 hello friends welcome to another session on polynomials and we were discussing graphs of different types of polynomial and today we are going to take up graphs of our cubic polynomial and see in our graphing tool that is geojibra that we use how does a cubic polynomial behave if we change the values of the coefficients now here is the cubic polynomial which is given right so it's ax cubed ax cubed bx square plus cx plus d this is the coefficient the polynomial in degree 3 right hence it is called a cubic polynomial you know this right and a b c d belongs to the set of real numbers that's how we represented and a obviously cannot be 0 why because if a is 0 then the the polynomial loses its identity of being a cubic polynomial now let us go take an example and then see how to plot a graph so let us say px is equal to x cube plus 3x square plus x plus 1 is the given polynomial okay now if I have to plot this polynomial that means if different values of x what is the corresponding response in y right how do I do that so we take random values of x and corresponding value of y's are found so x let us take this first one x is equal to 0 so if you put x equals to 0 you will get if you put 0 everywhere wherever you see x you will get y as 1 right so hence my first point to be plot is 0 comma 1 right we'll see how to plot this later now so when you take x equals to 1 you will get y is equal to 1 cube plus 3 1 square 3 times 1 square now don't get confused by this dot over here this is not a decimal this is to represent multiplication so multiplication is also represented by a decimal as in dot right so don't don't treat it as to be a decimal now so the value will be you can see it is 5 so hence the next point is 1 comma 5 or rather it will be 1 plus 3 4 plus 1 6 it's not be 5 it will be 6 actually so this one is 6 okay so this value is 6 right 1 plus 3 4 plus 1 5 plus 1 6 now if x is equal to minus 1 then what will happen so if you see x equals to minus 1 will give you minus 1 cube plus 3 times minus 1 square plus minus 1 plus 1 and this value will be minus 1 plus 1 goes this these two will cancel each other out now minus 1 cube is minus 1 and plus 3 here so that is 2 so this is the second point so hence third point rather so it is minus 1 comma 2 now if you take x equals to minus 3 then it is minus 2 right so now let us plot these points on the graph so let us say this this is x and y I have shown here x and y now let us plot the points so 0 comma 1 is the first point so 0 comma 1 would be this point so I'm encircling it 0 comma 1 then 1 comma 6 1 comma 6 will be 1 2 3 4 5 6 this is 1 comma 6 correct so this is 1 comma 6 and minus 1 comma 2 would be somewhere here right and minus 3 comma minus 2 will be minus 3 comma minus 2 is somewhere here let me zoom it a bit yeah so clearly these points do not look like they are you know straight line so if you join them free hand so this is how you'll have to join them something like this so this is what our cubic curve look like okay now if you see in this case the curve is cutting the x axis exactly at one point okay one point and it is if you see there are two turns of the curve two turns so curve is turning two times so one is this type first time and then this is the second time so if you see there is a local maxima here local maximum why local maximum because it's not the maximum value throughout the curve but in this region this is the maximum right in this surrounding area this is the maximum and if you also see this this point is local minimum or minimum we say okay so the curve is having two turns so you can see cubic equation will have two turns at max it need not be having two turns all the time but at max it can have two turns it will never have three turns that means what do I mean this will not cubic curve will never be like this okay so there are three turns in it isn't it this is a bi-quadratic curve actually so cubic curve will be either this or this like that or it will simply be like that or like that these are the different types of cubic curve and we'll go to geojibra tool you will see what are the different varieties of cubic curve but you can rest assured that it will never be of this shape never why because this is a three turns curve right and so on and so forth so it will also be not like this never only three turns at max is allowed or possible rather right so hence and it is cutting x-axis at one point but it can it can cut at three points all also for example another another type of cubic curve not three points other two points only so a cubic curve can cut x-axis at max at two points so like that if it is i'm sorry at max at three points right so one two and three this is one variety another variety could be this something like that now let us examine the same cubic curve in the geojibra tool so what I have shown here is a curve and you know you can see here y is equal to one x cube plus one x square plus one x plus four point three seven it says on the left hand side now what we are going to do is we have to see you know we're going to analyze this curve by changing the values of let us say a b c and d so in the first case let me just drag this d value back to one so you can see how does the curve look like if you can see exactly at one point it is cutting the x-axis now let me change the value of a so as you know the value of a is changing can you see now a is growing up and now it is coming down down down and you can see the entire you know the two limbs of the curve flip over the moment it becomes a becomes negative right so it just gets flipped over so this is how if a changes it behaves like this so let me put it back a as one let me take a to b one now let me see what happens with b is change now b is increasing so only one limb of the curve you know grows up and goes down and then secondly so at one time only one one limb one side of the curve only changes right most max the most of the changes happening only on the one side of the limb this is how b behaves or the curve behaves when the b value keeps on changing now let me stop it and then we'll change the value of c and c what happens to the curve okay so this is now c is changing now i'm changing the c value and now if you see it is appearing as if the curve is being stretched up and compressed the two sides of the curves right it's like stretched and compressed it becomes almost like a straight line or at one point right so the two peaks and the valley which is there one peak and one valley is there it is you know made more steep now that is what happens when the c value changes but you can see that at max at any given given moment or the curve doesn't intersect x-axis more than thrice it and the least number of cut is one also so hence number of intersection with x-axis is either one or three it cannot have two cuts and only two cuts that is yeah so either one or three so only odd number of times it is intersecting now i am changing the value to one okay so one now let us see what happens and b changes so like quadratic equation this time also the curve just simply gets translated up and down nothing else so if you change the value of d the curve simply gets translated up and down that's how the curve behaves so hence you can now understand what happens if different values of a and b all the coefficients are changed then how the curve behaves is it so what did we learn in this session we learned that a cubic curve or polynomial can have at max three cuts to the x-axis right and the minimum number of cut is definitely one so hence either it will be one cut or three cuts those points where the cubic curve intersects the x-axis are called the zeros of the polynomial there are two turns at max the curve can have so turns means thus the change in direction of the growth of the curve so it goes up becomes maximum then turns down goes down to minimum and then again turns up and goes towards maximum so this is how that is what i was trying to tell you so if you see yeah so as i change the value of a can you see yeah so this is what i was trying to talk about so as x increases y increases hit some maximum value then with increase in x y goes down and then again at some point after some point as x increases y increases and the vice versa the other way could also be possible this is what i was saying so as x is increasing y comes down becomes minimum then there is a phase when x increases y also increases and then after that as x increases y again starts decreasing so that's how a cubic curve will behave i hope you know you understood most of the properties of a cubic curve how to plot them and how to see or what are the behavior of the curve when different values of the coefficients are there you can also try downloading this app geojibra and write your own equation and try to understand different varieties of cubic curves thank you