 Hello and welcome to the session. In this session we will discuss amplitude, period, midline and graph of general form of trigonometric function involving transformation. Now in our earlier session we had discussed about the period, amplitude and midline. Now in this session we will discuss the general form of trigonometric function that is y is equal to a sin of b into x minus c the whole plus k and y is equal to b into x minus c the whole plus k. Now in our earlier sessions we have also discussed the graph of trigonometric functions we saw the graph of trigonometric functions is periodic. We have also discussed that a of f of x f of x plus a the whole are all transformations of the function whose transformation can be applied to sin and cosine functions. Now in order to write the periodic function we will make use of transformation of sin and cosine functions. Now equation of periodic function involving trigonometric functions can be of the following type. This type is y is equal to a sin of b x and y is equal to a cos of b x. Second one is y is equal to a sin of b into x minus c the whole of y is equal to a cos of b into x minus c the whole. And the next type is y is equal to a sin of b into x minus c the whole plus k and y is equal to a cos of b into x minus c the whole plus k. Now for the functions y is equal to a sin of b x and y is equal to a cos of b x. The amplitude is given by the absolute value of a. Period is given by 2 pi upon absolute value of b 60 degrees upon absolute value of b. The function y is equal to minus 6 all the 3 x. Now here a is minus 6 and b is 3. The amplitude will be equal to absolute value of minus 6 which is equal to 6 and period is equal to 2 pi upon absolute value of 3 which is 3. Now let us discuss what is the phase shift. Now the functions y is of b into x minus c the whole and y is equal to a cos of b into x minus of length horizontal. Now the curve is shifted to the right the curve is shifted to the left if c is less than 0. Now here you can see we have taken the graph of y is equal to a sin of b x that is this curve and here you can see when we shift the graph by length c your c is greater than 0 and we are shifting this graph horizontally to the right we get this green curve and the equation of this curve is y is equal to a sin of b into x minus c the whole and now when we shift the curve that is this pink curve by the length c and here c is less than 0 so here we are shifting this pink curve by length c horizontally greater than this blue curve and here equation of this blue curve will be y is equal to a sin of b into x plus c the whole and here c is less than 0 and here the equation of this pink curve is y is equal to a cos of bx when we are shifting this graph by length then we are getting this green curve and equation of this curve is y is equal to a cos of b into x minus c the whole and when we shift this pink curve by length c horizontally then we get this blue curve and equation of this blue curve is y is equal to of b into h plus c the whole for example the whole b is 1 so here h is equal to absolute value of a which is equal to absolute value of 2 that is equal to 2 of b that is absolute value of 1 and this is equal to 2 pi upon 1 which is equal to 2 pi now we know that phase shift is given by c so we can write x plus pi by 4 of minus pi by 4 minus pi by 4 where c is less than 0 therefore take to the left c is less than 0 now let us discuss what is the vertical shift now graph of trigonometric functions can be translated horizontally through a phase shift and they can be translated vertically through a vertical shift now in a trigonometric function let it be y is equal to sin x if it is added then the graph is shifted upwards or downwards y is equal to sin x plus k is greater than 0 and k is a constant which is added to sin x function so the graph shifts upwards now if we have then the graph shifts downwards now the midline changes when there is a vertical shift new midline is given by y is equal to k for downward shift new midline is given by y is equal to minus k now here we can see the graph of trigonometric function the midline is given by y is equal to 0 that is the x midline is the line about which the function oscillates above and below equally the equation is y is equal to the new midline that is the midline of the screen curve is given by y is equal to k so here you can see that the curve with equation y is equal to sin x and we have obtained a new curve that is the screen curve with equation y is equal to this is the curve without vertical shift now in general we have into x minus c the whole plus k and y is equal to a cos of b into x minus c the whole plus k so here midline is given by upwards for k greater than 0 and curve shifts downwards for k less than 0 now let us see an example here consider a function y is equal to 1 by 2 into theta 3 the whole and phase shift of this function let us compare this function with y is equal to a cos of b into x minus c the whole plus k comparing here we have a is equal to 4 b is equal to 1 by 2 theta 3 and k is equal to minus 6 so a that is absolute value of 4 which is equal to 4 upon absolute value of b that is absolute value of 1 by 2 this is upon 1 by 2 which is equal to 4 part is given by y is now here as k is minus the curve shifts downwards and here midline is given by y is equal to now phase shift is given by the value as c is pi by 3 is equal to pi by 3 c is greater than 0 so the curve is shifted to the right by pi by 3 now we can draw the graph of the function and here we draw number line with angles in radius on horizontal axis and number line now here midline we have drawn a horizontal line now period is equal to 4 pi and the graph is stretched till 4 pi from midline above y is equal to minus 6 and we get y is equal to as the maximum below y is equal to minus 6 and we get the minimum range at y is equal to minus 10 y is equal to minus 2 and at y is equal to minus 10 now here first of all we will draw the graph of y is equal to 4 in cause of 1 by 2 into theta minus 6 without root graph starts on y axis in the wave pattern minus 2 and from that point graph moves downwards according to midline and reaches minimum height then it returns back to the maximum completes one cycle of the function y is equal to 4 of 1 by 2 into theta minus 6 the curve to the right by we get this graph and this is the graph of the function of 1 by 2 into theta minus pi by 3 the whole minus given then the midline would have been x that is y is equal to 0 so in this session we have learnt amplitude, period, midline and graph of general form of trigonometric function in verbal transformation and this completes our session hope you all have enjoyed the session