 Hello and welcome to the session. In this session we are going to discuss how to graph polynomial functions identifying zeros when suitable factorizations are available and showing n behavior. First of all we are going to discuss about polynomial functions. The polynomial functions are of the form f of x is equal to a n x raised to power n plus a n minus 1 x raised to power n minus 1 plus and so on plus a 1 x plus a naught where n is a non-negative integer and a n a n minus 1 and so on up to a naught are coefficients which are real numbers. Now degree of the polynomial is the highest value of n where a n is not equal to zero. Now let us discuss how to graph a polynomial function. The first step is find zeros of the given function that is we need to find the x intercepts. Now in the second step we have to find the y intercept. Now in the third step we need to determine the nth behavior. For determining nth behavior you have to check degree n and leading coefficient a and here are the possible conditions. The first condition is when n is even and a is greater than zero then when x tends to plus infinity then f of x also tends to plus infinity and when x tends to minus infinity f of x tends to plus infinity. Now the second condition is when n is even and a is less than zero then when x tends to plus infinity f of x tends to minus infinity and when x tends to minus infinity f of x tends to minus infinity when x tends to plus infinity f of x also tends to plus infinity and when x tends to minus infinity f of x x also tends to minus infinity and the fourth condition is when n is odd and a is less than 0 then if x tends to plus infinity f of x tends to minus infinity and when x tends to minus infinity f of x tends to plus infinity and then we sketch the polynomial function by using above steps and by making table of values. Now let us discuss some examples. Now if you have to graph the function given by f of x is equal to x square plus 3x to graph this polynomial function we will follow some steps and in the first step we shall find x intercepts. Now to find x intercepts we equate this function to 0 and solve this function for x. Now taking x common from these two terms we get x into x plus 3 the whole is equal to 0 which implies that x is equal to 0 and minus 3. Now in the second step we shall find y intercept. Now we are given the function f of x is equal to x square plus 3x and we can rewrite this function as y is equal to x square plus 3x and to find y intercept we put the value of x as 0 and this implies that y is equal to 0 square plus 3 into 0 which implies that y is equal to 0 plus 0 that is equal to 0. So y intercept is equal to 0. Now in the third step we shall determine the end behavior of the function f of x is equal to x square plus 3x. Now in this function we see that n is equal to 2 and a is equal to 1. Now here n is even and a is equal to 1 which is greater than 0 and we know that when n is even and a is greater than 0 then as x tends to plus infinity f of x tends to plus infinity and as x tends to minus infinity f of x tends to plus infinity. Now in the next step we shall make table of values. Now here in the first column we have randomly taken x values as 0 minus 1 minus 2 minus 3 1 minus 4 and now we shall find out the values of y for these corresponding values of x. Now here we are given the function f of x is equal to x square plus 3x which can also be written as y is equal to x square plus 3x. So for x is equal to 0 y is equal to 0 square plus 3 into 0 which implies that y is equal to 0 plus 0 that is equal to 0. So for x is equal to 0 y is equal to 0. Similarly for x is equal to minus 1 y is equal to minus 1 whole square plus 3 into minus 1 which implies that y is equal to 1 minus 3 that is equal to minus 2. So for x is equal to minus 1 y is equal to minus 2. Similarly we shall find values of y for the remaining corresponding values of x and we have found for x is equal to minus 2 the value of y is equal to minus 2 for x is equal to minus 3 y is equal to 0 for x is equal to 1 y is equal to 4 and for x is equal to minus 4 y is equal to 4. Now we shall plot these points on the coordinate plane and we get the following graph. Now here we can see that the x intercepts are given by the points 0 and minus 3 also the y intercept is given by the point 0 also we can see that as x tends to plus infinity f of x tends to plus infinity and as x tends to minus infinity f of x tends to plus infinity and this is the required graph of the function f of x is equal to x square plus 3 x. Now let us consider another example. Here we have to graph the function f of x is equal to x cube minus 9 x. Here also we will follow the same procedure. So now in the first step we shall find x intercepts and to find x intercepts we equate this function to 0 and solve it for x. Now taking x common from both these terms we get x into x square minus 9 the whole is equal to 0 which further implies that x into now here applying the formula of a square minus b square that is equal to a plus b the whole into a minus b the whole we can write x square minus 9 as x square minus 3 square that is equal to x plus 3 the whole into x minus 3 the whole. So here we can write it as x into x minus 3 the whole into x plus 3 the whole is equal to 0 which further implies that x is equal to 0, 3 and minus 3. Now in the second step we shall find y intercept. Now here we are given the function f of x is equal to x cube minus 9x which can also be written as y is equal to x cube minus 9x and to find the y intercept we put the value of x as 0 in this equation and we get y is equal to 0 cube minus 9 into 0 which implies that y is equal to 0 minus 0 that is equal to 0. So y intercept is equal to 0. Now in the third step we shall determine the nth behavior of the given function that is f of x is equal to x cube minus 9x. Now here n is equal to 3 which is odd and a is equal to 1 that is greater than 0 and we know that when n is odd and a is greater than 0 then as x tends to plus infinity f of x also tends to plus infinity and as x tends to minus infinity f of x also tends to minus infinity and now in the last step we shall make table of values. Now here in column 1 we have randomly taken the x values as 3 2 0 minus 2 minus 3 and in column 2 we will write the y values for the corresponding x values. For x is equal to 3 y will be equal to 3 cube minus 9 into 3 which implies that y is equal to 27 minus 27 that is equal to 0. So for x is equal to 3 y is equal to 0. Similarly we shall find the remaining values of y for the corresponding values of x and we have found for x is equal to 2 y is equal to minus 10 for x is equal to 0 y is equal to 0 for x is equal to minus 2 the value of y is 10 for x is equal to minus 3 y is equal to 0. Now plotting these points we get this graph of the function f of x is equal to x cube minus 9x here we see that x intercepts are given by the points minus 3 0 and 3 and y intercept is given by the point 0 and we see that as x tends to plus infinity f of x also tends to plus infinity and as x tends to minus infinity f of x also tends to minus infinity. So this is the required graph of the function x cube minus 9x. Thus in this session we have discussed how to graph polynomial functions identifying zeros when suitable factorizations are available and showing end behavior. This completes our session hope you enjoyed this session.