 Welcome back. Today we're going to do our second set of quadratic applications and this is going to be a data application. So here we have the sales of DVD players in the US have increased over the years. So here are our times, our different years that we have, and then here are our different sales that we have. Now the one thing you also want to notice here is that it says in years T since 1990. Okay, so we're going to need to scale our years. So if we have 1997, let me go back over here, so I'm going to put a T. So that's going to make that one 7, that one 8, that one 9, that one 10, and that one 11 because it's years since 1990. Okay, so we're going to be finding a quadratic function that models this data. Okay, that means we're going to be using the calculator to do some regression. So let's go to the calculator. And if you remember how to do this, then we want to go to our stat. So stat, edit. Now in our L1 and L2, if you have anything in here, you can just highlight up and then hit clear and enter, and it'll wipe out any data points you have in there. Mine are empty because I cleared them before I started the video. Okay, so remember our inputs go in L1, our outputs go in L2. So that means my time is going to be my L1 and my sales are going to be my L2. Okay, so I'm going to put a 7, enter, 8, enter, 9, enter, 10, enter, and 11, enter. Okay, toggle over to the right to get into my L2. I'm going to put in my sales then. So 0.3, enter, 0.9, enter, 3.6, enter, 9.9, enter, and 16.0. So I don't have to type anything but just 16. Okay, kind of a gut check on this one is make sure your lists have the same length. And then since it's such a small data set, I'll probably just go through really quickly and make sure I typed everything in correctly. I think it looks pretty good. Okay, also remember that our L2, so this represents sales in millions, and our L1 represents years since 1990. Okay, so we're going to go back to stat. We're going to go over to calc. Now in the past, when we wanted a linear model, we would pick option number 4. That stands for linear regression, but this one asks us for a quadratic function. So we want number 5, so that's quadratic regression. So hit 5, all right, tell the calculator to run it, and there is our model. And that's ugly, so that's why it says round to two decimal places. Okay, so I'm going to have to toggle back and forth here. Half a t is going to equal, so my a was 1.04. Now we have to use a t in here, so that would be t squared. And if you forget what this looks like, your calculator puts it up here at the top for you, so that's pretty handy. My b is going to be negative 14.73, so this will be minus 14.73. Again, t, not x, because t's our variable, plus our c value, which is 52.17. So plus 52.17. Okay, fantastic. Sketch a rough graph of your model. Okay, so there's a couple of different ways we can do this. We can just type into our y equals what I put here. Or since we have a regression, we can now go through a fancy thing here. If you go to your y equals, don't type this in, but go to your vars menu. Toggle down to number five or click on five for statistics. Then you want to put the equation in, so you notice this third menu over here says eq. Regression equation, that's what this reg eq stands for, if that's what you want to pronounce it. Symbolically or phonetically here. So go ahead and click on that, and it'll put in the whole shebang here for you. So look at how many decimal places it puts in. But again, I said rough sketch here, so if it's not perfect, that's fine too. Another thing that we can do since we have some stats in here, some data, is let's go ahead and do a zoom. And we can play around with our window like I've been showing you what we've been doing. Or we can do zoom stat, so we can do number nine. So it should fit our window, but I might have my stat plot on. No, it's okay, this window's good enough. But if you didn't have a good enough window, you want to turn your plot on first, duh. And then I can go to zoom, number nine is your zoom stat. Oh, so this gives us a much different window. So here's our data. This has kind of a curve to it, but you notice it looked more like a parabola when we had the whole window on here. And it looks like it goes through most of the points, so I would say that's pretty good. But if it were me, I'm going to go ahead and sketch out the whole thing that we have here. So remember it kind of went like this, popped down, and then went back up again. Okay, all right, according to the model, how many DVDs were sold in 1990? Okay, so 1990 is our year. T remember is years since 1990. So that means we want to look at where T equals zero. Okay, so let's take a look at our table. So if I go to second table, since I can't see much of my graph here since I did my zoom stat. If I go back to zero, it gives me 52.16666. So yeah, so we can call that 52, we can say about 52. And then our Y was in millions, so we could say about 52 million. Okay, where is this on the graph? Well, I didn't label it very well, did I? So let's say this would be it right here. So this would be the point zero comma 52. All right, keep on checking. According to the model, what was the smallest number of DVDs sold? Okay, so let's go back to the calculator. Let's turn that stat plot back off because I think that's kind of distracting. So let's turn that off and let's change our window because I want to go back a little bit before this. So let's go ahead and do zero to, that should be fine, zero to 11. I want to see more of that parabola. There we go. That looks like more parabola now. What was the smallest number of DVDs sold? Well, that would be at our vertex right here, our minimum. So you want to go to second calc. We want the minimum, so that's option number three. Left bound, so we're on the left side. Let's hit enter. Toggle on over. That's pretty good. Okay, hit enter. And our guess is going to be somewhere in the middle, about right there. Okay, so the minimum is at 7.0, whatever, and it's going to be 0.14. So it wants the smallest number of DVDs sold. So again, that's our vertex. So the smallest number, that means it wants the y value. So the smallest is, was that 0.14? See, smallest is, I don't want to print it either. 0.14 million. Okay, when did that happen? That was in 1997. Because remember, our T was seven there-ish. If you want to add a little bit more to that, you could. But remember, it said 7.06, close enough to seven in 0.14. So that's our vertex. Okay, where is it on the graph? Well, we already said it's the vertex, but let's go ahead and label it. So I'm going to go back up here, take my green, and we'll label that about right there. So that's our smallest number. Okay, using your model, find f of 19 and answer with a sentence describing the meaning of the solution. Okay, so 19, because that's inside the parentheses, that's going to be my input. So that's my T, or in this case my x, if I go to look at my calculator. So I'm going to look at my table, since I'm too lazy to plug this in. And let's toggle on down to 19. Plus, with all these decimals, it's just going to be really messy. So I think it's a lot easier to let the calculator do the work for us. Okay, so at 19, we have 148.74. So f of 19 was 148.74. Let me double check that, make sure I remembered that correctly. Yep, looks good. Okay, answer with a sentence describing the meaning of the situation. Well, 19 is our T, so let's go back up here. Remember, T is years since 1990. So that would mean in 2009, there were, since it's in the past now, about, let's see, 148.74 million DVDs sold. Using your model, find T when f of T is 100. Okay, so because this is on the other side of our equals, that means this is our output. This is our number of DVDs sold. This is our y value, so to speak. Okay, so let's take a look at our table again. You think we're going to get lucky and find 100? Close. At 17, it's 103, so I'd say that's pretty darn good. But let's go back to our y equals. Let's toggle down into our y2, and let's go ahead and put 100 in. Okay, I'm going to graph it. And I'm probably not going to be able to see what's happening here. Okay, so let's go to our window. Let's change our max. What did I say? That was 17. So let's go ahead and make our x max 20 just to be safe. And we want our y value to be 100. We only had it at 18. Okay, so let's go ahead and change that to how about 120 just to be safe. So if I hit graph, I can hopefully see the intersection now. There is the, there she goes. Perfect. Okay, so then to figure out where that point is exactly, we know it should be somewhere around 17. We can go to second calc. We want intersection, so I want option number five. First curve. So that means let's go ahead and follow our first curve until we get close to that intersection point. Maybe, maybe. Okay, that's pretty good. Hit enter. Second curve. Well, we're already pretty on our, we're on a second curve pretty darn close to our intersection point. So hit enter. Guess sounds good. And our intersection is at 16.84. Now, remember, this is talking about real life things here. So we could say towards the end of 2006 would be fine. If you could say at the beginning of 2007 or right before the beginning of 2007, however you want to state it, but that's going to be our year for that one. Okay, so I think I'm going to say at the end. At the end of 2006, there were about a hundred. Oopsie. Let me fix this one too. I don't know what happened there. A hundred. And remember our outputs are in millions. A hundred million DVDs sold. Or you could also say in 2007, I think I'd be happy with that answer too. Okay, does there seem to be model breakdown? So this is kind of a gut check with this one. But yes, I mean, do you really think that the years, the selling years started here went down and then eventually went up? No, because remember, let me clear this out. Remember, we started our date at about seven. So that was about at this minimum. So what's happening before that? I don't think it was probably more. You know, I don't know when DVDs started, but yeah, so I would probably say yes. The years before 2007, I think that's about in there are too big. But that's why we call these models. They're not perfect predictors or anything else. And, you know, I doubt that the number of DVDs is going to get that large, but it might. You just never know with some of these things. So that's where, you know, you have to do some interpretation of stuff. All right. Good luck. Keep practicing these and have a great day.