 Hello and welcome to the session. In this session we will discuss translation of the functions and their graphs vertically and horizontally. Also known as vertical and horizontal shift. First of all we will discuss vertical shift or vertical translation. Now suppose we have a function y is equal to f of x then as repeatedly the vertical shift of the function is given by y is equal to f of x plus k where k is a constant. Now if k is greater than 0 then the graph of the function is shifted or moved upwards by k units and if k is less than 0 then the graph of the function is shifted or moved downwards by k units. Now let us consider the example of trigonometric function. Now we know that the graph of the trigonometric function y is equal to sin x is as follows. Now we see graph of the trigonometric function y is equal to sin x plus 3. Now this blue curve is the graph of the function y is equal to sin x plus 3. And this is of the form y is equal to f of x plus k where k is equal to 3 that is greater than 0. Now if we carefully examine the two curves we see that graph of the function y is equal to sin x plus 3 is same as the graph of the function y is equal to sin x. It seems that we have just shifted the red curve above by 3 units. Now the distance between each point of the function y is equal to sin x and the function y is equal to sin x plus 3 is of 3 units vertically. Thus here is vertical shift of 3 units above because here k is equal to 3 that is greater than 0. Now let us see the graph of the function y is equal to sin x minus 3. Now here this purple curve is the graph of the function y is equal to sin x minus 3 which is again of the form y is equal to f of x plus k where k is equal to minus 3 that is less than 0. Again this curve is similar to the curve of y is equal to sin x. It seems we have shifted the curve y is equal to sin x 3 units down. So here this red curve is shifted downwards by 3 units and we have obtained the graph of the function y is equal to sin x minus 3 and here as k is less than 0 so the graph is shifted downwards. Thus the vertical shift is represented by y is equal to f of x plus k and if k is greater than 0 then there is a upward shift by k units and if k is less than 0 then there is a downward shift by k units. And now let us discuss horizontal shift or we can say horizontal translation for suppose we have a function y is equal to f of x then as regularly the horizontal shift of the function is given by y is equal to f of x minus k the whole k is the constant. Now if k is greater than 0 then the graph of the function is shifted or moved to right by k units and if k is less than 0 then the graph of the function is shifted or moved to left by k units. Now let us consider the following function y is equal to x square which is of the form y is equal to f of x now here we have drawn graph of this function that is the function y is equal to x square. Now the graph of this function is a parabola with vertex at original. Now let us see graph of the function y is equal to f of x minus 2 the whole now we replace x by x minus 2 in the function y is equal to x square. And we get y is equal to x minus 2 whole square which is of the form y is equal to f of x minus k the whole where k is equal to 2 which is greater than 0. Now let us see its graph now this blue curve is the graph of the function y is equal to x minus 2 whole square. Now when we examine both the curves we see that blue curve is a vertical to the red curve but it is at a distance of 2 units from red curve. Now here we can see each corresponding point on both the curves is at a distance of 2 units. Thus it seems we have shifted this red curve by 2 units to the right and we have obtained this blue curve and this shift is called horizontal shift here k is equal to 2 that is greater than 0. So the shift is towards right now we see graph of the function y is equal to x plus 2 whole square this is of the form y is equal to f of x minus k the whole that is y is equal to f of x minus of minus 2 the whole. k is equal to minus 2 that is less than 0. So its graph will be like this that is this purple curve gives the graph of the function y is equal to x plus 2 whole square. Again it is identical to the graph of the parabola y is equal to x square that is this red curve but it seems that we have shifted the curve y is equal to x square to the left by 2 units and we have obtained this purple curve that is the graph of the function y is equal to x plus 2 whole square. So this is also horizontal shift back to the left because k is equal to minus 2 which is less than 0. So here when k is greater than 0 then we have horizontal shift to the right and when k is less than 0 then we have horizontal shift to the left. So in this session we have discussed about horizontal and vertical locations of functions and this completes our session hope you all have enjoyed the session.