Interpreting a quadratic in factored form

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 zero.

So 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 x's with zeros. So h of 0 is going to be equal to 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 4 times 0, which is 0, plus 2 times -18, thus h of 0 equals -18. So, that gives us our height.

Now let's see if I did that right. Yep, that sounds right!

Next, 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 zero.

So if you want to figure out how many seconds after launch, how many seconds is 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 2 times x minus eighteen. 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 2 equals zero. I got that from right over here. So if x plus 2 were equal to 0, then this equation would be satisfied, and that would be the situation when x is equal to negative 2. But remember, x is the number of seconds after the launch, so a negative x would mean going before the launch. So we can rule that one out.

Then we could also think about, well, x minus 18: if that's equal to 0, then this entire expression could be equal to 0. So, x minus 18 equals 0. If 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, 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 0 when x is either equal to 18 or x is equal to -2. So to answer this question, we just have to go halfway between -2 and 18.

So let's do that: -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 8, 8 seconds after launch. And so to figure out the height, then we just have to evaluate what h of 8 is.

h of 8, 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, 8 seconds after launch, I know I have maximum height.

To figure out that height, I just input it into the function. So h of 8 is going to be equal to negative 4 times 8 plus 2 times (8 - 18).

8 plus 2 is 10. 8 minus 18 is -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.