 Imagine a local hockey team plays in an arena with a seating capacity of 15,000 spectators. With a ticket price set at $14, average attendances at the recent games have been 9,500 people. Now let's imagine that because they do some market research, the survey indicates that every dollar they lower in price, that'll increase the attendance by 1,000 people. And so this is the idea of demand, right? There's a relationship between price and demand. And so if we lower the price, it increases the demand. And so we want to find a function that models the revenue in terms of ticket price. Now revenue in terms of finance is going to be the product of price and ticket sold. So if we say that X is going to be the number of tickets that we sell at a specific game, right? And then let P of X, our price function. This is going to be, this here is going to be the price, the price to then sell X tickets. Because the idea is as we increase the price, the X will go down. As we lower the price, the number of tickets will go up. So there's this relationship. So we have this so-called price function. Now the inverse relationship of price, X of P, this is what we call the demand function, the demand function here. And then the demand function tells us that, oh, when price is set at this level, this is how many tickets we're going to sell. And so when it comes to revenue, revenue is going to equal the number of tickets you sell times price. And so we can then treat our revenue in the following way. We can either think of price as a function of X or we could think of X as a function of price. And I'm actually going to stick with price as my variable in the situation because of the market research we have. So what we've seen is the following. When our tickets are sold at $14, we sell 9,500 tickets. So that gives us a data point. We get when price is $14, ticket sales is 9,500. So again, this is, we're thinking of price as our direct variable and then our indirect variable is ticket sales. We as the people who run and manage the arena, we control the ticket price. That's what we have direct control over. The number of sales is somewhat indirect. It's dependent upon our price, right? So we're going to think of price as the input variable and X as our output variable. Now we also see that as the price goes down by a dollar, that the sales will increase by 1,000. So this is actually giving us some type of rate, a rate that tells us that we'll increase by 1,000 tickets each time we go down by a dollar. And so this is right here a rate or if you want to, you can think of it as a slope. And what we're describing here is the demand function is a linear relationship. The demand is linear given this constant increase of sales as we lower the price. So our rise is a thousand tickets every time we lower the price by one. So then our demand function, X of P right here, if we think of it in terms of slope as a line, we actually would get that X of P minus 9,500 is equal to negative 1,000. That's our slope times X minus 14. And so then we can distribute the negative 1,000, we get negative 1,000 here. Oh, and that shouldn't be an X, that should be a P. Sorry about that. Just used to writing X as my input variable. It's a P there. So we get negative 1,000 P plus 14,000, like so. And then we're going to add the 9,500 of both sides. And so then we get here that X of P is going to equal negative 1,000 P plus 23,000. 23,500. So I want you to interpret this demand function right here. So the demand function tells us that if we gave our tickets out for free, we'd in charge emission, then we would expect there to be 23,500 people that want to come to these hockey games. Now, as we have a seating chart of only 15,000 spectators, that's way too much right, which admittedly is a bad business model anyways. We want to make money from ticket sales. But, you know, we couldn't even fit that many people here. So in terms of the domain, the domain is going to be a lot smaller. The domain would be what price could get us up to 15,000 and go from there. But this is our demand function. Revenue, remember, revenue is going to be X times P. So if we take our domain function, which is 23,500 minus 1,000 P, and we times that by P and distribute, we see 23,500 minus 1,000 P squared. Oops, I forgot the P before. This is our P right there. So 23,500 P minus, maybe I'm going to fix that minus sign. So it's a little bit easier to see. 23,500 P minus 1,000 P squared. This gives us the revenue so we can pick our price and then we can go from there. So this would help us with the question about finding a model for revenue. So this tells us how we can compute the revenue. Now what we want to do with this is actually can then consider what is the maximum revenue, right? We want to make the most money for our arena here. What is the best price to get the most money for this company here? Well, since revenue, remember, was 23,500 P minus 1,000 P squared, we can then compute the maximum revenue. This is going to be at the vertex, right? What is the vertex here? Well, we can find that by taking H to be negative B over 2A. This would then give us a negative 23,500 over 2 times negative 1,000, which that simplifies to just give us here where did it go? We're going to get, of course, there's a double negative so it cancels out. Since you're dividing by 1,000, you can move some decimal places. You'll end up with 11.75. This is supposed to be our optimal price right here. Now, be aware, does our graph go upward or downward? Does our parabola curve up or does our parabola curve down? Since we have a negative coefficient in front of the P squared, this is the picture we have right here. This is a concave down parabola and therefore it does have a maximum, a maximum revenue. And we claim that the maximum revenue will be obtained at this price of $11.75. If you try to figure out what is the associated K value, right? K would be plug-in 11.75 into this and then that would tell us the revenue, sorry. Yeah, that would tell us the revenue, the maximum revenue, which admittedly, if you're just in charge of the ticket box there, you don't actually need to know the maximum revenue is you need to know how much to sell the tickets for. And so with the demand function we have found, this arena would do itself a favor by selling their tickets for $11.75, which is a little bit cheaper than the $14 it's currently selling it for.