 Quadratic functions can be used to model projectile motion. Imagine we're trying to shoot some type of bird out of a giant slingshot to hit a pig in the distance or whatever, hypothetically speaking. The path of a projectile will be a parabola. This comes from the fact that the acceleration due to gravity is a constant. And therefore we can use quadratic equations, quadratic functions, quadratic graphs to model projectile motion in physics. So imagine a projectile is fired from a cliff that is 500 feet above the water. So we see something kind of like the following. We have this cliff and then there's some water below. And this cliff is 500 feet above where the water is below. And this is gonna be this, we're gonna shoot with an inclination of 45 degrees. We're not gonna worry about the trigonometry too much. The angle of 45 degrees here just represents that we're gonna get the maximum horizontal distance. You know, if you've ever played a game like Angry Birds or not, right? You know, the higher the incline, the lower incline can affect the horizontal distance. 45 would be the maximum there. So that's why that's chosen. And we're gonna shoot out of our cannon. We're gonna shoot our projectile out with initial muzzle velocity of 400 feet per second. This is an initial speed of our bullet or cannon ball or whatever we're shooting. Now in physics, the height of our projectile, so as this thing shoots and eventually, it eventually is gonna land in the water somewhere down here. If we could measure like the distance horizontally from the cliff, let's call that distance X, then we can measure how far above the water are we at any given moment. Let's call that distance H and therefore our typical point on our parabola would look like X comma H right here. If H represents the height of the projectile above the water, then just by some standard laws of physics, you get the following equation right here. H equals negative X squared over 5000 plus X plus 500. Over here, you get like the muzzle velocity. This was the initial height. Negative 32 has something to do with the acceleration due to gravity. This isn't a physics class. We don't have to go through all of this. What I want you to get out of this is that H equals negative X squared over 5000 plus X plus 500. This is a model you can use for the height of the projectile in this situation. So we might wanna ask ourselves, what is the maximum height? Like if you're just trying to shoot a pig with your angry bird, there might be like a wall in the way, you have to shoot over the wall. So the maximum height needs to be bigger than the height of the wall. So what would be the maximum height of this projectile? Well, using the formula, we really just wanna find out the vertex, right? We're looking for this point right here. We need to find the vertex here, which when you're looking at the vertex, there's two points always associated to it. There's the X coordinate and the Y coordinate. Do we want the X coordinate, the number that's going into the machine, or do we want the Y coordinate that's coming out? If we're looking for the height here, the height is supposed to be the Y coordinate of this function. So we want the maximum height. We need to figure out what is the Y coordinate of the vertex. Now to find the Y coordinate, we typically start with the X coordinate here, because we've seen in the past that X equals negative B over two A, which based upon the formula we have here, you're gonna get negative one over two times negative one over 5,000, which that's a double negative. It'll turn into 5,000 over two, which is then 2,500 feet. Now what is that 2,500 feet measuring? This is measuring the distance from the cliff to where the maximum height is obtained. So this is how far from the cliff we travel to find the maximum height. The maximum height though, to be computed, we have to then look at H of 2,500, which that turns into negative 2,500 squared over 5,000 plus 2,500 plus 500. Don't have to be a hero. You could use your calculator to help you calculate this thing. It's not too difficult here, but this will just crunch out to be 1,750 feet. And so that's the key takeaway here, that the maximum height is not gonna be, is not this X coordinate 2,500. The maximum height will actually be the Y coordinate associated to the vertex, for which our formula first finds the X coordinate of the vertex, and then we find the Y coordinate. So when you're looking at quadratic problems, you often wanna look at the vertex, as the vertex represents sort of like a maximum, in this case, the maximum height, which is that Y coordinate. Another thing I wanna mention in terms of optimization that is finding the maximum minimum of some things, sometimes you wanna find the X intercepts. So for example, if I asked how far from the base of the cliff will the projectile strike the water, coming back to our picture here, we're trying to figure out what's the farthest horizontal distance you could go, and this is gonna coincide with an X intercept right here. And so to find how far from the base do we go, we're really trying to ask ourselves, what is the X intercepts of this graph? For which the quadratic formula is probably appropriate here, we had Y equals negative X squared over 5,000, plus X plus 500. So the quadratic formula told us that X is gonna equal negative B, which is one plus or minus the square root of one minus four times negative one over 5,000 times 500. And this all sits above two times negative one over 5,000. So we crunched that number and get something out of it. This would be, again, if we simplify this, we can get something like 250, 2,500, excuse me, plus or minus 2,500 times 1.4, in which, again, estimates is all we need here. We don't need an exact value. This would give us negative 458 comma 5,458. Now this is one of the reasons why we should think about domain in such a situation. Coming back to our picture over here, does negative 458 even make sense? In terms of our parabola, X equals zero is kind of like the smallest value you would get. You're gonna have to get that X is greater than zero. And then the maximum values could be whatever this is out of here. And so negative don't really make any sense. Just be like you have a mole that's digging through the ground, digging, digging, digging, digging, digging, digging. Boom! And then you hit the cannon, it jumped into the cannon white when it got shot. Well, clearly that doesn't really make any sense, right? Negative values are gonna be outside the domain of this problem. So we can eliminate negative 458 from consideration. So the maximum value is gonna happen right here. We can shoot the, we can shoot our projectile. It'll go 5,458 feet from the site it was launched. And that's how far from the cliff it'll hit the ground.