 Hello and welcome to the session. In this session, we will describe quantitatively the functional relationship between two quantities by analyzing a graph. Now suppose Jean earns $20 in a month. Now if she has to shop in a mall, she can buy products in the mall whose total cost is $20 or less. So she will not buy anything which costs more than $20. This means she can buy less products and now suppose her income increases to $200 when she can buy more products. It means if her initial income was $20, her expenditure on products was less. But when her income was increased to $200, then her expenditure on products has also increased. There is a relationship between income and expenditure when income was less, expenditure was less and when income increases, expenditure also increases. So here expenditure depends on income. In this series graph, here we take expenditure on vertical axis and income on horizontal axis. This means dependent variable is taken along the vertical axis and independent variable is taken along the horizontal axis. Now here you can see the first point which shows when income is $100, when expenditure is $50 and when income is increased to $300, the expenditure also increases to $200. Similarly for the other points also you can see when the income is increased, the expenditure had also increased. So expenditure depends upon income and as expenditure depends on income, so expenditure is dependent variable and income is independent variable. So the value of variable that determines output is called independent variable. Now let us analyze some graphs. Now graphs are analyzed using their shapes. A relation can be determined without using scale on axis. For example, the relationship between the expenditure and income can be directly shown using graph without theta and this graph shows when income increases, expenditure also increases. Now the graph showing relationship between two quantities can be increasing, decreasing, constant or fluctuating. Now let us see how to analyze graph showing relationship between two quantities. Now a graph is not always a straight line moving upwards or downwards. There can be many changes or fluctuations in a graph. Now let us see these lines on graphs and see what they depict or represent. Now this first graph shows y increases as x increases. The second graph study of constant x increases and the third graph shows y decreases at concentrate as x increases and fourth graph shows y increases slowly as x increases. The upward movement of graph shows an increase in y and downward movement of graph shows decrease in y and when there is an immediate increase or decrease the graph is steeper and when the quantity increases or decreases slowly the graph is less steep. As here in the first figure there is an upward movement of graph which shows an increase in y. So here y increases as x increases. In the second figure there is a horizontal line which shows that there is no change in y that is where y remains constant. Similarly you can observe the other graphs also. Now you can observe the first and fourth graph where there is an increase in y but in the first graph y increases fastly and in the fourth graph y increases slowly because the first graph is steeper than the fourth graph. Now let us discuss distance time graphs. Now there are some graphs which show relationship between time taken and distance traveled for a particular journey when we represent this relationship between distance and time graphically when that graph is called distance time graph. Now let us see the relationship between time and distance or displacement on a graph. Now on vertical axis we take distance and on horizontal axis we take time. Now we know that speed is equal to distance covered upon time taken. Now let us examine this given distance time graph here. Jack goes for a bike drive from his home. Now here this point represents the starting point. So this graph shows his journey from home. Now here on vertical axis the distance is given in kilometers and on horizontal axis time is given in hours. Now let us give explanation of his journey after examining the graph. Now Jack starts from his home and he reaches this point which shows in one hour he traveled 22 kilometers. Now from this point to this point that this point be A and this point be B. Now from A to B he did not travel. Here we see that from point A to point B no distance is moved that is at point A distance was 20 kilometers and at point B also distance is 20 kilometers. So here no distance is moved from point A to B. Now at point B distance covered is 20 kilometers but time is 2 hours. So it means from point A to B he has stopped for one hour then from point B he reaches point C. Now point C shows in 3 hours he covers 30 kilometers from point B to C he has covered 30 minus 20 that is 10 kilometers in 3 minus 1 hour. From point C to D you can see there is another horizontal line which shows that no distance is moved from point C to D that means C to D no distance is covered in history that is point 5 which means 30 minutes. But he returned home without stopping that is from point D he moved to point E. Now at point D time taken was 3.5 hours at point E time taken is 5.5 hours it means from point D to point E time taken is 5.5 minus 3.5 that is 2 hours. Now we can also calculate average speed from the graph. Now in the first hour 20 kilometers and time taken is 1 hour. So speed in first hour is equal to distance covered upon time taken and on calculating this is equal to 20 kilometers but in the third hour he covered 10 kilometers in third hour is equal to 10 upon 1 which is equal to 10 kilometers per hour. Now distance is equal to 30 kilometers 2 hours is equal to 30 upon 2 which is equal to 15 kilometers per hour. Now we see that speed in first hour is 20 kilometers per hour and speed in third hour is 10 kilometers per hour and from the graph we see that from the return journey speed is so graph is less. The straight lines in time distance graph depict speed and in time distance graphs moving away from starting point or going up and downward movement shows returning back or going down. So in this session we have described the relationship between two quantities with the help of a graph and this completes our session. Hope you all have enjoyed this session. Thank you.