 Hello and welcome to the session. In this session, we will compare the properties of two functions, each represented in a different way, may be a graphically integral of gravity. Now a function is a relation in which each number of input is paired with exactly one output and now let us discuss linear functions. Now suppose we have given any two functions and we have to compare the two functions regarding the slope of two functions or which function has greater rate of change or which function has negative or positive slope. Now we know that a linear function is of the form y is equal to mx plus b where x is the independent variable, y is the dependent variable and coefficient of x that is n represents the slope and b represents the y-intercept that is where n represents rate of change and b represents initial value. Also we know how to find rate of change or slope of any linear function from graph or table. So we can easily make comparisons. Now let us compare two linear functions. For this let us see an example. Here function n is the function whose input x and output y are related by y is equal to 3x plus 1 and function 2 is given graphically. Now we have to compare these two linear functions and also we have to determine which equation represents greater rate of change. Now here we have these two linear functions where function 1 is given algebraically in form of equation y is equal to 3x plus 1 and function 2 is given graphically. Now in function 2 as the graph is a straight line so function 2 is also a linear function. So here both are linear functions. Now let us compare the rate of change of the two functions. Now for function 1 the equation is given to us as y is equal to 3x plus 1. Now this is a linear function and also it is the equation of straight line in slope intersect form that is y is equal to mx plus b and we know that here rate of change or slope is equal to the coefficient of x which is 3. Now in function 2 we have a straight line where we have to find rate of change. Now we know that rate of change is equal to change in y upon change in x and we can say in graph it is rise upon 1. Now for calculating rate of change consider any two corner points which lie on this line. Now here let us take these two points. Now this is the point with coordinates minus 1, 5 and this is the point with coordinates minus 2. Now let us name this point as a and this point as b. Now let us find the rise. Now from point A we are moving two units up and we have reached at this point. As we are moving two units the rise is positive which is equal to 2 and now from this point we are moving one unit to the right and we have reached at the point b. Now as we are moving one unit towards right will be positive and this is equal to 1. So rate of change is equal to rise upon 1 which is equal to 2 upon 1 and this is equal to 2. Now for function 1 rate of change is equal to 3 and for function 2 rate of change is equal to 2. So function 1 has greater weight of change. Now we can also find the rate of change with the help of coordinates of two points on the line in coordinate plane. Now here you can see that the points A and B are lying on this line and coordinates of point A are minus 2 3 and coordinates of point B are minus 1 5. Now weight of change is equal to change in rise upon change in h. Now change in rise will be equal to difference in the y coordinates of these two points which is 5 minus 3 and upon change in x will be difference in the x coordinates of these two points which is minus 1 minus of minus 2 and this is equal to minus 3 which is 2 upon minus 1 plus 2 and further this is equal to 2 upon 1 which is equal to 2. So by this method also you can find the rate of change. Now this question can also be represented in statement form and in table. Here function 1 is written in statement form and in function 1 it is given that cost of pizza is 3 dollars per pizza with additional charges of 1 dollar for the lending and we have to write the rule of the total cost y of ordinary pizza as a function of the number of pizzas x or b. And function 2 is given in table form. Now we have to determine which function represents greater rate of change. Now first of all we will write the rule in form of function for function 1. Now here the total cost is represented by y and number of pizzas are represented by x. Now cost of pizza is 3 dollars per pizza with additional charges of 1 dollar for the lending. Total cost of pizza will be equal to additional charges for the lending. Therefore this pizza that is y is equal to 3 x plus additional charges for the lending that is 1 dollar. So this is the rule for total cost y as a function of number of pizzas x. Now this is a linear function and also this is the equation of the straight line in slope intercept form and here the rate of change or slope is equal to coefficient of x which is 3 and now let us find a rate of change for function 2. Now we know that rate of change is equal to change in y upon change in x. Now here from the table we can find the rate of change. Now from the table we can see that change in y is minus 2 minus 1. So rate of change is equal to minus 2 upon minus 1 which is equal to 2 upon 1 and this is equal to 2. Therefore rate of change is equal to function 1 rate of change is 3 and for function 2 rate of change is 2. Therefore function 1 is the rate of change. So in this fashion we have compared properties of two functions each represented in a different way and this completes our session. Hope you all have enjoyed the session.