 Hello and welcome to the session. In this session we will derive equation of line of the form y is equal to mx plus b from graph and compare it with non-linear functions. Now in our last session we had derived equation y is equal to mx plus b and we saw that it was a linear equation and here m is the slope and b is the y-intercept made by the line on y-axis it means b determines where the line meets y-axis. Now we know a linear function is that function whose graph is a straight line. Now let us derive equation y is equal to mx plus b from graph. Now suppose we are given a graph can we write a linear function that relates y to x. Now let us understand it with this example. Here we are given a graph on which points are plotted if we join the points then we get a straight line thus the points lie on a straight line which shows there is a linear relationship between x and y and now we will find that relationship. Now we know that a straight line has a slope so first of all we will find the value of the slope. Now a straight line has constant rate of change so first note we will take any two points which lie on this straight line. Now let us take the points 0 minus 2 and 1 1. Now slope is equal to change in y upon change in x or you can say rise upon 1 which is equal to now change in y means difference in y coordinates so this is 1 minus of minus 2 whole upon change in x means difference in x coordinates so this is equal to 1 minus 0 which is equal to 1 plus 2 whole upon 1 which is equal to 3 upon 1 that is equal to 3. On the graph you can see the run is 1 and rise is 3 so slope is equal to rise upon 1 which is equal to 3 and also you can see that this line crosses or cuts the y axis at the point 0 minus 2 is equal to minus 2 so the relationship between x and y is given by y is equal to mx plus b. Now here slope m is equal to 3 and y intersect b is equal to minus 2 so putting the values of m and b here this implies y is equal to 3x plus or minus 2 which further gives y is equal to 3x minus 2 so this is the required equation which describes the relationship between y and x. Now here as we have moved to the right of intersect for run and then up for rise it means here both run and rise are positive that is why here the slope is positive. For sign of slope we see the direction in which we move now for rise if we move up the sign will be positive if we move down the sign will be negative and for run if we move right sign will be positive and if we move left then the sign will be negative. Now let us determine the equation y is equal to mx plus b from tables we are given input output table of x and y values and we have to find the linear relationship between x and y we will make use of the above procedure only the point x is equal to 0 and y is equal to b will give us the value of y intercept. Now suppose we have the following table values of x are given to us and the corresponding values of y are also given to us first of all we will see that there is a constant rate of change or not first of all let us see the change in x now here 0 minus of minus 1 is equal to 0 plus 1 which is equal to 1 then next 1 minus 0 is again 1 and here 2 minus 1 is also 1 so here change in x is constant now let us see the change in y here minus 2 minus of minus 5 will be equal to minus 2 plus 5 which is equal to 3 similarly 1 minus of minus 2 is 3 4 minus 1 is also 3 so change in y is again constant so there is a constant rate of change so it is linear let us find the slope now slope is equal to change in y upon change in x now change in y is 3 and change in x is 1 so 3 upon 1 is equal to 3 now we know that the point x is equal to 0 and y is equal to b will give us the value of y intercept the y intercept is given by the point 0 minus 2 so here that is b is equal to minus 2 so the required equation is y is equal to mx now here m that is slope is equal to 3 so y is equal to 3x plus minus 2 which implies y is equal to 3x minus 2 now let us compare linear and nonlinear functions now a linear function shows a constant rate of change so its graph is a straight line whereas a nonlinear function does not have constant rate of change so its graph is not a straight line now consider these two tables now for table 1 let us see the change in x now here you can see that change in x is a constant now let us find change in y now here it is 36 minus 45 which is equal to minus 9 then again 27 minus 36 is minus 9 and 18 minus 27 is again minus 9 so change in y is also constant this is a linear relationship table 2 we will see change in x in this table now here it is 4 minus 2 which is equal to 2 then next it is 6 minus 4 which is again 2 and then it is 8 minus 6 which is also 2 so here change in x is constant now let us find change in y now here it is 16 minus 4 which is equal to 12 then 36 minus 16 which is equal to 20 and then minus 36 which is equal to 28 so here change in y is not constant so it is a nonlinear relationship now graphically we can show a nonlinear relationship now consider a function y is equal to x square now here let us put different values of x for which we will obtain different values of y for x is equal to minus 1 y will be equal to minus 1 whole square which is equal to 1 then for x is equal to 0 y is equal to 0 then for x is equal to 1 y is equal to 1 and for x is equal to 2 y is equal to 2 square which is equal to 4 now let us plot all these points on the graph first of all let us plot the point minus 1 1 on the graph now we will start from 0 then here x coordinate is negative so we will move 1 minute to the left of 0 and we will reach at this point then y is equal to 1 which is positive so we will move 1 unit upwards and we reach at this position now here this point represents the order pair minus 1 1 now let us plot the point 0 0 on the graph now this is the point 0 0 now let us plot the point 1 1 on the graph now this is the point whose coordinates are 1 1 lastly let us plot the point 2 4 on the graph now here this is the point whose coordinates are 2 4 now after joining these points we see the 4 points do not lie on one straight line and if we join freehandedly we see they form a curve and not a straight line so it shows nonlinear relationship between x and y now from the above discussion we can say few have given any function showing relationship between x and y then this function is linear if it can be written in the form y is equal to mx or y is equal to mx plus v it cannot be written in slope intercept form then that function is nonlinear for example we have a function y is equal to 4 minus 3x square now it can be written as y is equal to minus 3x square plus 4 it cannot be written in the form y is equal to mx plus v because here we have the right equation of straight line of the form y is equal to mx plus v from the graph and also we have compared it with nonlinear functions and this completes our session hope you all have enjoyed the session