 Hello and welcome to the session. In this session we will discuss how to use method of completing a square in a quadratic function to show extreme values and symmetry of the graph. Now we know how to make a perfect square in a quadratic equation, a square plus bx plus c is equal to 0. Now we will use the same method to make a perfect square in a quadratic function given by y is equal to ax square plus bx plus c where a is not equal to 0 and hence we will find the extreme values that is maxima or minima given by the vertex and the axis of symmetry. Now graph of everything function given by y is equal to ax square plus bx plus c is upward or downward facing parabola having maximum or minimum point at the vertex and axis of symmetry. Now here this green line represents axis of symmetry. Now let us just recall. Now suppose we have a graph of quadratic function f of x is equal to minus x square plus 4x minus 1. Now this is a graph. Now see the coefficient of x square is negative thus the graph is a downward facing parabola. Here the highest point or maximum point is located at this point that is point 23 and this point of maxima is called vertex. Also this line is axis of symmetry Now here you can see that the vertex lies on the axis of symmetry thus the x coordinate of the vertex will give us the equation of axis of symmetry. Now here the x coordinate of vertex is 2. So equation of axis of symmetry is x is equal to 2. Now we will find vertex and symmetry using perfect square. Now let us take the above example only. Here we have y is equal to minus x square plus 4x minus 1. Now let us take minus 1 common on the right hand side to make coefficient of x square positive. So we have y is equal to minus 1 into x square minus 4x plus 1 double or we can write it as y is equal to minus of x square minus 4x plus 1 Now let us take this expression that is the quadratic expression and then we will complete the square. Now we have x square minus 4x plus 1. Now to complete x square we will add subtract the square coefficient of x. Now here you can see coefficient of x is minus 4. So it is half is equal to 1 by 2 into minus 4 which is equal to minus 2. So let us find its square is equal to minus 2 whole square which is equal to 4. Now here let us add and subtract 4. So it will be x square minus 4x plus 1 plus 4 minus 4 which is equal to. Now let us write it as x square minus 4x plus 4 and let us combine these two terms by 1 minus 4 is minus 3. Now here we have completed the square. Now x square minus 4x plus 4 can be written as x square minus 2 into x into 2 by minus 3. Now we need a minus b whole square is a square minus 2ab plus b square. Now here a and b is 2. So using this formula it will be x minus 2 whole square minus 3. So here we have used the formula of a minus b whole square and we have combined these three terms. So by this method we have completed the square. So this implies y is equal to minus of now x square minus 4x plus 1 and completing the square is equal to x minus 2 whole square minus 3 the whole. Now let us open the brackets. First of all let us open the outer bracket. So it will be y is equal to minus of x minus 2 whole square plus 3. Thus we have made x for x square. Now we see this is of the form y is equal to a into x minus h whole square plus k which is called vertex form where vertex is given by coordinates h a and axis of symmetry is given by equation x is equal to h also if a is less than 0 then graph will open downward. Thus vertex is maximum point and if a is greater than 0 then graph will open upward thus vertex is minimum point. Now comparing this with vertex form we have vertex having coordinates 2 3. Now we know that vertex is given by coordinates h k. Now we are comparing h is equal to 2 and k is equal to 3. So vertex is having coordinates 2 3 and axis of symmetry is given by the equation x is equal to 2. Also comparing this with vertex form we have a is equal to minus 1 which is less than 0. So graph is downward facing this maximum point. Now I have given any point on the graph of quality function and the vertex. Then we can find the equation of the quality function shown in the graph using vertex form. Now let us see one example. Here we have given any point on the graph of quality function that is the point with coordinates 0 3 and the vertex is given as and at this point is the vertex having coordinates minus 1 2. Now let us find equation of this quality function by using vertex form. So the equation of perfect function will be y is equal to a into x minus h whole square plus k. Now here we know that vertex is given by coordinates h k. So here h is minus 1 and k is 2. So let us put h is equal to minus 1 and k is equal to 2 in this equation. So we have y is equal to a into x minus of minus 1 whole square plus 2 which implies y is equal to a into x plus 1 whole square plus 2. Now we have to find, now here this point that is the point with coordinates 0 3 is lying on the graph of this quality function. Now let this equation be 1. So let us put x is equal to 0 and y is equal to 3 in 1. So we have 3 is equal to a into 0 plus 1 whole square plus 2 which implies 3 is equal to a into, now 0 plus 1 is 1 and 1 square is 1. So we get 3 is equal to a into 1 plus 2 which implies 3 is equal to a plus 2. This further implies a is equal to 3 minus 2 therefore we have a is equal to 1. Now let us put a is equal to 1 in equation 1. So we have y is equal to x plus 1 whole square plus 2. So equation of this quality function is y is equal to x plus 1 whole square plus 2. So in this session we have discussed how to use method of completing square in a quality function to show extreme values and symmetry of the graph and this completes our session. Hope you all have enjoyed the session.