 This video we'll talk about polynomial function applications. So we have this graph talking about water levels over a six month period. And it wants to know what is the minimum degree? Oh, we've got one turn, two turns, three turns. So three turns means that I can have one more in my degree, so it's degree four. How many months was the water level below normal in the six month period? Well, this is normal. The x-axis is normal. So it's got, here's one month and then it's above for two to three and three to four. And then from four to five, that's below and from five to six is below. So that gives us two. So we have a total of three months. And then in the same graph here, at the beginning of the water level is 36 inches. That's going to be important to us later. And we want to then construct a model for this particular graph. You find your x-intercepts and that's going to be x minus one and x minus two. And then we have another one at four. So x minus four and we have another one at six. So x minus six. But here is where this comes in. We don't know what our A is, but we do know that it is 36 inches when the M, or in our case, the x is zero. So if I plug zero in here and zero in here, zero minus four and zero minus six. We're going to have 36 is equal to, if you multiply all that out, you get 48 A. And when you divide 36 and 48, you end up with three fours. So our real function here is get my eraser. We're going to say not three, not A, but we want to have three fours. And there we have our function. The second problem, we have a posh restaurant opens up at 10, closes at nine. And this is what the customer count is and all that's good stuff. So we need our calculator because it says use a collected data to look at the scattergram and then determine the regression form and then find it. Well, I've already put all that data in here, but I will show you come over here to my calculator. I've put all that data in and I went into y equal and I turned my plot on. And this is really blank the way I have it set up. So we're just going to look at our zoom nine and you can see you have one, two, three, four, five, six data points, just like we have one, two, three, four, five, six data points there. Don't miss those ones that are near the x axis. So it really looks like kind of like an M. So that makes me think that I have one, two, three turns. So three turns means degree four and degree four is what we call a quartic. A cubic is degree three and a quartic is degree four. So go back over to stat and calculate and then court reg down here is seven. And you find out you get this wonderful equation that I have plugged in. I'll write it for you here. It's negative 0.357 is what I use t to the fourth plus 7.535 t cubed minus 49.77 because it's 699 t squared plus 111.687. And let me get rid of my quartic because I already told you that I can write it somewhere else later. And that's t and then plus 0.025 when we round. So we found a quartic. And when we go and look at our graph here, if I come down and I arrow over to the equal sign and press enter, it will now show me my graph on top of my scattergram. And I get this wonderful little graph. So it says use this equation and its graph to find what time it reaches its morning peak and its evening peak. Well, those are going to be the maximum points here. So second trace and max is four. And we need to be to the left of one of our maxes and then to the right of it. And when we do that, it's going to give us a nice number. Enter and then go to the other side and enter, enter for the guess. And it says 1.1.7. Oh, what happened to my pen? Hang on, let me get rid of that. One. Okay, this happened to me before, but it's 1.72. And 1.72 is approximately, if I take 60 and divide it by 0.72, I can figure out how many minutes. Oh, wrong place. And if I take 60 and divide it by 0.72, I find out that it's about, no, I did the wrong thing. 60 times 0.72 is equal to 43. So we're going to have here, what was it? I had a lot of seven hours and now 1.72. 1.72, I'll type that in. It gives us 1143. I'm just going to erase that and I will type all these numbers in since my pen isn't working. And then if we go do the other one, second trace four and we go over to the other one, we're going to find out that that one is 9.13. And we find out that this one is 9.1314. Six, let's go to our screen again. 60 times 0.14 is going to give us about 708 for the time. And now we'll have to type that one in too. And then I give you the graph so that we can, I have to necessarily work with the calculator, although we can. What time does the business is the slowest? So that's going to be this point down here. And we better go to my graph since my calculator is doing funny things. So second trace, we went the minimum this time. So come back and go to the left of that minimum and enter and go to the right of that minimum and enter and then enter for the guess. And we find out that that is 4.98. So we could do 4.98 or we could just round it approximately five, approximately five hours, which translates to 3 p.m. And then we want to know how many customers there are, that would be 32.9. So let's please call that 33. And then the last one, I want to come back to my calculator, Y equal. And enter down to Y2 and call that Y 100. Customers are my Ys. So I'm going to look at this graph and see the intersections. And second trace 5, enter, enter, enter will get me the first one. And that tells me that it's at about 7.87 hours, or let's call that eight hours. I think the book doesn't round it, but we're going to round it. And then second trace 5 and I want to go to the other one. So enter, enter, enter. And we find out that there are also 100 customers at 10 hours later. So we can say between the hours of 6, which is the eight hours later, and eight, which is the 10 hours later, they would have 100 customers or more. Because it does go above that between those two hours.