 Hello and welcome to the session. In this session we will discuss how to choose trigonometric functions to model periodic phenomena with specified amplitude, frequency and midline. Now in our earlier sessions we have discussed about the graph amplitude period midline of trigonometric functions y is equal to a sin of b into x minus c the whole plus k and y is equal to a s of b into x minus c the whole plus k and now we will use these trigonometric functions to model periodic phenomena. The first starting let us recall the meaning of a b c and stay in the general form of trigonometric functions given by these forms. Now here a sets the vertical stretch and amplitude and it is equal to modulus of a and here b affects the horizontal stretch and period and is equal to upon modulus of b and c refers to the horizontal shift or phase shift the graph is shifted to the right if c is greater than zero and it is shifted to the left if c is less than zero and here k refers to the vertical shift given by y is equal to k the graph is shifted vertically upwards if k is greater than zero and it is shifted vertically downwards if k is less than zero and now let us discuss how we can use these trigonometric functions to model periodic phenomena. Now suppose we have given a verbal description or a table and we have to write the periodic function it will be of the form y is equal to a sin of b into x minus c the whole plus k or y is equal to of b into x minus c the whole plus k we shall check sin or cosine the graph moves from maximum point to minimum point then we choose a function to the maximum point either from the minimum point nor maximum point but near midline then we can choose either function according to our convenience. Now let us consider the polynomial table and we will find the periodic function from the given table now let us plug these points on the graph now the first point that we will plug on the graph is one taking one and three on y axis this is the coordinates we get this curve now here you can see that the graph is starting from your point that is the point which coordinates one three that is the point which coordinates thirteen three and we know that if the graph moves from the minimum point to maximum point then we choose so we will choose sin function the periodic function will be y is equal to a sin of b into x minus c the whole plus k now here the maximum value is y is equal to five and minimum value is y is equal to one so we can find the midline by using the formula that is minimum value plus maximum value whole upon two so this will be equal to now minimum value is one plus maximum value is five whole upon two equal to three will be at y is equal to on this graph at y is equal to three we will draw midline here we have drawn a midline y is equal to that is the equation of this midline is y is equal to three the amplitude which is equal to maximum value minus minimum value whole upon two so this will be equal to minus one whole upon two which is equal to four upon two and this is equal to two as we can see that cycle x is equal to one thirty and after x is equal to thirty the cycle repeats thus our coverage from x is equal to one till x is equal to thirty now upon b b this implies z is equal to twelve trans is equal to two power upon b which implies b is equal to two power upon twelve now two into six this implies b is equal to power by six b is equal to by b is equal to two is equal to two equation of midline is given by y is equal to k and here equation of midline is y is equal to three k is equal to three all these values in this periodic function y is equal to into x minus c double to the horizontal shift or phase shift horizontal shift that is when c is equal to zero we have y is equal to two that is five by six into x plus three now if we take x is equal to zero here then we have y is equal to two sine zero degree is y is equal to now sine zero degree is zero y will be equal to zero plus three that is three so the x is equal to zero y is equal to three graph of this periodic function starts from the point zero three f will be like this hold on and then this graph x is equal to zero to x is equal to one horizontal shift and we know that c refers to the horizontal shift so here c will be equal to one minus zero that is one so this is the graph that is c is equal to one so for c is equal to one the periodic function will be y is equal to into x minus one level thus the periodic function is y is equal to x into x minus one level plus now we can make use of graphic calculators to graph a function when its equation is known we can graph the periodic function from graphic calculators now the final graph will be made from the function y is equal to two sine of five minus six into x minus one level plus three with the help of graphic calculator also we can predict values of x and y using graphic calculators here for y is equal we have x is equal to two we have y is equal to four point seven three so using graphic calculators we can predict the values of x and y we have learned how to choose to model periodic phenomena with specified amplitude frequency and midline and this contains our session hope you all have enjoyed the session