 Hello and welcome to the session. In this session we discussed the following question which says sketch a graph for the function y equal to x square minus 4x. Let's proceed with the solution. The function given to us is y equal to x square minus 4x let this be equation 1. Now we have to sketch a graph for this function. For this we will follow certain steps. First of all let us see the symmetry of this function. Now the function that we have is y equal to x square minus 4x. Now in place of x if we put minus x we get y is equal to minus x whole square minus 4 into minus x. This gives us y equal to x square plus 4x. That is on changing x to minus x in this function our function is also changed. So this is our new function which is y equal to x square plus 4x. This shows that the curve is not symmetrical about y axis. Now let's see if the graph is symmetrical about x axis. Our function is y equal to x square minus 4x. Let us now replace y with minus y. So our new function would be minus y is equal to x square minus 4x. So on changing y to minus y our function is also changed. This shows that the curve is not symmetrical about the x axis. Now we change x to minus x and y to minus y to check if the curve is symmetrical about the origin or not. So on doing this we get minus y is equal to x square plus 4x. So on doing this operation that is changing x to minus x and y to minus y our function is also changed and thus the curve is not symmetrical about the origin. Now we can check if the curve is symmetrical about the line y equal to x. For this we will interchange x and y. So on doing this we get x is equal to y square minus 4y. So the function is changed on doing this. So this curve is not symmetrical about the line y equal to x also. Now we will check if the curve is symmetrical about the line y equal to minus x. For this we change x to minus x and y to minus y. So we get minus y is equal to x square plus 4x that is the equation is changed on doing this. So it is not symmetrical about the line y equal to minus x also. And so we can say that the curve is not symmetrical about any lines. Now let us check if the curve passes through the origin or not. For this we will put x equal to 0 and y equal to 0 in equation 1. So we get 0 is equal to 0 square minus 4 into 0 that is we have 0 is equal to 0. So we can say the point 0 0 satisfy the given equation 1. Therefore we say that the curve passes through the origin. Now in the next step we find the points of intersection of the curve with the axis. First of all putting y equal to 0 in equation 1 that is in this equation we get 0 is equal to x square minus 4x that is x into x minus 4 the whole is equal to 0 which gives us the value of x as 0 and 4. So thus we will get two points of intersection 0 0 and 4 0. Thus we can say that the curve is the x axis which coordinates 0 0 and 4 0. This point that is 0 0 is also the point of intersection of the curve with the y axis. In the next step we find the intervals in which the curve is increasing or decreasing. For this we need to find dy by dx. So differentiating the equation y equal to x square minus 4x with respect to x we get dy by dx is equal to 2x minus 4 that is equal to 2 into x minus 2 the whole. Now for finding out the intervals in which the curve is increasing we take dy by dx greater than 0 that is 2 into x minus 2 the whole is greater than 0 which gives us x greater than 2. Now for finding the intervals in which the curve is decreasing we take dy by dx less than 0. This means that 2 into x minus 2 the whole is less than 0. So from here we have x less than 2. Thus we can say that the function is increasing for greater than equal to 2 and decreasing for x less than equal to 2. In the next step we find the turning points of the curve find the turning points of the curve we take dy by dx equal to 0. We have dy by dx is equal to 2 into x minus 2 the whole. So here we have 2 into x minus 2 the whole is equal to 0 and from here we get x is equal to 2. Now x equal to 2 in equation 1 that is this equation y equal to x square minus 4x we get y is equal to 2 square minus 4 into 2 that is we have y is equal to 4 minus 8 which is equal to minus 4. So when we have x equal to 2 y is equal to minus 4 from this we can say that the tangent at the point 2 minus 4 is parallel to the x axis. Then in the next step we find the minimum point on the curve for this we find d2y by dx2. Now as we have y is equal to x square minus 4x so from here we have dy by dx is equal to 2x minus 4 and from here we have d2y by dx2 is equal to 2 and this is greater than 0. Now from dy by dx equal to 0 we have x equal to 2 so d2y by dx2 x equal to 2 will be equal to 2 only and this is greater than 0. So this means that y is minimum when we have x is equal to 2 and for x equal to 2 y was equal to minus 4 so we can say 2 minus 4 is a minimum point on the curve. Now in the next step we prepare a table of values to find few points on the curve for the function y equal to x square minus 4x we will find different values of y by putting different values for x in this equation like when we put x equal to minus 3 in this equation we get y as 21 for x equal to minus 2 the value of y is 12 for x equal to minus 1 the value of y is 5 by putting x equal to 0 we get the value of y as 0 by putting x as 1 the value of y is minus 3 by putting x as 2 the value of y is minus 4 by putting x as 3 the value of y is minus 3 for x equal to 4 value of y is 0 for x equal to 5 value of y is 5 for x equal to 6 value of y is 12. Now we will plot these points first we have the point minus 3 21 so this is the point which coordinates minus 3 21 now our next point that we have to plot is minus 2 12 so this is the point minus 2 12 next we have a point minus 1 5 this point shows minus 1 5 next is the point 0 0 this is the point that is the origin which is 0 0 then we have a point 1 minus 3 1 minus 3 is this point next point that we are supposed to plot is 2 minus 4 this point represents minus 4 next we have 3 minus 3 so the point this represents 3 minus 3 next point that we have to plot is 4 0 this is the point that represents 4 0 then we have 5 5 this point represents 5 5 then next we have 6 12 so this point shows 6 12 now we will join these points by a free hand so we will get the shape of the curve this is the curve for the function y equal to x square minus 4 x now this curve is not symmetrical about any lines and as you can see it passes through the origin then it cuts the x axis at the points 0 0 and 4 0 this curve is increasing for x greater than equal to 2 and it is decreasing for x less than equal to 2 now if we draw a tangent at the point 2 minus 4 that is at this point we get that tangent parallel to the x axis the point 2 minus 4 is a minimum point on the curve so this completes the sketching of the curve for the given function so hope you understood the solution of this question