 Hello and welcome to the session. In this session we will discuss how to graph rational functions identifying zeros and asymptotes when suitable factorizations are available and showing end behavior. First of all let us see what is a rational function. Now a rational function is a function that can be written as f of x upon g of x where g of x is not equal to 0 and where f of x and g of x are polynomials. Now let us discuss steps to graph a rational function of the form y is equal to f of x upon g of x. In the first step find the vertical asymptotes by finding zeros of denominator g of x. Now these zeros will be vertical asymptotes if they are not the zeros of numerator f of x. In the next step we find the horizontal asymptote by using the following facts. Now if degree of f of x is less than degree of g of x then there is a horizontal asymptote at y is equal to 0 and degree of f of x is greater than degree of g of x then there is no horizontal asymptote. If degree of f of x is equal to degree of g of x then y is equal to a upon v where a is the leading coefficient of f of x and b is the leading coefficient of g of x. Now after finding the vertical asymptote then horizontal asymptote in the next step we find x intercepts by finding zeros of f of x these zeros will be the x intercepts if they are not the zeros of g of x. Then in the next step find y intercepts by putting x is equal to 0 in the given function. Now the vertical asymptotes will divide the number line into regions. Graph at least one point in each region. This point will tell us where the graph will be above or below the horizontal asymptote each we can plot several points to determine the general shape of the graph and then in the next step we will sketch the graph and you must note that if there is a common zero of f of x and g of x this represents a hole in the graph then simplify the given fraction and put the x value that is the common zero the obtained point is the point where hole exists in the graph and vertical asymptotes cannot be crossed because they describe where the graph is undefined horizontal asymptotes will be crossed only what happens to the graph as x gets very large or very small. Now let us discuss an example for the clock of the function y is equal to 2 x plus 4 divided by x plus 1 of the form upon g of x which is less than f of c which is less than f of c when the function is maximum at x is equal to c then is the vertical asymptote where the support is we will make use of these facts which we have discussed earlier. Now here we have this function and here f of x is equal to 2 x plus 4 and g of x is equal to x plus 1 now here degree of f of x is 1 and degree of g of x is again 1 in the horizontal asymptote as degree of f of x is equal to degree of g of x upon b where a is leading coefficient of f of x and b is leading coefficient of g of x now here leading coefficient of f of x will be coefficient of x that is 1 so horizontal asymptote will be at y is equal to b upon b that is equal to that is y is equal to 2 now in the next step finding zeros of a point numerator is equal to 0 so we have 4 is equal to 0 which implied is given by the ordered pair minus 2 0 now in the next step and we get y is equal to 2 into 0 plus 4 0 plus y is equal to 4 so y intercept is given by the ordered pair 0 4 that is given by x is equal to minus 1 so we have drawn this vertical asymptote this graph in two regions that is the horizontal asymptote y is equal to 2 flag these two points on the graph that we can clearly determine its behavior now equal to minus 3 in this function we get y is equal to 1 x is equal to minus 4 we get y is equal to 1.3 for x is equal to minus 0.5 we get y is equal to 6 then for x is equal to 1 we get y is equal to 3 and for x is equal to 4 we get y is equal to 2.4 these points on the graph plotted all these points on the graph finding all these points we get this graph left region minus 1 is to minus infinity is to minus infinity right region infinity y tends to infinity so this is the end to the behavior of the function y is a and this version we have learnt identifying zeros and asymptotes and suitable term hope you all have enjoyed this session