 as here the value of a is 1 by 2 so the new coordinates will be that is the vertically compressed curve will pass through points given by the coordinates now the point with coordinates minus 2 0 is transformed to the point with coordinates minus 2 0 into 1 by 2 that is 0 so we have the coordinates minus 2 0 now minus 1 minus 1 that is the point with coordinates minus 1 minus 1 is transformed to the point with coordinates minus 1 now minus 1 into 1 by 2 is minus 1 by 2 so we have the coordinates minus 1 minus 1 by 2 now the point with coordinates 0 0 is transformed to the point with coordinates 0 now 0 into 1 by 2 is 0 so we have the coordinates 0 0 now point with coordinates 1 1 is transformed to the point with coordinates 1 now we multiply 1 with 1 by 2 and we get 1 by 2 so we have the coordinates 1 1 by 2 similarly point with coordinates 2 0 is transformed to the point with coordinates 2 0 into 1 by 2 that is 0 so we have the coordinates 2 0 now we shall plot all these points on the coordinate plane so here we have plotted these points and now we join them in similar shape as of the given curve so here this is the transformed curve and the equation of this blue curve is y is equal to 1 by 2 f of x now we horizontally stretch this blue curve by factor 2 we know that in horizontal stretch by factor 1 upon b the coordinates x y are transformed to the coordinates 1 by b into x y here 1 upon b is equal to 2 now to obtain new coordinates of horizontally stretched curve we multiply the x coordinates of the points line on this blue curve by factor 2 points line on the blue curve are given by the coordinates minus 2 0 minus 1 minus 1 by 2 0 0 1 1 by 2 and 2 0 now the new coordinates after horizontal stretch are given by the coordinates first we take minus 2 0 and we multiply the x coordinate by factor 2 so we obtain minus 4 0 similarly the point with coordinates minus 1 minus 1 by 2 is transformed to the point with coordinates minus 2 minus 1 by 2 similarly the point 0 0 is transformed to the point 0 0 point 1 1 by 2 is transformed to the point 2 1 by 2 and point 2 0 is transformed to the point 4 0 now we shall plot these points on the coordinate plane and join these points in the similar shape as of this blue curve here we have plotted these points and now we join these points by a free hand curve so this green curve indicates horizontal stretch by factor 2 and equation of this green curve will be 1 by 2 into f of 1 by 2 into x now we have the value of b as minus 1 by 2 which is less than 0 so there will be reflection of green curve in y axis that is now we will reflect this green curve in y axis we know that the point with coordinates x y on reflection in y axis is transformed to the point with coordinates minus x y as we know that the points on this green curve are given by the coordinates minus 4 0 minus 2 minus 1 by 2 0 0 2 1 by 2 and 4 0 so on reflection in y axis the point with coordinates minus 4 0 is transformed to the point with coordinates 4 0 similarly the point with coordinates minus 2 minus 1 by 2 is transformed to the point with coordinates 2 minus 1 by 2 similarly the point 0 0 is transformed to the point 0 0 the point 2 1 by 2 is transformed to the point minus 2 1 by 2 and the point 4 0 is transformed to the point minus 4 0 So, again we plot these points on the coordinate plane, so here we have plotted these points on the coordinate plane and now we join these points by a free hand curve. Now here this dotted line purple curve shows the reflection of green curve in y axis. Now equation of this curve becomes y is equal to 1 by 2 into f of minus of 1 by 2 x. Lastly we translate this dotted curve 1 unit right and 1 unit up. So now all the points are to be moved 1 unit right and 1 unit up. So when we move this point with coordinates minus 4 0, 1 unit right and 1 unit up we reach this point with coordinates minus 3 1. Similarly when we move this point with coordinates minus 2 1 by 2, 1 unit right and 1 unit up we reach this point with coordinates minus 1 3 by 2. Similarly when we move this point with coordinates 0 0, 1 unit right and 1 unit up we reach this point with coordinates 1 1. Similarly when we move this point with coordinates 2 minus 1 by 2, 1 unit right and 1 unit up we reach this point with coordinates 3 1 by 2. Similarly when we move this point with coordinates 4 0, 1 unit right and 1 unit up we reach this point with coordinates 5 1. Now we join these points to obtain a free hand curve and this is the final curve after all the transformations and the equation of this curve is y is equal to 1 by 2 into f of minus of 1 by 2 into x minus 1 the whole plus 1. So here we have shown the transformation of the given function y is equal to f of x to the function y is equal to 1 by 2 into f of minus of 1 by 2 into x minus 1 the whole plus 1. This is the required answer. This completes our session. Hope you enjoyed this session.