 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that given the graph of the function y is equal to f of x where f of x is equal to square root of 4 minus x square and this is the graph of the function f of x is equal to square root of 4 minus x square sketch the graph of the function given by y is equal to 2 into f of x and y is equal to f of 1 by 2 into x. We know that the vertical stretch or compression is given by y is equal to k times f of x. If k is greater than 1 it moves points of y is equal to f of x further away from the x axis. It is called vertical stretch by factor k if 0 is less than k is less than 1 then it moves points of y is equal to f of x closer to the x axis. It is called vertical compression by factor k here on coordinate plane the point with coordinates x y is changed to the point with coordinates x k into y that is x ky similarly the horizontal stretch or compression is given by y is equal to f of k into x that is f of kx. If k is greater than 1 it moves points of y is equal to f of x closer to the y axis. This is called as horizontal compression by factor 1 upon k if 0 is less than k is less than 1 it moves points of y is equal to f of x further away from the y axis. This is called as horizontal stretch by factor 1 upon k on coordinate plane the point with coordinates x y is changed to the point with coordinates 1 by k x y with this k idea let us proceed to the solution we are given a function y is equal to f of x where f of x is equal to square root of 4 minus x square the whole we are given its graph and we have to draw graph of the function y is equal to 2 into f of x and y is equal to f of 1 by 2 x if we see the graph of the function f of x is equal to square root of 4 minus x square the whole we see that its curve is a semicircle now first we sketch graph of the function y is equal to 2 times f of x or this can be written as y is equal to 2 into square root of 4 minus x square the whole now see it is a form y is equal to k into f of x now from the key idea we know that it is a vertical stretch where k is equal to 2 which is greater than 1 now see the points with coordinates minus 2 0 2 0 and 0 2 lie on this curve we know that in vertical stretch the point with coordinates x y is transformed to the point with coordinates x k y so here the point with coordinates minus 2 0 is transformed to the point with coordinates minus 2 k y that is 2 into 0 which is 0 so we have the coordinates minus 2 0 similarly the point with coordinates 2 0 is transformed to the point with coordinates 2 2 into 0 that is 0 so we have the coordinates 2 0 and the point with coordinates 0 2 is transformed to the points with coordinates 0 2 into 2 that is 4 so we have the coordinates 0 4 here we should note that we have taken the value of k as 2 now let us plot these points on the coordinate plane here are the three points with coordinates minus 2 0 0 4 and 2 0 now we join these points in the same shape now here is the required curve of the function f of x is equal to twice of f of x which is equal to 2 into square root of 4 minus x square the whole now we draw graph of the function given by y is equal to f of 1 by 2 x so here we replace x by 1 by 2 into x in this given function and we get y is equal to square root of 4 minus 1 by 2 into x whole square the whole which implies that y is equal to square root of 4 minus x square by 4 the whole now the function f of 1 by 2 x is of the form y is equal to f of kx where k is equal to 1 by 2 and the value of k that is 1 by 2 lies between 0 and 1 from the key idea we know that the function given by the equation y is equal to f of kx gives the horizontal stretch if 0 is less than k is less than 1 it represents horizontal stretch by factor 1 upon k so it is a horizontal stretch by factor 1 upon k that is equal to 1 upon 1 by 2 which is equal to 2 and on coordinate plane the point with coordinates x y is transformed to the point with coordinates 1 by k into x y here we have k is equal to 1 upon 2 so the point with coordinates x y is transformed to the point with coordinates 1 by k into x that is 1 upon 1 by 2 into x which is equal to 2 x y so the point with coordinates x y is transformed to the point with coordinates 2 x y again if we see the graph of the function square root of 4 minus x square the whole we see that the points with coordinates minus 2 0 0 2 and 2 0 lie on this curve so the point with coordinates minus 2 0 is transformed to the point with coordinates 2 x that is 2 into minus 2 which is minus 4 0 so we have the coordinates minus 4 0 again the point with coordinates 2 0 is transformed to the point with coordinates 2 into 2 is 4 0 so we have the coordinates 4 0 and the point with coordinates 0 2 is transformed to the points with coordinates now 0 into 2 with 0 and 2 so we have the coordinates 0 2 now we plot these points on the coordinate plane here are the points with coordinates minus 4 0 0 2 and 4 0 now we join these points here we get this blue curve which represents the graph of the function given by the equation y is equal to f of 1 by 2 x or it is also given by the equation y is equal to square root of 4 minus 1 by 4 into x square the whole it represents the horizontal stretch by factor 1 upon k where k is equal to 1 by 2 so stretch by factor 2 this is the required graph this completes our session hope you enjoyed this session