 Hello and welcome to the session. In this session we will discuss transformation of functions that is reflection, symmetry, even and odd functions. First of all we shall discuss reflection of function given by y is equal to f of x. We know that reflection is a transformation that flips a graph across a line creating a mirror image. Given a graph of function y is equal to f of x then the graph of y is equal to f of minus x is a horizontal reflection across the y axis and the graph of y is equal to minus of f of x is a vertical reflection across the x axis. Let us consider the function y is equal to e raise to power x. Now we replace x by minus of x then this function becomes y is equal to e raise to power minus of x. Now it is of the form y is equal to f of minus x. Let us see the graph of both the functions. This red curve is graph of y is equal to e raise to power x and blue curve is graph of y is equal to e raise to power minus x. Now see the corresponding points of both the curves are equidistant from y axis. So we can say that blue curve is mirror image of red curve. Thus graph of y is equal to e raise to power minus x is the reflection of graph of y is equal to e raise to power x in y axis. Thus when we replace x by minus x in any function we get the reflection of the original curve of the given function in y axis. This is called horizontal reflection. Now let us see vertical reflection that will be in x axis. We have the function y is equal to e raise to power x. Let us consider the function y is equal to minus of f of x then the given function becomes y is equal to minus of e raise to power x. Now let us see graph of both these functions. Here this blue curve is of the graph y is equal to minus of e raise to power x and this pink curve is of the graph y is equal to e raise to power x and we see that the corresponding points on both the curves are equidistant from x axis. This blue curve is the mirror image of pink curve in x axis. Thus we see that y is equal to minus of e raise to power x is reflection of y is equal to e raise to power x in x axis. So when we multiply any function with a negative sign we get the reflection of the given curve in x axis. Now we are going to discuss symmetry of a function y is equal to f of x and even and odd functions. Let us start with symmetry about y axis. The curve y is equal to f of x is symmetric about y axis. If we replace x by minus of x in the given function and we get the same function that is f that is if f of minus x is equal to f of x we say that the curve is symmetric about y axis. Let us take an example let y is equal to x square. Let us find f of minus x for this we replace x by minus x in the given function and we get y is equal to minus of x whole square which is equal to x square. So we have y is equal to x square which is again f of x that is the given function y is equal to x square. So we get f of minus x is equal to f of x thus the given function is symmetric about y axis also the function f of x will be an even function. Let us see the graph of y is equal to x square. This is the curve of the equation y is equal to x square. Here we see that all the points are equally distant from y axis on both the sides and for every point with coordinates x y on the graph there exists another point with coordinates minus x y on the graph. If we take this point then this point has coordinates minus 1 1 then there must exist another point on this graph with coordinates 1 1. Thus we say that the function is symmetric about y axis and it is an even function. Now we are going to discuss symmetry about origin. The curve of function y is equal to f of x is symmetric about origin if for every ordered pair x y on the graph of the function ordered pair minus x minus y also lies on the curve or simply we say that if f of minus x is equal to minus of f of x then the given function has symmetry about origin and the function will be an odd function. Now let us consider the example of y is equal to x cube. Let us check algebraically whether the function is symmetric about origin or not. Let this be equal to f of x so f of minus x will be equal to minus of x whole cube which is equal to minus of x cube which is equal to minus of f of x thus f of minus of x is equal to minus of f of x so the given function is symmetric about origin. Let us see its graph and we can see for every point with coordinates x y lying on curve the ordered pair minus x minus y also lies on the same curve. So it is symmetric about origin if we see this point with coordinates 1 1 on the curve then we see that the point with coordinates minus 1 minus 1 is also on this curve since f of minus x is equal to minus of f of x so the given function is an odd function so we summarize if the function is symmetric about y axis then the function is even function and algebraically f of minus x is equal to f of x if the function is symmetric about origin then the function is odd and algebraically f of minus x is equal to minus of f of x this completes our session hope you enjoyed this session