 Hello and welcome to the session. In this session we shall explain what a point x, y on a graph of proportional relationship means in terms of situation with special attention to the points 0, 0 and 1, r where r is unitary. We will see what does the point x, y means of the graph. Then we will see what is the relationship between the points x, y and 1, r. Now let us see this graph. In this graph x axis represents the number of days and y axis represents the number of hours a person is left. Now here let us consider the ordered pair 2, 18. Now in this order pair represents the number of days and 18 represents the number of sleeping hours. So we can say that in two days a person slept for 18 hours. So every ordered pair x, y on this graph means for x days person slept for y hours. Here we can see another ordered pair. Let us consider the ordered pair 4, 36 which means in four days a person slept for 36 hours. So the meaning of the ordered pair x, y depends on the situation. Now let us see the ratio of each ordered pair. So we have the ratios 9 upon 1 which is equal to 9. Then the next ratio will be equal to 18 upon 2 which is again equal to 9. Then the next ratio is 27 upon 3 which is equal to 9. And the next ratio is 36 upon 4 which is equal to therefore 9 upon 1 is equal to 18 upon 2 is equal to 27 upon 3 is equal to 36 upon 4 is equal to 9. For the proportional relationship here the constant of proportionality it means y upon x is equal to 9 upon 1. So y is equal to 9x will be its equation. Now from the given equation we can answer this question which says did the person sleep in five days. Now let this be equation number one. So putting equation number one we get y is equal to 9 into 5 which is equal to 45 therefore y is equal to 45 hours. So a constant slept for 45 hours is the input and the equation gives us the relationship between the two variables y and x equal to 9x in a function whose graph clearly if x is equal to 0 then y is equal to 9 into 0 which is equal to 0. So the graph the origin that is the point 00. Now let us study the mean of the point 1r where r is the unit rate and its relation with the point xy. Now we know that when a rate it has is called now in this example which we have discussed earlier unit rate means how much a person sleeps per day which is equal to 9 upon 1 that is 9 hours per day. The ordered pair 1 line can be written as the ordered pair 1r. Now here the points are proportional and there is an increase of 9 hours with an increase of 1 day on comparing with unit rate we mean every one unit wrote in x-axis 9 units in y-axis. The pair xy on the graph of proportional relationship gives the same ratio which is equal to it means y upon x is equal to r upon 1 which implies y is equal to this means y coordinate is r times the x coordinate where r now consider the point xy on the graph as the point 218 means in relation to the point 1r that is the point 19 it means when we moved 1 unit further on the x-axis that is we have reached the point 2 on the x-axis we have moved 9 units vertically and reached the point 18 on the y-axis which is 9 times the x coordinate is 9 into 2 which is equal to 18. Graphically we can find the point xy now here we moved 4 units from 1 horizontally and we reached the point 5 on the x-axis that is y is 9 times that is 45 for the point xy to 9 into 5 that is equal to 45 the point xy is equal to the point it coordinates 5 that is x is equal to 5 y is equal to 45. Now here also weight of change between xy that is 535 which means the ordered pair 545 the point 1r that is the ordered pair 1 9 as r is equal to 9 is in y 9 minus 9 upon change in x that is 5 minus 1 which is equal to 36 upon 4 now 4 into 1 is 4 and 4 into 9 is 36 so here this is equal to 9 upon 1 of 4 days that is from 1 we have moved which is of 5 minus 1 that is 4 units of sleeping were increased by 36 we have reached to 45 minus 9 which is equal to 36 is the point with coordinates 00 the weight of change between the point xy that is the point with coordinates 545 plus 0 upon 5 minus 0 which is equal to 45 upon 5 into 9 is 45 also equal to 9 upon 1 that is y minus 0 upon x minus 0 is equal to thus we see in a proportional relationship ratio of change in y upon change in x is same the point 1r 00 weight of change which is equal to unit weight group chose an increase of 9 hours when a day is increased by 1 where we can see as the days are increased by 1 then the corresponding are increased by 9 now keep in mind the following points that is from a graph of a proportional relationship can be concluded weight of change between any two values and note that that is the point with coordinates 00 the relationship is y upon x we have learnt that a proportional relationship always passes through the origin so it means the weight of change between any ordered pair xy and 1r in a proportional relationship the weight of change between the ordered pair 00 is also same in short the ratio never changes y is equal to hope you all have enjoyed the session.