 Hello and welcome to the session. In this session we will discuss how to sketch graphs showing key features for a given verbal description. Let us consider the following verbal description. A rocket is launched from 180 feet above the ground at time t is equal to 0, the function that models this situation is given by h is equal to minus of 16 t square plus 96 t plus 180 where t is time in seconds and h is the height above the ground in feet. Here we will determine the y-intercept and what does it mean in the context of the question. Also we shall determine the maximum height obtained by the rocket and we will determine the intervals where the given function is increasing and decreasing. What do these intervals interpret? And we will determine the time at which the rocket hits the ground. We will solve it both algebraically and graphically. Let us first solve it algebraically. Here we are given the function that models the height of the rocket at any time t and the function is h is equal to minus of 16 t square plus 96 t plus 180 where h is the height in feet and t is the time in seconds. Firstly we have to find the y-intercept. We know that the y-intercept of a function is given when x-coordinate is 0 that is the y-intercept is given by the coordinates 0, y. So here the y-coordinate is represented by the variable h and x-coordinate is represented by variable t. So y-intercept will be given when t is equal to 0 that is ordered where given by 0 h represents y-intercept. So we put t is equal to 0 in the given function that is h is equal to minus of 16 t square plus 96 t plus 180 and we get h is equal to minus 16 into 0 square plus 96 into 0 plus 180 which implies that h is equal to minus 16 into 0 square is 0 that is minus 16 into 0 plus 96 into 0 is 0 plus 180 which further implies that h is equal to 0 plus 0 plus 180 that is equal to 180. So h is equal to 180. Thus we say that y-intercept is given by the ordered pair 0, 180 it interprets the initial height of the rocket before being launched and it is equal to 180 feet. Now we shall determine the maximum height obtained by the rocket. Now see that we are given a function h is equal to minus of 16 t square plus 96 t plus 180 which is a parabola. We know that in parabola given by the equation y is equal to ax square plus bx plus c the maximum height is at the vertex. Also the vertex lies on the axis of symmetry given by the equation x is equal to minus of b upon 2a since variable t represents x. So the maximum height is reached at t is equal to minus of b upon twice of a here a will be equal to minus of 16 and b is equal to 96. So time t will be equal to minus of 96 upon 2 into minus of 16 and this is equal to 16 into 1 is 16 and 16 into 6 is 96. So this is equal to 6 upon 2 that is equal to 3. So time t is equal to 3 seconds. So we say that maximum height is reached at t is equal to 3. So we put t is equal to 3 in this equation we get h is equal to minus 16 into 3 square plus 96 into 3 plus 180 which is equal to minus 16 into 9 plus 96 into 3 is 288 plus 180 and this is equal to minus of 144 plus 288 plus 180 which is equal to 324. So we get h is equal to 324 thus maximum height reached by rocket is 324 feet. Now first we shall determine the time at which the rocket hits the ground and then we shall find out the intervals where the given function is increasing and decreasing and what do these intervals interpret. Now we know that when rocket will hit the ground the height of rocket will be 0. So we put h is equal to 0 in the given function that is h is equal to minus of 16 t square plus 96 t plus 180 and we get 0 is equal to minus of 16 t square plus 96 t plus 180 all we can write it as minus of 16 t square plus 96 t plus 180 is equal to 0 which is a quadratic equation in t. Now we will solve it for t. Now we multiply this given equation by negative sign and we get 16 t square minus 96 t minus 180 is equal to 0. Now dividing the equation by 4 we get 4 t square minus 24 t minus 45 is equal to 0. Using quadratic formula we get t is equal to minus of b plus minus square root of b square minus 4ac whole upon 2a and here we get t is equal to minus of b that is minus of coefficient of t that is minus of minus 24 plus minus square root of b square minus 4ac. Now b square will be equal to minus 24 whole square minus 4 into a that is the coefficient of t square that is 4 into c that is the constant term that is minus of 45 whole upon 2 into a that is 2 into 4 and this implies that t is equal to 24 plus minus square root of minus 24 whole square that is 576 minus 4 into 4 into minus of 45 is plus of 720 whole upon 8 and this further implies that t is equal to 24 plus minus square root of 1296 whole upon 8 which implies that t is equal to 24 plus minus of 36 whole upon 8 that is t is equal to 24 plus 36 whole upon 8 and t is equal to 24 minus 36 whole upon 8 which implies that t is equal to 60 upon 8 and minus 12 upon 8 which further implies that t is equal to 60 upon 8 that is 7.5 and minus 12 upon 8 that is minus of 1.5 but time cannot be negative so we take t is equal to 7.5 thus we say that the rocket will hit the ground in 7.5 seconds now we have to determine the intervals where the function is increasing and decreasing now we have found that as t is equal to 3 we reach maximum point so t is equal to 3 is the critical point also rocket reaches the ground at t is equal to 7.5 so when t belongs to the open interval 0 to 3 the function is increasing and when t belongs to the open interval 3 to 7.5 the function is decreasing this means that height of rocket is increasing for t less than 3 and decreasing for t greater than 3 now let us see its graph now this is the graph of the function h is equal to minus of 16 t square plus 96 t plus 180 this is a downward facing parabola clearly we can see the y intercept is made at the point with coordinates 0, 180 also we see that maximum height is reached at the vertex and this height is 324 feet also we see that the curve is increasing for t less than 3 and decreasing for t greater than 3 and lastly we see that when h is equal to 0 then t is equal to 7.5 so the rocket hits the ground in 7.5 seconds thus we can identify the key features of the graph from the verbal description thus in this session we have discussed how to sketch graphs showing key features for a given verbal description this completes our session hope you enjoyed this session