 Hello folks welcome again to another session on polynomials and in continuation with whatever we were doing We will be doing graphs of linear trial polynomials today in the previous sessions we have seen definition of polynomials different types of polynomials and in the last session we discussed zeros of a polynomial, isn't it? Now in this session we are going to discuss graphs of linear polynomials. How do we draw graphs and You must have you know learned drawing graphs in your previous grades where we will be doing graphs on a two-dimensional plane and it is called Cartesian coordinate system which we will be using and We will in this session try to learn how to plot linear polynomials on a graph Now what is the linear polynomial linear polynomial? I have shown three forms here So one is px equals to ax plus b. This is the way you Describe a linear polynomial y is equal to mx plus c is another way of describing linear polynomials which is in coordinate geometry terms we call it slope intercept form slope intercept intercept form and This is the most standard form this one ax plus by plus c is the standard form of a linear polynomial standard Standard form standard form and it is also the standard form of a linear equation Isn't it so let us now understand how to plot Linear polynomial like px equals to ax plus b now to do that What we do is we first of all take some values of x and the corresponding values of px We find out so we in the previous sessions we have learned how to find out values of polynomials so hence you take some some values of x and some values of you know corresponding of values of y Which is nothing but our px so you find the values of px and hence we will be getting ordered pair Which will be the coordinates of the points lying on this polynomial Representation on the graph. So let us start with x equals zero. So let us say if x equals to zero Then clearly px is equal to b It will be better Let us say if we take an example and then describe and then we can always Generalize so let me say our equation is or our polynomial is px is equal to 2x minus 3 Okay, so hence let us now plot some points Let us say this is x and this is my px or which is nothing but y So let us say x is equal to 0 then what is px px is clearly minus 3 Now, let us say if x equals to 1 then px is clearly minus 1 and if x is Let us say 2 px is 4 minus 3. That is 1. So hence if you see For every change of one unit in x we are getting two unit of change in y or px So here again, the difference is 1 and here again the difference is 2. This is the Peculiar behavior of a linear polynomial. Okay, you can try some negative values as well So let us say x is equal to minus 1 Then if you see what will be the value of px px will be nothing but Minus of 5, right? So now how to do that? So So this is how we plot you have already learned how to plot Graph. So let us say this is my y axis y axis and this is x axis. Okay, this is x and this is y and let us say This is how we are So let us say one unit is or two units equals one unit here So I'm just drawing it roughly and we can definitely go to a tool like geojabra and plot it Accurately as well. So let us first see how this curve would behave or this linear polynomial will behave So I interchangeably use the words like curves and polynomials and all that So what is it polynomial and algebra can be seen as a curve in geometry? Yeah, so they are You know same or similarly explained now So this point is 0 comma 0. So, you know the origin is 0 comma 0 now 0 comma minus 3 So 0 is the x coordinate minus 3 is y coordinate. So if you see this is minus 1 minus 2 Minus 3. So let us say this is negative negative y, isn't it? So hence somewhere here is 0 comma minus 3 And then there is one point like 1 minus 1 comma minus 1. So where are those points like? 1 comma minus 1 where are where is this point? So this is 1 And minus 1 is somewhere here. Let us say Minus 1 would be somewhere Some 1 1 comma minus 1 is this Similarly 2 comma 1 would be somewhere here and Minus 1 comma minus 5 Would be somewhere here. So if you see if you now join them with freehand You will see that all of them fall On a line and hence this is called This is called a linear polynomial, right? So this is a polynomial for px is equal to 2x minus 3 Okay, so let us see this exactly in our geojibra tool. So please remember we have to plot 0 comma minus 3 So these are the points 0 comma minus 3 1 comma minus 1 2 comma 1 and minus 1 comma minus 5. Okay So let us go to the geojibra tool and plot them there and see what kind of graph we get So here is the geojibra tool and now we are going to plot The points which we discussed. So let us point let us plot the points and the points are nothing but Let me so this is how you should be actually able to plot So points are What were the points like when 0 comma minus 3? Isn't it 0 comma minus 3? Okay, so this is the first point Let me press enter and we have got one point. It is here a can you see that a correct? Let us plot the other points. Let us say it was 1 comma minus 1. So 1 comma minus 1 Okay, and here is point B. Let us plot now 2 and 2 comma 1, right? So point is 2 comma 1, right? So 2 comma 1 is c. So you see it is there then let us plot 3 minus 1 comma minus 5 is it the last point was minus 1 minus 1 comma minus 5 Right. So if you see it looks like all of these lines are falling on straight line So let me try and join them. So with the line. So I am joining these point Sorry, it got displaced. So wait a minute. Yeah, so now let me join this They select yeah, so 1 and 2 if you see I selected these two points and if you observe All the points so made all the points are made are now lying on this this particular line But once again, I'm going to draw it this one and this one. So here here it is, right? So if you see this is the line on line Which is passing through all the four points which we just found out. So this is how Linear equation or sorry linear polynomial will behave So if you put if you see another observation is as x is As x is very very small that is goes towards minus infinity y is also very very small And as x is increasing y is increasing and The concept of zero is also there. So if I have to find out the concept of the zero zero is here point e E where the value of px is becoming zero, right? So point e here is the zero of this polynomial and its value is 1.5 if you see it, right? No now, so this is how polynomial Polynomial behaves when shown in a graph now another thing which I wanted to show you is In a general case, let us say if I had a general case, for example, I'm now going to Show you a general case, right? So if you see I had shown you y is equal to ax plus b Now what happens if I change the value of a and b? So let us first change the value of a and see how the curve behaves, isn't it? So our our intention is to see How the curve behaves when I change the value of a so look carefully now I am increasing the value of a so if you see the moment I increase the value of a What happens the curve or the line starts rotating around a point, isn't it around the point now? Let me also Point out this so this point is g As I am changing the value of a the line is rotating around the point g, right? This is what happens. So if you see Actually, this point a is closed the slopes called the slope So slope is nothing but how much is inclined is the line with the x axis So if you see as I'm increasing the slope value a The inclination of the line with the x axis is increasing and as I'm decreasing the value of a The inclination of this line with x axis is decreasing. So that's that's how a line will behave now Secondly, if I now change the value of b what happens? So if you see I'm increasing the value of b So its distance from y axis is actually increasing and I am and I am as I'm reducing the value of B the distance or it is, you know, just the curve is getting shifted upwards and downwards depending upon the value and sign of b Now one special point is when b is equal to zero if you see b I am making b is equal to b is equal to zero. So what happens when b becomes zero? So let me just make Yes, b is zero. So if you see now b is equal to zero in this case The line is passing through origin if you see line is passing through origin actually the b here the parameter b is called the y intercept y intercept So y intercept is nothing but the distance Of the point where the line is cutting the y axis from the origin. So if you see If you see now b is one that means b is one means it is one unit The point where the line is cutting the y axis that is g is one unit away from the x axis, right? So this is how so now I'm going to show you if both a and b are moved together So what happens, right? So see now both a and b are moving together and this is how the line is behaving Right, this is how the line is behaving the line is changing the direction of the rotation as in It is rotating first of all and then it is also going up and down Up and down in along the y axis, right? So this is what what happens when you when you Change the value of let us say a and b I hope you understood what are the parameters a and b what are their importance In fact in general slope intercept form. It is called y is equal to mx plus c if I have to make a note here So y is equal to actually mx mx plus c where m here in this case is a So this is ax plus b in this case we have taken a and b But in terms of coordinate geometry in that language we talk about m and c where m is a and c is b And this m is called slope, right? So as you change the value of slope the inclination of the line from the x axis keeps changing And this particular thing is called y intercept y inter intercept where It is the distance of the point of intersection of y axis with the line from the origin