 Hello and welcome to the session. In this session, first of all let us discuss quadratic function. Now an algebraic function of the form a squared plus bx plus c is called a quadratic function of f, a is not equal to 0 and a, b, c are the constants. Now let f of x is equal to ax squared plus bx plus c. Now let us draw the graph of f of x. Now let us discuss the procedure for the graph of the quadratic function f of x. Now in the first step, make a table of the corresponding values with the function. Then in the second step, graph these points on a pair f of x, except that in the third step, join these points to obtain for the given quadratic function. Now the graph of the quadratic function which we have taken as f of x is a parabola which direction changes that is the vertex of the given parabola which is called some points which help us to make the graph easily. The first point is of the quadratic function f of x. First is some k of x when a is greater than 0 and the second one is downwards when a is less than 0 which is the coefficient of x squared. Secondly, that's the x of x when the roots the equation real and if it touches the x of x when in that case roots are real and equal if both of these conditions doesn't work then in that case the roots are given imaginary. Now let us discuss the graphs when the coefficient of x squared that is p is greater than 0. Now you can see here the graphs when a is greater than 0 that is when a is positive. Then here in the first graph that is the graph of the quadratic function cuts the x-axis in the second the graph and in the third the graph has no points in common with the x-axis. So for the first case since the graph cuts means the roots of the given equation are real and unequal which means the discriminant is d is greater than 0 and the roots are given by the distance of p. So here the minimum value of the function that is the minimum value is equal to minus b squared whole upon 4 a which is at the root x. The value of x is equal to the distance where s which is equal to minus b upon function which is the roots of the equation are real and equal which means the discriminant d which is p squared minus 4 a c is equal to 0 and here equal which are given by p. Now in the which means the discriminant d is less than 0 the root has the lowest point p squared whole upon 4 a that is the lowest point of the given curve above. Now let us discuss the graphs when a is less than 0. Now when a is less than 0 then again we can have 3 less. So we know that when the graph cuts the x-axis in that case the roots are real and since d is greater than 0 the distance of p and the distance of q the maximum value for a c minus b squared whole upon 4 a the vertex. And here the value of x is equal to which is equal to minus b upon 2 a. Now in the second case of the quadratic function that is the x-axis it means the roots are real equal. So d is equal to 0 and the roots are given by the distance of p. So the roots are real and equal and they are given by the distance of p. The graph has no points in common with the x-axis this means the roots are imaginary and this means d is less than 0 that is the discriminant is less than 0. x q equal to 4 a c minus b squared whole upon 4 a plus the sign of quadratic square plus b x plus c be the roots of the quadratic equation a x squared plus b x plus c is equal to 0 then plus c will be identical to a into x minus p the whole into x minus q the whole. Now there will be 3 if the discriminant d which is b squared minus 4 a c. Now when d will be less than 0 then we will get other graphs that is the first graph when a is greater than 0 and we will get the second graph when a is less than 0. Now when a is greater than 0 then in that case f of x that is the quadratic function f of x which is equal to a x squared plus b x plus c is always positive a is greater than c f of x is always negative theoretically. Now when d is less than 0 this means b squared minus 4 a c is less than 0 plus b x plus c can be written as a into x squared plus b over a into x plus c over a the whole. When completing the squares will be equal to a into x plus b upon 2 a whole square minus b squared minus 4 a c that is b squared minus 4 a c whole upon 4 a squared the whole and this complete whole that means b squared minus 4 a c is negative. So here b squared minus 4 a c is negative so negative into negative will be positive any quantity is again positive this complete expression will be positive for real values of f so as this complete expression is positive this means the sign of the expression that is really depends upon the sign of a. So whatever will be a sign of a the same will be the sign of when d is equal to 0. So let us discuss the sign of the quadratic function when d that is the discriminant is equal to 0 which implies b squared minus 4 a c is equal to 0. Now this is the graph of the quadratic function when d is equal to 0 that is the graph that is the quadratic function f of x is always greater than f of x is less than equal to c by taking the expression a x squared plus b x plus c. Now where the roots are equal now we have taken the roots as p at q so where p is equal to q therefore a x squared plus b x plus c can be written as a into x minus p whole square quantity is a positive therefore x minus p whole square is a positive expression. So the sign of f of x will be the third case when d is greater than 0 that is when b square minus 4 a c is greater than 0. Now in this case we know that roots are real or unequal so where let p is less than q. Now here two types of graphs now these are the two graphs greater than 0 and the second one is for a less than 0 and since d is greater than 0 for this case therefore the graph of the function cuts the x axis. This is not lie between p and first case when x lies on the right of q x is greater than q then x will be also greater than p. So for this x minus p both will be positive x lies on the left of p that means when x is less than when x is less than p this means x is also less than q. Therefore x minus that is the product of x minus p and x minus this b x plus c which is f of x is identical to a into x minus p the whole into x minus q the whole. Now the product in x minus q is positive for these two cases this means the sign of f are in between p and q. Now when x is in between p and q then in this case will be that is the product of x will be negative opposite signs will be in between p and q. So here for the first case we have got 0 then in this case a is less than 0 greater than 0 it will be less than 0 p. We have learnt about graph of the quadratic function hope you all have enjoyed the session.