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The sales $ y $ (in billions of dollars) for Harley-Davidson from 2000 through 2007 are shown in the table.

(a) Use a graphing utility to create a scatter plot of the data. Let $ x $ represent the year, with $ x = 0 $ corresponding to 2000.(b) Use the regression feature of the graphing utility to find a quadratic model for the data.(c) Use the graphing utility to graph the model in the same viewing window as the scatter plot. How well does the model fit the data?(d) Use the trace feature of the graphing utility to approximate the year in which the sales for Harley-Davidson were the greatest.(e) Verify your answer to part (d) algebraically.(f) Use the model to predict the sales for Harley-Davidson in 2010.

a) See plotb) $y=-.0408 x^{2}+.7151 x+2.8196$c) See graphd) 2008e) 2008f) 5.89

Algebra

Chapter 2

Polynomial and Rational Functions

Section 1

Quadratic Functions and Models

Quadratic Functions

Complex Numbers

Polynomials

Rational Functions

Missouri State University

McMaster University

Harvey Mudd College

University of Michigan - Ann Arbor

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So we have some data and we have three years and we want to use three years after 2000. So we have zero all the way up through seven, and then we have some sales. I believe they are. And so you'll want to put those into your calculator in list one and list, too, and then go to a stat plot. Most of you probably have a T I product a stat plot in turn plot one on. And also, once you get all that data in there and you have the stat plot on double check and make sure your wife's of one is clear, sometimes people will forget to do that. Then it doesn't graft correctly, but then go go to zoom Stat. Zoom staff will give you the window you want for your data so you don't have to put the window in yourself. And when I did that, I'm seeing a nice little scatter plot here on my screen. And it does appear that these dots, they're doing this and about like that and let's see and then next we want to. We want to use a regression feature on our calculator with Quad regression. So we're going to go to stay at and then calculate, and then we want to go to quad Regression and let's do that. Let's that's supposed to be a G. So let's get that quad regression. So I got to hit it, Stat calculate and quad regression on my calculators Number five and once. And I have a list one list, too, and we're going to calculate. And when I dio I get my quadratic is why is equal to negative 0.4077 x squared and then we have plus 0.715 x and then we have plus 2.819 and we'll go six Now, on the next question, we want a graph that I believe I'll just pick out the question again. We want a graph that onto our our graph. So now I'm going to go for part C. I'm going to go. Why someone on my graph work? And one thing I can dio is Aiken type all this in. But if I go to variables and then I go over to why variables? I'm sorry, go to variables and then go to statistics number five and then go over to e Q. And when you go over to E Q, it will say wreg equation. And if I hit that, it will actually type that equation in with all the decimal places, and I don't have to type it in myself. And then I could just grab and it should grab that right on top. And when I grab that feature, you know it's it's pretty darn close. It comes right up and kind of goes through a couple of points there, but it's very close, so it looks like it's a pretty good model. Uh, now the next part, we were asked to use the trace feature to find where that high is so we can trace along. And if we hit the trace button on our calculator, we can trace along and find out where that peak is on our calculator. We can also use the max value on are calculated to find that, and then it asks us. I believe thio actually verify that Algebraic lee. So let's verify Algebraic Lee what that peak is going to be where that maximum is going to be. And so we should recall that when we get Thio, go back to black when we have a quadratic in this form of, uh, a X squared plus b x plus c. And there's are quadratic that we're gonna have a access of cemetery. Yeah, at negative Be over to a. So if I take negative b the opposite of B over two times this little I and get out of there my calculator. So I have negative 0.715 divided by left parentheses. E two times negative, 20.4077 I find out that that access the cemetery is taking place at 8.77 So after 8.77 years around my screen, it may not have actually shown that that I was having that maximum value was going to take place, actually beyond my data, because my data on Lee went up to seven. So it was going to take place after that, and then we could take this value, and I'm actually in a kind of I'm going to store this hit the STO button as ex and then hit Enter. So you hit stow and X and enter. And now I have that started X And so now I'm gonna plug that into this equation. I'm going to type in negative 0.4077 times X squared. And I'm literally using X squared because I've stored that value into it, plus 0.715 times X and then plus 2.8196 And when I hit enter on that. I get the corresponding why value is 5.95 so that maximum point appears to be this x value. And this appears to be the high for the Y value. And then we had to use our model to find where we have the, uh, predict the sales for 2010. So we need to plug in and find what the functional value is when we plug in 10. Because we are 10 is Todd 10 years after 2000. And so, like wise, I can go back and I can take 10 and I can store it, hit the stove button as ex and hit an ER, and then I could go back and actually on my calculator, I can just go back to that teacher that I had already typed in when I had extort as that five point. Ah, no, the 8.77 and I can just go back in second entry and I find out that this becomes five point 89 to 6, and I think that was in probably in millions of dollars. Lett's back up on the screen and look billions of dollars. Oh, my wrong billions of dollars. This was in billions of dollars. And so hopefully that helps you with acute you calculator tricks.

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