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CCSS.Math: , ,

we are told a rocket is launched from a platform its height in meters X seconds after the launch is modeled by H of X is equal to negative 4 times X plus 2 times X minus 18 now the first thing they ask us is what is the height of the rocket at the time of launch pause the video and see if you can figure that out well what is X at the time of launch well X is the number of seconds after the launch so at the time of launch X would be equal to 0 so the height of the rocket when X is equal to 0 they're essentially saying well what is H of 0 and to figure out H of 0 we just have to go back to this expression and replace it all the X's with zeros so H of 0 is going to be equal to negative 4 negative 4 times 0 plus 2 which is just going to be 2 times 0 minus 18 which is just going to be negative 18 and so let's see this is going to be negative 8 times negative 18 negative 8 times negative 18 which is the same thing as negative 8 times negative 9 times 2 so this is going to be positive 72 times 2 which is 144 so 144 meters did I do that right let's see we're going to have yep that sounds right that's right how many seconds after launch will the rocket hit the ground so pause this video again and see if you can answer that well what does it mean for the rocket to hit the ground that means that the height is equal to 0 so if you want to figure out how many seconds after launch how many seconds that's X so we want to figure out the X when our height is equal to 0 so we can set up an equation let's make our height H of X equal to 0 so 0 is equal to negative 4 times X plus 2 times X minus 18 well if you have the product of three different things being equal to 0 you the way you get this to be equal to 0 is if at least one of these things is equal at least one of these 3 things is equal to zero well negative four can't be equal to zero so we could say X plus two equals zero I got that from right over here so if X plus two were equaling zero then this equation would be satisfied and so that would be the situation when X is equal to negative two but remember X is the number of seconds after the launch so a negative X would mean be going before the launch so we can rule that one out and then we could also think about well X minus 18 if that's equal to zero then this entire expression could be equal to zero so X minus 18 equals zero you add 18 to both sides you get X is equal to 18 so 18 seconds after launch well we're going forward in time 18 seconds after launch we see that our height is zero we have hit the ground next question how many seconds after being launched will the rocket reach its maximum height pause the video again and see if you can figure that out well the key realization here is if you have if you have a curve if you have a parabola in particular and it's going to look something like this if you're gonna have a parabola that looks something like this you're going to hit your maximum height right over here between your two zeros or between the two times that your height is zero so if you figure out this this x value and this x value the average of the two will give you your x value the time after launch when you're at your maximum height well we already figured out what this x-value is and what this x-value is we know that H of X is equal to zero when X is either equal to 18 so that is X is equal to 18 or X is equal to negative 2 so that is X is equal to negative 2 so to answer this question we just have to go halfway between negative 2 and 18 so let's do that so negative 2 plus 18 divided by 2 gets us what that's going to be 16 over 2 which is going to be equal to 8 so this is right over here this is x equals 8 seconds the rocket is at its maximum height last question what is the maximum height that the rocket will reach once again pause the video and try to answer that well we already know from the previous question that we reach our maximum height when X is equal to eight eight seconds after launch and so to figure out the height then we just have to evaluate what H of eight is H of eight remember that's what this function does you give me any x value any elapsed time after launch and it will give me the height so eight seconds after launch I know I have maximum height to figure out that height I just input it into the function so H of eight is going to be equal to negative four times eight plus two times 8 minus 18 8 plus 2 is 10 eight minus 18 is negative 10 and so you have negative 4 times negative 100 so that's going to be positive 400 and H is given in meters so that's its maximum height 400 meters