 Hello and welcome to the session. In this session, we shall learn to graph proportional relationship and interpret unit rate as the slope of related linear function. First of all, we will learn the concept of slope. Now slope is the rate of change between two points on a line. In the slope of a line, we will find the ratio of change in y that is the vertical change and the change in x that is the horizontal change. So slope of a line will be equal to change in y upon. Now this is the graph of a straight line that is the points 33 and 77 that is the points whose coordinates are 33 and 77. Now initially we have taken y coordinate as 3. Then we move four steps vertically and reach the point 7. y coordinate is 7. The x coordinate is 3 horizontally and reach the point 7. Now the x coordinate is 7. Now we know that slope of a line is equal to change in y upon change in x. Now here in the graph you can see that initially we have x is equal to 3 and we have reached to the point 7 on the x axis. In x will be equal to 7 minus 3. Change in y will be equal to 7 minus 3. In the given graph of the line is equal to change in y that is 7 minus 3 upon change in x which is again 7 minus 3. So this is equal to 4 upon 4 upon 1 which is equal to 1. Therefore slope of the line is equal to 1. Now the constant rate of change is unit rate in a proportional relationship. Now we have already studied that we can conclude a proportional relationship. There is a constant rate of change in x and y on the graph and if y upon x is a constant, constant rate of change, the ratio of y upon x differs then that the two quantities are not proportional. Let's say 0, 0 then unit rate is equal to y minus 0 that is the change in y upon change in x that is x minus 0 which is equal to y upon 0. Note that the steepness of the line is equal to change in y, change in x will be equal to change in y that is y through minus y1 upon change in x that is x through minus x1. Slope of the line is equal to upon 1. Now the following graph represents the distance travelled while driving on a highway and we will use the graph to find the constant rate of change. Now to find the rate of change we pick any two points on the line and let us pick the point 150 and 200 that is the points whose coordinates are 150 and 200 which lie on the given line. Now the rate of change will be equal to in y that is the change of miles upon change that is the change in hours representing the hours and y axis is representing the distance travelled in miles. Now here the change of miles will be equal to the change in y that is equal to 100 minus 50 and the change in hours will be equal to the change in x that is equal to 2 minus 1. So the rate of change will be equal to 100 minus 50 upon 2 minus 1 which is equal to 50 upon 1 which is equal to 50 which increases by 59 in 1 hour. So this is the unit rate of change per hour. So the unit rate of travelling is per hour and you can also see that this line passes through the original so the quantities are proportional. Now here for these three points we will see the ratio of y upon x. Now here the ratio y upon 1 is equal to 50, 100 upon 2 is equal to 50, 150 upon 3 is also equal to 50 therefore we know the ratios 50 upon 1 is equal to 100 upon 2 is equal to 150 upon 3 which is equal to now we know that they are proportional if the ratio of y upon x is a constant. Now here also these quantities are in proportion as here the ratio of y upon x is a constant which is 50 is called the constant of proportionality. So the two quantities x and y are in proportion if y upon x is equal to k where the constant of proportionality. Now here in this case we are getting 50 upon 1 is equal to 100 upon 2 is equal to 150 upon 3 is equal to 50 where 50 is a constant of proportionality. So we can say that this is a proportional relationship and are equal to the constant of proportionality. Now this relation can also be written as y is equal to a linear function. The two quantities are related in such a manner when they are proportional also see it is a linear relationship between two quantities and its graph will always pass through the origin. So any equation of the type y is equal to kx to the origin is 00 that is the point whose coordinates are 00 which is one other pair now at this point v and this point v x to y2. Now we know that slope of the line is equal to that is y2 minus y1 upon change in x that is x to minus 0 upon x minus 0 which is equal to y upon x which is equal to k relationship unit rate is y. Now we can also compare the graphs of two proportional relationships using the slope. Now we know that slope tends as the steepest of the line steeper the line and we know that slope is a unit rate in proportional relationship. So we can compare the two proportional relationships by using the slope for example making shops sell cakes and other baked items. Now this given graph shows the number of cakes sold by both shopkeepers on y axis and number of days on xx. Now to find the slope of bakery A consider these two points which lie on this line. Now for bakery A the slope is equal to change in y upon change in x that is 8 minus 4 upon 4 minus 2 to 4 upon 2 which is equal to 2 which is equal to 2 cakes. Now to find the slope for bakery B these two points with coordinates 00. Now for bakery B the slope is equal to change in y that is 6 minus 0 upon change in x that is 2 minus 0 which is equal to 3 upon 1 which is equal to 3 which is equal to 3 cakes. So bakery B sells both cakes note that bakery B than bakery A. From the graph you can also see that bakery B has a steeper line than bakery A. Bakery B has higher slope than that of bakery A. So in this session you have learnt to graph proportional relationship and interpret unit rate as slope of related linear function. So this completes our session hope you all have enjoyed this session.