 Hello and welcome to the session. In this session we discuss the concepts of amplitude, period and midline from graphs of periodic functions. Now we know a function is called a periodic function where is the number a such that is equal to f of x plus a in the domain of the function. Now the least positive value of a for which f of x is equal to f of x plus a is called the period of a function. And now we will discuss a real life situation which has a characteristic of a periodic function. Now when you take a file we write you move up and down and again up and down many times with time. You may be at a height of 4 feet above ground in 4 seconds, in 10 seconds and so on. It means you oscillate between the maximum height and the lowest height at different times. Now if a ferris wheel takes in a given time then you will reach maximum height 10 times and minimum height 10 times at a ferris wheel. The height you are above the ground varies periodically as a function of time. Now consider the height of the center of the ferris wheel to be the starting point. Left a particular wheel has the diameter of 38 feet and travels at a rate of 4 revolutions per minute. Now suppose you note the height where you have reached in 10 minutes and plot the graph. Now since height reached depends on time so we will take time t on horizontal axis and height h on vertical axis. So we get the following graph since center of this ferris wheel is considered as starting point so when time t is 0 the height h is also 0 and since diameter of this wheel is 38 feet so 19 feet will lie above the center and 19 feet will lie below the center of this ferris wheel. So maximum height reached from starting point that is 0 is 19 feet. This means 19 feet above the starting point that is 0 and minimum height is 19 feet below the starting point 0. Also this graph is oscillatory and here we can see the maximum and minimum height reached is repeated several times. See what is a period? Now from the graph you can see in the time period 0 to 15 the height reached is not repeated. We have again reached the starting point that is 0 and after that we can see that the height is repeated. So here the period is of 15 seconds in which no repetition of height is there. It means after 15 seconds the cycle is repeated of a periodic function the length of one cycle gives us the period of function. Now we can also find the period from verbal description here we have given that wheel takes the curve repeats. So time taken for one revolution is equal to that is 1 by 4 into 1 minute which is equal to 1 by 4 into now 1 minute is equal to 60 seconds and we know that 4 into 15 is 60 so this is equal to 15 seconds. So time taken for one revolution is equal to 15 seconds. Now let us discuss what is a midline. Now midline is the horizontal line drawn at the central value of the function. Now here this horizontal line is called midline equally. Now in this graph of ferris wheel the middle line is drawn at the central level which is taken when h is equal to 0. Now midline now we can also use formula for finding the point from where midline is drawn. Here midline is equal to maximum point plus minimum point whole point 2. Now in this case of ferris wheel midline is equal to maximum point now here maximum point is 19 feet above 0. So we will take it as plus 19 plus now here minimum point is 19 feet below the starting point. So we will take it as minus 19 and this is 19 minus 19 whole point 2 which is equal to 0 point 2 and this is equal to 0. So here line is drawn is equal to 0. Now let us see what is amplitude is the distance between the maximum minimum height at the midline ferris wheel. You can see the maximum height is when h is equal to 19 and midline is at h is equal to 0. Now the difference is equal to 19. So it means amplitude is equal to 19. Also use the following formula for finding the amplitude is equal to maximum point minus minimum point whole point 2. Now we know that in this case maximum point is plus 19 and minimum point is minus 19. So putting the values amplitude is equal to plus 19 minus of minus 19 whole point 2 which is equal to 19 plus 19 whole point 2 and this is equal to 38 upon 2 which is equal to 19. So amplitude is equal to 19. If you get negative value then always take absolute value of amplitude. Thus periodic functions have occurred similar to waves. Now here the following graph shows the related concepts of periodic function. Here we have the maximum point and the minimum point. Then we have the midline that is the line about which the function oscillates above and below. amplitude which is the distance between the midline and maximum point or minimum point and then period that is length of one complete cycle which repeats itself after the same period of interval. In this session we have learnt about related concepts and this completes our session. Hope you will have enjoyed the session.