Analyzing model in vertex form

An object is launched from a platform. Its height in meters, x seconds after the launch, is modeled by h of x is equal to negative 5 times x minus 4 squared plus 180.

So normally, when they talk about seconds or time, they usually would use the variable t, but we can roll with x being that. 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.

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 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. Then we're going to launch this projectile, and it's going to go in the shape of a parabola.

It's going to be a downward opening parabola. You might say, well, how do you know it's going to be a downward opening parabola? It's going to look something like that. I didn't draw it exactly perfectly, but you get hopefully the point. The reason why I knew it was 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.

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 is going to be x squared plus something else plus something else, and 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. 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? I encourage you to pause the video and try to figure that out. 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 the platform is, we essentially just have to evaluate h of 0.

So that's going to be negative 5 times negative 4 squared plus 180. I just substituted x with 0. See, negative 4 squared is 16, negative 5 times 16 is negative 80 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. The x value of that would tell us how long after takeoff, how long after launch do we hit that maximum height?

I'm trying to set a color so you can see. What is this x value right over here? 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 the maximum height.

Well, it's going to be the x coordinate of our vertex. 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.

To appreciate that, we just have to see the structure in the expression. It's 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 the whole thing is going to be non-positive, and 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. 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? Well, x equals four will make this zero.

So if I were to write h of 4, this term is going to go to 0, and you're going to be left with the 180. So there you go. 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? 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 0.

Once again, pause the video and see if you can solve this. All right, we could subtract 180 from both sides. You get negative 5 times x minus 4 squared is equal to negative 180.

We can divide both sides by negative 5. We get x minus 4 squared is equal to 36. Let me scroll down a little bit, and then we can take the, well, we could take the plus and minus square root, I guess you could say.

So that will give us x minus 4 could be equal to 6 or x minus 4 is equal to negative 6. So in this first situation, add 4 to both sides, you get x is equal to 10. Or you add 4 to both sides here, you get x is equal to negative 2.

Now we're dealing with time here, so negative 2 would have 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. We want the positive time value.

And that's right over here: that is when x is equal to 10. Ten seconds after takeoff, our height is going to be equal to zero. And if the ground is at a height of zero, if it's at sea level, I guess, then that's when our projectile is going to hit the ground.