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Video transcript

- [Instructor] 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 four times x plus two 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 zero. The height of the rocket when x is equal to zero, they're essentially saying, "Well, what is h of zero?" To figure out h of zero, we just have to go back to this expression and replace all the xs with zeros. H of zero is going to be equal to negative four, negative four times zero plus two, which is just going to be two times zero minus 18, which is just going to be negative 18. Let's see, this is going to be negative eight times negative 18, negative eight times negative 18, which is the same thing as negative eight times negative nine times two. This is going to be positive 72 times two, 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? 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 zero, 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 zero. We can set up an equation. Let's make our height h of x equal to zero, so zero is equal to negative four times x plus two times x minus 18. Well, if you have the product of three different things being equal to zero, the way you get this to be equal to zero is if at least one of these three 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. If x plus two were equaling to zero, then this equation would be satisfied. 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. We can rule that one out. 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." 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 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. If you figure out 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 two, so that is x is equal to negative two. To answer this question, we just have to go halfway between negative two and 18, so let's do that. Negative two plus 18 divided by two gets us what? That's going to be 16 over two, which is going to be equal to eight. This is right over here. This is x equals eight 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. 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 eight minus 18. Eight plus two is 10. Eight minus 18 is negative 10. You have negative four times negative 100, so that's going to be positive 400. H is given in meters, so that's its maximum height, 400 meters.