 Hello and welcome to the session. In this session we will discuss curve sketching of simple curves. Now let us discuss the graph of a given function y is equal to f of x. Now we will follow the following procedure. First of all let us discuss the symmetry and first is the symmetry about the origin. Now if we replace x by minus x, y by minus y and the equation of the curve remains to be symmetrical about the origin. For example the equation y is equal to x cube. Now here we can see that this equation remains uncertain if x is replaced by minus x and y is replaced by minus y. Therefore this graph of the equation y is equal to x cube is symmetrical about the origin o. Now we can also check this. Now put y is equal to minus y and x is equal to minus x in the given equation. Further this will be minus y is equal to minus x cube which implies y is equal to x cube. So here we can see that by replacing y by minus y and x by minus x the equation remains unchanged. Now second is the symmetry about the x axis. Now here we replace y by minus y and the equation of the curve remains unchanged. Then the graph of the given equation is said to be symmetrical about the x axis. Now check of the equations x is equal to y square and x is equal to minus y square. Now for the equation x is equal to y square I am replacing y by minus y that is here. I am replacing y by minus y the given equation will remain unchanged. So we can say that the graph of the equation is symmetrical about the x axis and also for the second case this is the graph of the equation x is equal to minus y square. And here also y by minus y the equation will be x is equal to minus y square. This means that the given equation remains unchanged when y is replaced by minus y. So we can say that the graph of the given equation is symmetrical about the x axis. So the first graph is symmetrical about the positive x axis and the second graph is symmetrical about the negative x axis. So we can say that the symmetrical about the x axis occurs whenever the even pass of y occurs in the equations. Similarly we can discuss the symmetry about the y axis. The symmetry about the y axis by these two examples. Now if we replace x by minus x in the given equation and the given equation remains unchanged when we say that the graph is symmetrical about the y axis. For example if the equation y is equal to x square if we replace x by minus x then the given equation remains unchanged. So we say the graph of the given equation is symmetrical about the y axis of the equation y is equal to minus x square symmetrical about the negative y axis. Now let us discuss the symmetry about the line y is equal to x. Now if in the given equation x is replaced by y and y is replaced by x that is x and y are interchanged then if the given equation remains then the graph of the given equation is said to be symmetrical about the line y is equal to x. This is the graph of the equation root x plus root y is equal to 1. Now here on replacing x by y and y by x in an equation remains that is here we have replaced the values. All we can say we have interchanged the values of x and y and the given equation remains unchanged. So we can say that the graph of the given equation is symmetrical about the line y is equal to and now let us discuss the symmetry about the line y is equal to minus x. Now if in the given equation we replace minus y and y by minus x and if the given equation remains then the graph of such equations are symmetrical about the line y is equal to minus x. For example this is the graph of a circle the coordinates of whose center are minus 2, 2 and radius is equal to 2. Now the equation of the given circle is x square plus y square plus 4x minus 4y plus 4 is equal to 0. Now in this equation I am replacing by minus y and y by minus x we can see that the given equation will be y square plus x square minus 4y plus 4x plus 4 is equal to 0 which is the equation that is the given equation x square plus y square plus 4x minus 4y plus 4 is equal to 0. This means that the given equation remains unchanged when we have replaced x by minus y and y by minus x. We can say that the graph of the given equation is symmetrical about the line y is equal to minus x. Let us discuss few examples. Now the graph of the equation x square plus y square is equal to 1 is symmetrical about both the axis. It is also symmetrical about the line y is equal to x y is equal to minus x. Now this is the graph of the equation x square plus y square is equal to 1 which is the circle center 0, 0 and radius is equal to 1. Now this graph of the equation x square plus y square is equal to 1 is symmetrical about the x axis as I am replacing y by minus y and unchanged also the graph of the given equation is symmetrical about the y axis as I am replacing x by minus x. The given equation will remain unchanged and it is also symmetrical about the line y is equal to x as on interchanging that is by replacing x by y and y by x. The graph is also symmetrical about the line y is equal to minus x on replacing by minus y but y by minus x the given equation again remains unchanged. And the graph of the equation x square minus y square is equal to 1 is symmetrical not about the line y is equal to x and y is equal to minus x. Now let us see how to check that the curve passes through the original or not. 0, 0 satisfies the given equation then will pass that is the curve of the given equation will pass through the original otherwise not. Now let us take an example now the curves y is equal to x square x is equal to y square the point 0, 0 these equations and as the point 0, 0 does not satisfy the given equation that is the equation x square plus y square is equal to 1 and x y is equal to 2. So the curves of these equations do not pass through the original the intersection with the axis. Now let us discuss it with the help of an example. Now for the equation y is equal to x square minus 1 is equal to 0 in the given equation we get the point of intersection of the curve of the given equation with the y axis. So here we get y is equal to 0 minus 1 which implies y is equal to minus 1 therefore the curve 0 minus 1 when y is equal to 0. Therefore putting y is equal to 0 in 1 we get minus 1 is equal to 0 which implies x square is equal to 1 which gives x is equal to plus minus 1. So the curve cuts the x axis letters 1, 0 and minus 1, 0. Now let us discuss how to find the turning points of the curve. Now for turning points we know that f dash of x is equal to 0 and on solving this we get the turning points. Now we have considered this example here f of x is equal to x square minus 1. So for turning points f dash of x is equal to 0 and on solving this we get the turning points. Now f dash of x will be equal to for turning points f dash of x is equal to 0 which implies 2x is equal to 0 which further gives x is equal to 0. Now for x is equal to 0 from the given equation we get y is equal to minus 1. Therefore the tangent at the point 0 minus 1 is parallel to the x axis. Now let us discuss the intervals in which the given curve is increasing or decreasing. Now for increasing curve f dash of x is greater than 0 and for decreasing curve f dash of x is less than 0. Now for y is equal to x square minus 1 for finding the increasing interval f dash of x is greater than 0 which means now here f of x is equal to x square minus 1. So f dash of x will be equal to 2x that is 2x is greater than 0 which implies x is greater than 0. Add for decreasing f dash of x is less than 0 which implies 2x is less than 0 which gives x is less than 0. Therefore the function increasing for x is greater than equal to 0 and decreasing for x is less than equal to 0 that the function has the minimum point x is equal to 0 because before x is equal to 0 the function is decreasing and after x is equal to 0 the function is increasing. Therefore the minimum value y is equal to x square minus 1 is y is equal to 0 square minus 1 which implies y is equal to minus 1. Therefore 0 minus 1 is the lowest point on the curve. Now let us draw the graph of y is equal to x square minus 1 for this first of all let us draw a table for the different values of x and y. Now for the different values of x as 0, 1, 2, 3, minus 1, minus 2 and minus 3 the corresponding values of y are minus 1, 0, 3, 8, 0, 3 and 8. Now we will plot these points on the graph so we have plotted all the points on the graph. Now on joining all these points we are getting the curve of the equation y is equal to x square minus 1 and here you can see that this curve does not pass through the origin o. Secondly the curve is symmetrical about the y axis as on replacing x by minus x the equation of the curve remains unchanged. That is here on replacing x by minus x the equation of the curve remains unchanged and the curve cuts the y axis at the point 0 minus 1 the x axis at the point 1, 0. The lowest point on the curve is 0 minus 1 and you can also see that before x is equal to 0 the function is decreasing. Now before x is equal to 0 you can see that as x increases y decreases before x is equal to 0 the function is decreasing. x is equal to 0 x increases y also increases therefore x is increasing after x is equal to 0 and the turning point of the curve is 0 minus 1 so the tangent at the point 0 minus 1 is parallel to the x axis. So in this session you have learnt about the curve sketching of simple curves and this completes our session hope you all have enjoyed this session.