 Welcome back to our lecture series Math 1050, College Algebra for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In this section, we are in section, well, we're starting section 2.2, which is about linear modeling, which as you mentioned, the word modeling or applications, these are sort of like politically correct terms. So we don't trigger people with words like story problems. Ooh, these are kind of scary things at times, but it is actually a very important part of studying functions. When we first learned about functions at the beginning of this series, I mentioned there was four ways of representing functions. We had numerical representations, often as a table, right? We had geometric representations as a graph, algebraic representations by some formula. And the last one we haven't talked a whole lot about these verbal descriptions. This is what a story problem is all about. We have a relationship between quantities that are described in words, in common language that we use. We'll of course use English in this course, right? And so be able to decipher what is the function relationship given a description is a very important concept. Difficult, don't get me wrong. It's like translating from French to Spanish or something like that. Some people might be able to do it easier than others, but in general, it's still a difficult process. And that translation will be difficult. Your typical story problem is, but I mean, it's very, very much for your life type stuff, right? Your typical story problem will often work in the following way. We will be given a function described to us via some type of story, some written or verbal description. We then need to translate that verbal description into an algebraic representation and then use the tools of algebra to answer the question posed by the problem and then translate it back into some statement about the verbal description. We can't just end with a number and be like, great, the answer's seven. What does that mean? We have to interpret things here. And that is a little bit challenging, but it's very important that we do such a thing. So in this first video about linear modeling, because this is the type of modeling we're gonna start off with, linear functions are the simplest of all type of functions. That way we can focus mostly just on the verbal descriptions and not so much on the algebra itself. One type of use linear functions has is the idea of population modeling. We can model the growth of various quantities, whether those are people or animals or money or other things. We can talk about population models and just means that stuff is growing over time. Again, oftentimes we think of people or other things and linear growth is an acceptable way of modeling. Now, when it comes to actual nature, like you always talk about the spread of infectious diseases or population growth of people in cities or in animals and ecosystems, linear model's not exactly the right model there. We'll see some better models later on. Linear model is appropriate when there are sort of like a restriction, like if you take the student body at SUU, for example, the university actually has control on how many students will we admit each year? And so because of that control, the university could say something like, oh, we're gonna allow an extra 100 students each year come into the school in terms of emissions. And that in that regard, we actually, since we're sort of controlling how many people are coming in, then we actually do get a linear growth. Or another situation, if you're like pumping water into a tank, oh, if we're pumping at a rate of 10 gallons per minute, then that's this constant inflow. And that would be a sort of like a quote unquote population growth for which we would use linear growth. Linear growth happens when there's a constant increase per unit of time. So for example, let's say we have a fictitious town population that's growing linearly. That just means that every unit of time it grows the same amount of relevant of outside factors. In the year 2004, the population of this small town was 6,200. And then in 2009, the population grew to 8,100. Assuming this trend continues, can we predict what the population would be in 2013? So how would we predict this to be? Well, one thing we could do here is we wanna figure out essentially how many people is the town growing per year? That essentially is finding the slope of this thing. If this thing is growing linearly, basically what we're trying to do is we're trying to find a linear function f of x equals mx plus b. The m right here is gonna be the rate of growth. And so we have to make this rise over run. What's the change of population? The change of population with respect to time, right? So it went up to 8,100. It came from 6,200. And this happened over the timeframe 2009 to 2004, which is a five-year span, right? So the denominator here is a five. If you take the difference on top, you get 1,900. So over that five years, the people gained 1,900. And then if you buy 1,900 by five, you get 380. So what we wanna think of here is right here is what is this 380? Our population is increasing, 380 people. Per year. And so it's oftentimes important to start mentioning units when you work with storage bombs. This isn't just some arbitrary number 380. It has a real-life interpretation. The city increased 380 people per year. Keep the units in track there. And as slope is a ratio, it's a rate of change. Your rate of change should have two units, one unit divided by another. This is a rate of people per year. The fact that this is a positive 380 per year means the city is growing. We had a negative 380. That means that the city would actually be decreasing. Maybe there's an exodus out of the city for whatever reason. Now you could then try to figure out like, okay, how much farther from 2009 is 2013, right? So we could very quickly determine that this is four years later, in which case then you're gonna take four times 380 added to the current population, go from there. That's perfectly fine. I do wanna just kinda show you a slightly different approach, which is basically the same thing, but this strategy is gonna be helpful in more general situations. Because after all, when you give it a story problem, your goal as a student is not actually to answer the story problem, right? Do we need you to decide how many people are gonna be in this city in 2013? No, I mean, one, the city is fake. It's not a real city. And even if it is, the city has people to do this type of job. City planners and things like that, right? That's their job. Why are you doing this? The reason you are doing this is not to make the prediction, but to understand this mathematical analysis. How that you can use functions to model data and thus make predictions, make forecast based upon that data. That is a very relevant skill for you that you could use in the future if we know how to do it. So don't focus on finding the answer. Focus on how do you get the answer? That might seem a little bit different, but the goal is not to actually find the prediction. The goal is to understand a strategy that'll give us to that prediction reliably here. And that's where this modeling functions is gonna come into play here. When you're coming to modeling functions, you have to decide your X coordinates. And your X coordinates do not necessarily have to coincide with the numbers that you're given here. Because we have the years 2004, 2009, 2013, you might think that, oh, I'll just set X to be the current year of X 2004, 2009, 2013. That's perfectly acceptable. But what you can also do is the following. You could then set X equals zero to be 2004. There's no reason you can't start your timeframe there. Which would tell us that 2009 is actually year five with this model. And then we get that 2013 will be year nine in terms of this model. Like if we start graphing the population growth, are we gonna start at year one AD? Or are we gonna probably start it at 2014? So in some essence, our projection starts at 2014. And so that's time zero. That's gonna be our X equals zero right there. Which that does modify our rate of change right here. Our denominator would look like five minus zero. But notice five minus zero is still five, right? When it comes to the slope, the actual location doesn't matter. It's the change that matters. This is a rate of change. And whether the starting value is X equals zero or X equals 2004, that's a difference of five years. So that won't change in the denominator irrelevant of that. The slope is still 380. That does affect of course the X intercept. One of the advantages of setting X equals zero, sorry, it would change the X intercept. I meant to say it would change the Y intercept. One of the advantages of changing X equals zero to be corresponded to 2004 is that the Y intercept would then be that value, 6,200, right? So by convenience, I have the slope intercept form basically immediately. My function F of X equals 380 X plus 6,200 right here. This didn't give me a model for this city's population. For which case then, I wanna look at F of nine. I'll get 380 times nine plus 6,200. In which case, 380 times nine, that is 3,420. We add that to 6,200 and we see that the predicted population will be 9,620. And so it's good to actually explain what's going on here. So we would actually probably say something like the following in the year, in the year 2013, we expect, let me make this a little bit bigger for us. We expect the town's population to be 9,620 people. When it comes to a story problem, it's really best practice to have a story answer. There should be some sentence that explains the context of what that number means. Now, when you work with like a website, like WebAssign or MyMathLab or whatever, they often just have a box that you type in the answer and so it does all this writing for us. And so therefore we forget that we should actually be interpreting what does this number mean and the number is our prediction. We predict to be about 10,000 people in the year 2013. And I should mention that this number right here is probably gonna be wrong, right? I mean, at the time of the recording of this video where seven years past 2013, we could actually check if we had this town data, was this right or not. One doesn't expect the answer to actually be perfect. Modeling is not about being perfectly right. It's about being estimates, right? People complain about the weatherman like, he said the temperature high today was gonna be 79 degrees and it was only 77, right? Sure, sure, off by two degrees Fahrenheit. It's not so significant, right? But if they predicted the temperature's gonna be 79 degrees and then it's like, oh, it turned out to be 47 degrees outside, that's a big difference, right? You might've wanted to wear a coat when you went to school today. So these are supposed to be predictions. They're not exactly right, but they're supposed to be close. And so that's what modeling's all about. Another important question would be, so like we've answered the question, when in 2013, what do we expect the population to be? We expect to be 9,620. But other times we have to answer questions like, when will the population reach 15,000? Maybe 15,000 represents some critical infrastructure point for this town, right? If the population gets bigger, we expect more people will be driving, more water will be used, more trash will be created. How do we compensate for that, right? Well, we might need to figure out, when is it gonna reach 15,000? So how much time do we have to plan how to update the roads or the dump or the water or whatever the infrastructure is? This is a very important question. So if you wanna identify the year in which the population will reach 15,000, we're trying to answer the question. We're trying to solve the equation f of x equals 15,000. We have to solve this equation. But once we have an algebraic formula for f of x, this doesn't become too much of a chore. We take 3,800 times x plus the 6,200. This equals 15,000. We then have to solve this equation. So we're gonna subtract the 6,200 from both sides. This is gonna give us 3,800x is equal to, 15,000 takeaway 6,200, which is gonna be 8,800. Then we need to divide both sides by 3,800. This will cancel over here. We get 3,800. And so then x would turn out to be, that turns out to be, if you wanna be precise, that's gonna be 440 over 19. But I think a decimal approximation is a little bit more fruitful here. You'll get 23.16. So basically this tells us approximately 23, I'm just gonna round to the next year right here. So 23 years later, 23 years later from what from, so 23 years later from our starting point, which was 2004. So then with that estimate in mind, we'd probably say something like the following. So we expect, we predict, again, let's make this text box a little bit bigger. We predict the town's population will reach 15,000 around, let's see 2004 plus 23, that would be 2027. So again, this video was recorded in the year 2020. So our town hasn't reached that point yet. And so we would expect about seven years from now it will reach that critical infrastructure point of 15,000. So our town will have to compensate based upon whatever it needs to do to prepare for that population of 15,000, but it'll happen around the year 2027. And so this is the basic idea of how we do linear modeling. We can use modeling to predict at a certain point in time, what will the population be? But we can also use the linear modeling to predict when in the future will we reach some specific number of interests to us. So this comes down to function evaluation as part A, but also comes down to solving equations, which we did on part B here. And linear models come into play when we expect the growth or decay of the quantity to be constant with respect to time.