 Hey friends welcome again to another session on polynomials and like we did linear polynomials in the last class we are Carrying forward and in this session. We are going to discuss quadratic polynomials and how to draw graphs for them Okay, so let us start with you know your definition of the polynomial as we have been doing so far so polynomials are defined or you know expressed as you know P of x is a n x n Plus a n minus 1 x to the power n minus 1 plus a n minus 2 x to the power n minus 2 and so on and so forth and Finally a 1 x plus a naught and you know that a 1 a 2 a 3 a naught all our real numbers and n is a non-negative integer Okay Now if n is equal to 2 then we know that the polynomial is called a quadratic polynomial, so if you see the degree here is 2 So this is a quadratic polynomial where a b c are real numbers and a cannot be 0 Why because if a becomes 0 then the polynomial in this particular polynomial will be reduced to a Linear polynomial, right? So let us understand with a few examples. So p x is Y equals x square minus 3 x plus 2 This is an example of a quadratic polynomial where degree of px as I have mentioned is 2 and a is equal to 1 b is equal to minus 3 and c equals to 2 mind you guys Many students make this mistake while you know figuring out what is the value of b here b is minus 3 So the sign is always included. Okay, so this is an example of quadratic polynomial now like we drew the graph of linear polynomial in the last session this session also we are going to draw the graph of quadratic polynomial so hence we have taken an example x square minus 3 x plus 2 and As described in this table, we have taken some values of x and corresponding values of y That is corresponding values of px. Isn't it? So 0 comma 2 minus 1 will give you 6 minus 2 x equals 2 minus 2 will give you 12 And so on and so forth. Now, let us try and plot these points So 0 comma 2 is nothing but this point if you see this is my 0 comma 2 minus 1 comma 6. So this is minus 1 and Minus 1 comma 6 would be this point, isn't it minus 1 comma 6 Now minus 2 comma 12 will be, you know, let's ignore that value because in this scale We are not we will not be able to fit in 12 over here and let us know now go further Let us say 1 comma 0 this point. I'm going to plot 1 comma 0 So clearly this point is here. Similarly, 2 comma 0 is here And 3 comma 2 3 comma 2 is somewhere here Isn't it? Now in in no way we see that this is going to be a straight line. So drawing this curve Using freehand, let us say you are drawing like that Okay, so this will be This will be the The shape of the curve, isn't it? So if you see From negative infinity of x or this this side is negative infinity, isn't it? So as you are going from High negative values of x y sorry negative values of x towards positive value of x if you see The value of y continuously decreases, isn't it? It is decreasing decreasing And as x is approaching zero it has decreased to this value Which is 2 here and then further down as you keep on increasing x the value of y keeps on decreasing Then it becomes zero At one point at x equals to 1 and hence we said this point is called zero of px Zero of px, isn't it? And then as you moved away from this zero If you notice the y value hits a minimum, isn't it? It is not going beyond the point and then starts increasing Increasing again hits a zero. So this is another zero of the polynomial of zero of px So we have so far seen two zeros and then Then y values keep on increasing xx as x increases y value keeps on increasing, isn't it? And then at let us say as x tends to positive infinity Here y value also tends to positive infinity, isn't it? So this is what A quadratic curve or quadratic polynomial looks like Now there are a few things here. So the lowest point in this case Here this point is called the vertex of this curve vertex of this curve Okay. Now technically this curve shape is called Parabola, okay So you will learn in conic section and most of you would be knowing This is a parabola. This is the same curve Which is traced by a projectile when it is in flight And let us say a shell when Fired from a cannon or a cricket ring ball When it is hit by a bat or a football when it is kicked by a footballer All these are example of parabola Okay, and hence the curve trace will be nothing but a quadratic polynomial Now you must be wondering that uh, you know, how come this kind of a curve is a You know, whatever examples we took in that case actually it is In all those cases, which we just discussed the curve looks like this, isn't it? So if you see this in exactly Uh, mirror image of such kind of a curve or if you flip it then you will get a Uh trajectory which we just discussed in those examples. So now it's turn to see Uh and evaluate using the geojibra rule and maybe we'll be able to get some more insights there. So let's switch on to our geojibra tool So here is our geojibra and let us now try to Try to get a quadratic Equation so I'm typing y is equal to x squared Can you see the moment I typed x square? You can see we have got The nature of curve which we wanted anyways, but let me make it more general. So I'm writing y is equal to a uh x squared x squared so it is x squared plus b b x and then plus Plus c that is what we discussed as a so this is This is the quadratic curve which we are talking about. So here if you see a x square plus b x plus c all a b and c values are one And hence it looks like this particular curve Now you must be wondering that it doesn't match with whatever example we had taken. So let us match this So let us say in that example where b was three, isn't it? Yeah, so b is Three and c all I believe two. Yep Yeah, so if you see In fact, no b was minus three b was minus three. So let me take it to minus three Yeah, so b was minus two and if you see exactly same curve which we had drawn in our Notes, right? So hence, let me Find out these points. So this is the this is these are the points where the curve is intersecting the x axis And as you know, these points are called this point is called what? zero Zero right zero number one and this is the second zero of the polynomial So it is cutting the x axis into places and now if you see this is uh opening upwards and it looks very symmetrical, isn't it? So other thing is it's very symmetric Very symmetric, isn't it and you can you can see that There can be a line here if you see this line is there This line this line is the axis of symmetry. Isn't it this line is Axis of symmetry and my dear friends this point here is called c This c is called the vertex This c is nothing but This c is nothing but the vertex actually it's not sitting over there. So maybe I'll have to yeah this point here is So let me draw this point Let this point let this point be c. Yeah, this point d now is so let me Just Delete e and c. Yeah, so if you see this point d is the Is the vertex of the parabola? What is this point d d is the D is the Vertex of the parabola and if you see 1.5 x value of d is if you see here x value of d x value of d is 1.5 here And uh y value is negative 0.24. Okay. Now, uh, actually if you see 1.5 is nothing but the arithmetic mean of 1 and 2 which are the zeros of zeros of uh, this this polynomial let us say this value was alpha And this value was beta x equals to alpha and beta. So this point here is nothing but x is equal to alpha plus beta by Two, isn't it? So this point is x So hence x equals to alpha plus beta by two And if you see the minimum possible value exists at that value of x only So this value is the minimum possible value of This polynomial given polynomial, isn't it? Now, this is one observation And let us now try to play with the three coefficient and see what exactly And how exactly it impacts It impacts our Curve Yeah, so now what I'm going to do is I'm going to change the value of a and b and c and see what happens to the curve So as I increase the A value you can see now a value is being increased So as the a value is being increased The curve is winding up winding up winding up and actually it it flips if you see. Yeah, this will give you a better view So this is This is what happens now a is increasing. So as a increases Oh, wow. So hence what happens is the curve just Changes the direction. So whenever a is positive. It is it is you can now see when a is positive It is facing upwards. Isn't it opening upwards as I am reducing a Reducing a it is spreading. Isn't it spreading and at zero it becomes a straight line has to be why Because now coefficient of x squared is zero. So it has become a straight line And as I change it to negative value the curve Now is opening downwards. This is our new New observation, isn't it? So hence, what do we see as you? Decrease the value of a so for a to be less than zero it is Opening downwards as a is increased towards positive value The curve opens upwards. This is one observation, right and and And what now what I'm going to do is I'm going to change the value of v and let's say what happens when v changes So as we the value of b changes The curve just swings Just swings and you know If you notice the vertex the vertex is tracing another curve itself, isn't it? Isn't it? So this is what happens if b value increases and decreases Right now. What happens when c value increases and decreases if you see right now In this particular case the curve Polynomel is not cutting x axis at all. So hence there are No zeros, right Since it the value of the polynomial is always above zero. So if you notice here the value of All the y's are always above zero. So hence, there is no possibility of this being zero. So there exists no zeros But I can have I can have a condition where there is exactly one zero. So if I Change this value of c. So if you notice value changing the value of c Shifts the curve Shifts the curve up and down. Now, you see the curve is now just touching the x axis Just touching the x axis. There's only so it is cutting the x axis exactly at one point So that is the case where we have only one zero for this curve And as I reduce this curve value of c now, you can see there are two distinct different zeros, right exactly two points The curve is cutting the x axis. So hence my friends We have three cases here one When the curve is cutting the x axis in two points And there is one point and there is a case when it it cuts the x axis exactly at one point exactly at one point like this like this and then We have condition where the curve doesn't cut the x axis at all. So this is an observation So hence we can say that the quadratic polynomials can have at max no No zeros one zero. So one zero was this case one zero Right one zero was yeah one zero and it can have No zeros, sorry couple of zeros as well, right? But at no under no circumstances it will have three zeros and also if you notice the curve Has one turn one turn and the lowest point is minima in this case Minima in this case. So whenever a is greater than zero You can say the curve opens upwards and hence it will have a minimum value It it doesn't have a maximum value because it is tending towards infinity And when a is negative my friends, so it does have a Maximum value isn't it? Can you see this is the maximum value which it can attain and the curve cannot go beyond that and And and the in in this case also there is no zero But hence I can increase the value of c and you can see Now the curve has a maximum value as well as there are two zeros So please go through this session slowly and understand and I would recommend that you download this Software geojibra and try to write this equation And try to play with the values of a b and c and see how the graph responds This will give you a very clear idea of the behavior of the quadratic polynomial So we learned what if a changes If a value changes then The curve curve flips if b value changes the vertex traces another Another path right and if c value changes the curve just shifts up and down up and down this is what We learned okay, so hope this session was useful in the next session. We will be taking some other type of curves