 Hello and welcome to the session. In this session we will discuss transformation with stretches or compresses vertically and horizontally the graph of the function. First we discuss vertical stretch and compression. Suppose we have a function y is equal to f of x. Now if we multiply this function by a constant k such that k is greater than 0 we get the function y is equal to k times f of x. The effect of k is to vertically stretch or compress the function by a factor of k. If k is greater than 1 then it moves points of y is equal to f of x further away from x axis. This is called as vertical stretch and if 0 is less than k is less than 1 then it moves points of y is equal to f of x closer to x axis and this is called as vertical compression also on coordinate plane ordered pair x y changes to the ordered pair x k y. Let us consider an example of a trigonometric function that is y is equal to sin of x the graph of sin of x is as shown. Now let us multiply the given sin function by constant 4 we get the function y is equal to 4 into sin of x. Now let us see the graph of the function y is equal to 4 sin of x which is of the form y is equal to k into f of x where k is equal to 4 and is greater than 1. Here this blue curve is the graph of the function y is equal to sin of x. If we carefully examine these two curves that is this red curve which is the graph of the function y is equal to sin of x and this blue curve it seems that we have stretched the red curve vertically. Now see at x is equal to pi by 2 the y coordinate of y is equal to sin of x is 1 and at the same point the y coordinate of y is equal to 4 into sin of x is 4. Thus each value in y is equal to 4 into sin of x is now 4 times of the value as in the original graph that is of y is equal to sin of x or on coordinate plane we have the coordinates x y transforms to the coordinates x 4 y. Thus there is a vertical stretch in the graph of y is equal to sin of x by factor k that is equal to 4. Now let us see the graph of y is equal to k times sin of x when 0 is less than k is less than 1. Let us compare the graph of y is equal to 1 by 2 into sin of x where k is equal to 1 by 2 with the graph of y is equal to sin of x. Now here this green curve is the graph of the function y is equal to 1 by 2 into sin of x. If we carefully examine the two curves that is this green curve with this red curve it seems that we have just brought down the red curve vertically to form green curve. Also see at the point pi by 2 the y coordinate of y is equal to sin of x is 1 and at the same point the y coordinate of y is equal to 1 by 2 into sin of x is 1 by 2. Thus each y value in y is equal to 1 by 2 into sin of x is now half of the value as in the original graph of y is equal to sin of x or on coordinate plane we have the coordinates x y which transforms to the coordinates x 1 by 2 into y. Thus there is vertical compression in the graph of y is equal to sin of x by factor k that is equal to 1 by 2. Thus the graph is compressed vertically by factor k when 0 is less than k is less than 1 and on coordinate plane the point with coordinates x y transforms to the point with coordinates x k y. Now we shall discuss horizontal stretch and compression. Suppose we have a function y is equal to f of x now we replace x by kx in the function where k is a constant such that k is greater than 0. So we get the function y is equal to f of kx the effect of k is to horizontally compress or stretch the graph by factor 1 by k. Now if k is greater than 1 then it moves points of y is equal to f of x closer to the y-axis. This is called as horizontal compression and if 0 is less than k is less than 1 it moves points of y is equal to f of x further away from y-axis. This is called as horizontal stretch also on coordinate plane the ordered pair x y is transformed to the ordered pair 1 by k into x y. Again let us consider the function y is equal to sin of x which is of form y is equal to f of x. Now let us replace x by 2x in the given function and we get y is equal to sin of 2x which is of the form y is equal to f of kx where k is equal to 2 which is greater than 1. Now let us see the graphs of these two functions Now here this red curve is the graph of the function y is equal to sin of x and this blue curve is the graph of the function y is equal to sin of 2x. Now if we compare the two curves it seems that we have just compressed the red curve horizontally to form blue curve. Now let us compare the x coordinates of both these functions at y is equal to 1 the x coordinate in y is equal to sin of x is pi by 2 and in y is equal to sin of 2x is pi by 4 which is half of the value in y is equal to sin of x thus each x value in y is equal to sin of 2x is now 1 by 2 that is half of the value as in the original function y is equal to sin of x also the ordered pair xy transforms to the ordered pair 1 by 2x y thus this is horizontal compression thus there is horizontal compression by factor 1 by k where k is greater than 1 and on coordinate plane the point with coordinates xy transforms to the point with coordinates 1 by kx y. Now let us see horizontal stretch now again we consider the same function that is y is equal to sin of x and we replace x by x by 2 so we get the function y is equal to sin of x by 2 which is of the form y is equal to f of kx where k is equal to 1 by 2 and lies between 0 and 1 let us see the graphs of both these functions now here red curve is graph of the function y is equal to sin of x and green curve is graph of y is equal to sin of x by 2 if we compare the two curves it seems that we have just stretched the red curve horizontally to form green curve let us compare the x coordinate in both the functions at y is equal to 1 the x coordinate in y is equal to sin of x is pi by 2 and in y is equal to sin of x by 2 it is pi which is twice of the value in y is equal to sin of x thus each x value in y is equal to sin of x by 2 is now twice of the value as in the original function y is equal to sin of x algebraically on coordinate plane the point with coordinates x y transforms to the point with coordinates 1 by k into x y here the point with coordinates x y changes to the point with coordinates 1 upon now k is 1 by 2 into x y or this can be written as the point with coordinates x y transforms to the point with coordinates 2 x y thus this is horizontal stretch so we say that there is a horizontal stretch by factor 1 by k where 0 is less than k is less than 1 on coordinate plane the point with coordinates x y transforms to the point with coordinates 1 by k into x y the discussed stretches and compressions hold for all the functions like exponential parabolic logarithmic and trigonometric or any other polynomial function thus in this session we have learnt transformations which stretch or compress vertically and horizontally the graph of the function this completes our session hope you enjoyed this session