 Hello and welcome to the session. In this session, we will see the graphs of the function when transformations like translation, compression or stretch, reflections are combined together. In our earlier sessions, we had discussed about translation of graphs vertically and horizontally. Then we discussed vertical and horizontal stretches and compressions and we also discussed about reflections. Now let us combine two or more transformations and form their graphs. 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 a is less than 0 that is there is a negative sign before the given function then the graph is reflected over x axis. Now 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 of the function is reflected over y axis. Now c causes the graph to translate horizontally. Also if c is greater than 0 then curve moves to right and if c is less than 0 then curve moves to left. Similarly d causes the graph to translate vertically if d is greater than 0 then the curve moves upwards and if d is less than 0 then the curve moves downwards. And we should note that reflections and stretching always happen first and the translations are always last. Now let us consider an example. Suppose we have a function given by the equation y is equal to f of x and we want to write the function which is vertically stretched. By a factor of 2 reflected in x axis horizontally translated 2 units to the right. Now c we are given that the function is vertically stretched by a factor of 2. So a will be equal to 2 and we also know that the function is reflected in x axis so we put a negative sign before a thus a will be equal to minus of 2 and there is a horizontal translation of 2 units to right. So c will be equal to 2 as there is no horizontal stretch so b is equal to 1 as there is no vertical translation so d is equal to 0. Now let us put all these values in the equation given by y is equal to a into f of b into x minus c the whole plus d. So we get the function y is equal to minus 2 which is a into f of b into x minus c the whole so we have 1 into x minus 2 the whole that is x minus 2 the whole. So we have f of x minus 2 the whole plus d that is 0. So we get the function as y is equal to minus of 2 into f of x minus 2 the whole plus 0. So y is equal to minus 2 into f of x minus 2 the whole. Now if y is equal to x square and we want to write it using these transformations as given in this example so we will get the function y is equal to minus 2 into f of x minus 2. And we write this function as y is equal to x square which is equal to f of x. So here in this function we replace x by x minus 2 and we get x minus 2 whole square that is we have replaced x by x minus 2 and multiplied it with minus 2. So required transformed equation of the given function is y is equal to minus 2 into x minus 2 whole square. Now let us graph the transformed function. This is the curve of the graph y is equal to x square. Now we have to draw the curve of y is equal to minus 2 into x minus 2 whole square. So first we stretch it vertically by factor 2. Now in vertical stretch by factor a on the graph the coordinates change from xy to xay that is x coordinate is same but y coordinate is multiplied by factor a. Now we see we have this graph of the equation y is equal to x square. Let us see few coordinates on this curve. Now here we have this point with coordinates 00 this point with coordinates minus 11 and this point with coordinates 11. Now we have this point with coordinates minus 24 and similarly this point is with coordinates 24. Now to draw vertical stretch by factor 2 the new coordinates will be x to y that is point with coordinates 00 is transformed to the point with coordinates. 0 and 2 into 0 will be 0 so we have the coordinates 00. Similarly coordinates 11 transforms to the coordinates 1 2 into 1 that is 2. Coordinates minus 11 transforms to the point with coordinates minus 1 2 into 1 that is 2. Similarly point with coordinates 24 is transformed to the point with coordinates 2 2 into 4 that is 8. So we have the coordinates 28 and point with coordinates minus 24 is transformed to the point with coordinates minus 2 2 into 4 that is 8. So we have the coordinates minus 28. Now let us plot these points on the coordinate plane. Now we join these points with same shape as of y is equal to x square. Now we see that this red curve is vertical stretch of given curve by factor 2. Now we reflect this red curve in x axis. On these graph the coordinates of reflection in x axis are given by xy is transformed to x minus y. So the coordinates of stretched graph will be reflected. We should note that we can change the sequence of transformation that is we can reflect first and then stretch. Then also our graph will be same. Now we reflect the coordinates of this stretched graph and here are the coordinates of the stretched graph. The point with coordinates 00 on reflection in x axis is transformed to the point with coordinates 00. Similarly the point with coordinates minus 12 is transformed to the point minus 1 minus 2 on reflection in x axis. Similarly the point with coordinates 12 on reflection in x axis is transformed to the point with coordinates 1 minus 2. Point with coordinates 28 is transformed to the point with coordinates 2 minus 8. And point with coordinates minus 28 is transformed to the point with coordinates minus 2 minus 8. Now we shall plot these points on the coordinate plane. Here we have plotted these points on this plane and now we join these points by a freehand curve and here we get this curve that is we have reflected this red curve in x axis. Lastly we translate this curve horizontally two units to write. So we shift all the points on the reflected image by two units to write. Now when we shift this point with coordinates 00 two units to write we reach this point with coordinates 20. Similarly when we move this point with coordinates 1 minus 2 two units to write we reach this point with coordinates 3 minus 2. When we move this point with coordinates minus 1 minus 2 two units to write we reach this point with coordinates 1 minus 2. Similarly when we move this point with coordinates minus 2 minus 8 two units to write we reach this point with coordinates 0 minus 8 and when we move this point with coordinates 2 minus 8 two units to write we reach this point with coordinates 4 minus 8. Now we join all these points by a freehand curve. So here we get this blue curve which is the translated image of this purple curve. So here we get the final graph of the transformed function given by the equation y is equal to minus 2 into x minus 2 whole square of the function given by the equation y is equal to x square. This completes our session. Hope you enjoyed this session.