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CCSS.Math: , ,

an object is launched from a platform its height in meters X second after the launch is modeled by H of X is equal to negative five times X minus four squared plus 180 so normally when they talk about seconds or time they usually would use a variable T but we can roll with X being that so let's think about what's going to happen here let me just visualize it so let me draw an H axis for our height and let me draw an x axis so in the x axis so at time X is equal to zero we're on a platform so we're already going to have some height at time X is equal to zero I'm usually I'm used to saying time T equals zero but at time X is equal to zero we're already going to have some height because we're on some platform and then we're going to launch this projectile and it's going to take it's going to go in the shape of a parabola it's going to be a downward-opening parabola and you might say well so how do you know it's going to be a downward-opening parabola it looks something like that I didn't draw it I didn't draw it exactly perfectly but you get hopefully the point and the reason why I knew as a parabola in particular downward-opening parabola is when you look at what's going on here this is written in vertex form but it's a quadratic and in vertex form you have an expression with x squared and then you're multiplying by negative 5 right over here this tells us that it's going to be downward opening if you were to multiply this out if X minus 4 squared it's going to be x squared plus something else for something else so then you have to multiply all those terms by negative 5 your leading term is going to be negative 5 x squared so once again it's going to be a downward-opening parabola that looks something like that so given this visual intuition that we have let's see if we can answer some questions about it the first one I'd like to answer is how high is the platform how high is the platform how high is the platform and I encourage you to pause the video and try to figure that out what is what is that value right over there well as you can see we are at that value at time X equals zero so to figure out how high is the platform we essentially just have to evaluate H of zero so that's going to be negative five times negative four squared plus 180 I just substitute as X with zero see negative four squared is 16 negative five times 16 is negative eighty plus 180 so this is going to be equal to 100 so the platform is 100 meters tall remember everything is given in or the height is given in meters now the next question I have is how many seconds after launch do we hit our maximum height so our maximum height if we're talking about a downward-opening parabola it's going to be our vertex it's going to be our maximum height and so the x value of that would tell us how long after takeoff how long after or launch do we hit that do we hit the maximum height kind of a color you can see what is this x value right over here so once again pause the video and see if you can figure it out all right so we're trying to answer how long after launch is max is the maximum height well it's the it's going to be the x-coordinate of our vertex well how do we figure that out well this quadratic has actually been written already in vertex form which makes it sound like it should be relatively easy to figure out the vertex over here and to appreciate that we just have to appreciate we have to see the structure in the expression one way to think about it let's think about what's going on you have this 180 and then you have this other term right over here anything squared is going to be non-negative so X minus 4 squared is always going to be non-negative but then you always multiply that times a negative 5 so it's going to be not this whole thing is going to be non positive so it will never add to the 180 so your maximum value is when this term right over here is going to be equal to zero and when is this term going to be equal to zero well in order to make this term equal to zero then X minus four needs to be equal to zero and the only way to get X minus four to be equal to zero is if X is equal to four so just by looking at this you say what makes this zero for x equals four will make this zero so this is right over there if I were to write H of four this is going to be this term is going to go to zero and you're going to be left with the 180 so there you go this right over here the maximum height is 180 it happens four seconds after launch now the last question I'll ask you is how long after launch do we get to a height of zero so for what X makes our height zero well to do that we have to solve H of X is equal to zero or we can write H of X as negative five times X minus four squared plus 180 is equal to zero and once again pause the video and see if you can solve this alright look at subtract 180 from both sides you get negative five times X minus four squared is equal to negative 180 we can divide both sides by negative five we get X minus four squared is equal to 36 let me scroll down a little bit and then we can take the B well we could take the plus and minus quick root I guess you could say and so that will give us X minus four could be equal to 6 or X minus four is equal to negative six so in this first situation add 4 to both sides you get X is equal to 10 or you add 4 to both sides you get X is equal to negative 2 now we're dealing with time here so negative 2 has been in the past if it wasn't sitting on the platform and if it was just continuing its trajectory I guess you could say backwards in time but that's not the X that we want to consider taking a consideration we want the positive time value and that's right over here that is when X is equal to 10 10 seconds after takeoff our height is going to be equal to zero and if the ground is at height of zero if it's at sea level I guess then that's when our projectile is going to hit the ground