 Hello and welcome to the session. In this session we will learn how to construct a linear function and to determine its initial values and weight of change. Here we will also see how to obtain values from a given function and make a table of values. Now we know a linear equation is of the form y is equal to mx plus b and equation is in the function form if it is solved for y. Let us see an example for this. Now 2x plus 3y is equal to 4 is not a function. Now further we can write this equation as 3y is equal to minus 2x plus 4 which further gives y is equal to minus 2 upon 3 into x plus 4 upon 3. So when this equation is solved for y then in this form it becomes a function. Now it is showing a linear relation so it is a linear function. Now here we can put values of x and solve it for y. So we can say that y is a function of x is denoted by y is equal to minus 2 upon 3 into x plus 4 upon 3. Now suppose we have to find f of 1 that is to find value of y when x is equal to 1 equation number 1 supporting in equation 1 we get x is equal to minus 2 upon 3 into 1 plus 4 upon 3 and this is equal to minus 2 upon 3 plus 4 upon 3 which is equal to minus 2 plus 4 whole upon 3 which is equal to 2 upon 3 is equal to 1. Then we have y is equal to 2 upon 3 x is equal to 2 upon 3 we can find other values like f of 0 f of 0 will be equal to minus 2 upon 3 into 0 plus 4 upon 3 which is equal to 4 upon 3. This means for x is equal to 0 y is equal to for these values a table with linear function we can mean for the different values of x and y. Here input means the value of x and output means the value of y is equal to 1 we are getting y is equal to 2 upon 3 equal to we are getting y is equal to 4 upon 3. Now this table is called table of values is equal to 2 upon 3 then we have the ordered pair as 1 2 upon the second ordered pair is 0 4 to graph a linear function we graph it in a similar way as we have graphed linear equation of graphing or we can use input output table for graphing we will plot these ordered pairs on the graph to obtain the graph of now in a linear function of the form is equal to mx plus v or y is equal to mx plus v coefficient of x that is m represents weight of change that is slope v represents the y indices. This is how to form a linear expression and I have considered the following example. Here the table below shows the cost in charges $45 a day for the car as well as charging 1 time $55 fees for cars medication system while expression for the cost in dollars c as a function of number of days d. Solution in which cost is a function of number of days that is of d which means c is a function of d here c represents cost in dollars and d represents number of days the change in both the variables c and d in d now is equal to m then 3 minus 2 is equal to 1 and here again is equal to 115 minus 70 is minus 115 is again 45 minus 160 is again 45. The relation between c and d is linear and linear function will be of the form y is equal to mx plus v. Now per day charges are $45 for d number of days $45 for medication system that is $25 for medication system. Now for d number of days of per day charges will be equal to $45 of d that is the total cost for d number of days will be equal to 45d charges for navigation system c is equal to 45d plus. So this is the required linear function in which cost is the function of number of days. Now let us discuss initial value and rate of change. Now in linear function rate of change is given by slope that is at what rate y value changes with respect. Now in this example which we have discussed earlier in both the variables where different values of c represent values y values of d represent values is equal to change in y upon change in x is 1 equal to 45 upon 1 is equal to 45 thus slope m is equal to 45. Now initial value y is that value which we obtain when the other quantity x is minimum and mostly it is 0. Sam had 5 cards he decides from where onwards he will purchase every week from his done relationship between number of cards y and number of weeks x is given by y is equal to 2x plus 5. Now initially he had started with 5 cards now minimum number of weeks can be 0. Now we have taken number of weeks as x so then number of weeks are 0 that is when x is equal to 0 y is equal to now putting x is equal to 0 in this equation we get y is equal to 2 into 0 plus 5 which is equal to 5. So this implies y is equal to now you will know that if represents y intercept and coefficient of x that is 2 represents the slope. Now here we have obtained y is equal to 5 it means y intercept gives us the initial value. For most of the problems we take initial value of y at x is equal to 0 but initial value depends on the overall description of the question. Like in example of rental cost of a car we cannot say initial charges are 25 dollars because according to this question we have to hire a car for at least so here we will get initial value when number of days d is equal to 1. Now let us put d is equal to 1 in this equation we have c is equal to into 1 plus 25 which is equal to 45 plus 25 which is equal to thus initial charges are 70 dollars. For this example we touch the initial charges when d is equal to 1 we can say here we get the initial value of y at x is equal to we have learnt how to construct a linear function and to determine its initial values and rate of change. So this completes our session hope you all have enjoyed the session.