tag:blogger.com,1999:blog-6651514617266100245.post7617820944862854725..comments2024-01-03T04:58:04.221-05:00Comments on I Hope This Old Train Breaks Down...: Numerous Connections to Quadratic FunctionsUnknownnoreply@blogger.comBlogger6125tag:blogger.com,1999:blog-6651514617266100245.post-39133876757627757942011-10-09T14:19:33.414-04:002011-10-09T14:19:33.414-04:00In fact I've taught this using finite differen...In fact I've taught this using finite differences...<br /><br />Given the points (2, 7), (3, 9), (4, 12), (5, 16), you can see the y-values increasing: 7, 9, 12, 16. The first differences of these values are 2, 3, and 4, so we can say that the second difference will be 1. This means that the function is quadratic and has a leading term of (1/2)x^2.<br /><br />The points for (1/2)x^2:<br />(2, 2), (3, 4.5), (4, 8), (5, 12.5)<br /><br />Original points:<br />(2, 7), (3, 9), (4, 12), (5, 16)<br /><br />Make the connection to (1/2)x^2<br /><br />x= 2, y= 7 = 2 + 5<br />x=3, y= 9 = 4.5 + 4.5<br />x=4, y= 12 = 8 + 4<br />x=5, y= 16 = 12.5 + 3.5<br /><br />Put some parts together and you have:<br />x= 2, y= 7 = 0.5(2)^2 + 6 - 0.5(2)<br />x=3, y= 9 = 0.5(3)^2 + 6 - 0.5(3)<br />x=4, y= 12 = 0.5(4)^2 + 6 - 0.5(4)<br />x=5, y= 16 = 0.5(5)^2 + 6 - 0.5(5)<br /><br />So, y = 0.5x^2 + 6 - 0.5x.<br /><br />Side note: I believe solving linear systems is considered a high school thing in the USA.<br /><br />JoeAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-89871477452201404592011-09-29T14:29:32.752-04:002011-09-29T14:29:32.752-04:00The Lagrange method also offers an explanation of ...The Lagrange method also offers an explanation of why any "other" matching polynomial would need to be degree-n, because you could add a secret y4 that was always outputting zero. Then (for your data)<br /><br />y4 = A(x-2)(x-3)(x-4)<br /><br />where A can be any number or even another function.<br /><br />I often ask "What is the smallest-degree polynomial we can fit this data to?"<br /><br />You also forgot Method #7, finite differences ;)<br /><br />Good stuff!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-69803715343825101122011-09-26T09:59:43.549-04:002011-09-26T09:59:43.549-04:00Hi Mimi,
I had never use the Lagrange method wit...Hi Mimi, <br /><br />I had never use the Lagrange method with quadratics too. All of this is very interesting. <br /><br />Thank you !The Dudehttps://www.blogger.com/profile/09582233217452159286noreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-15563453216495276082011-09-26T09:54:22.627-04:002011-09-26T09:54:22.627-04:00Hi Mimi,
Oh, I agree -- I'm just commenting o...Hi Mimi,<br /><br />Oh, I agree -- I'm just commenting on the prospects of verifying if a function is quadratic using four or more points (i.e. the observation your student made). You can potentially rule out a quadratic (if the fourth point doesn't fit), but you cannot confirm with certainty that it is a quadratic, regardless of how many redundant points you have. I think this kind of asymmetry can be useful to know.<br /><br />DanielDanielhttps://www.blogger.com/profile/13494412064284186178noreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-65643775553791141592011-09-26T00:04:58.116-04:002011-09-26T00:04:58.116-04:00Hi Dan,
Yep - but at some point you have to look ...Hi Dan,<br /><br />Yep - but at some point you have to look at the utility of teaching that to young kids (like in high school). It's much less confusing to let them think that n points probably will fit into a regression function LESS than n-degrees. I think the stretch into understanding that n points CAN be regressed into an n-polynomial is a small one, AFTER they first understand how to find smaller functions like quadratics and cubics and quartics. :) The problem is that most kids never get past the quadratic part, let alone thinking about n-degree polynomial!<br /><br />Cheers (and hope you are well!),<br />Mimiuntilnextstophttps://www.blogger.com/profile/15285583728476473117noreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-17526458504700248652011-09-25T17:42:40.177-04:002011-09-25T17:42:40.177-04:00Hi Mimi,
Even if you have n points (and say n >...Hi Mimi,<br /><br />Even if you have n points (and say n > 3) to which you have fit a quadratic function, you still cannot rule out that the actual regression function is a degree-n polynomial.<br /><br />DanielDanielhttps://www.blogger.com/profile/13494412064284186178noreply@blogger.com