Advanced Math/Finding the right quintic function
QUESTION: Dear Clyde,
I'm hoping to find a quintic function that can be used to find the Y values that result from X values that are in between the following numbers: 0; 50; 75; 79,6875; 81,25; 100. I have a system that I use for how I plan my days and it involves 79,6875 as the perfect number (100%). I converted the six most important elements of my system to coordinates so I might be able to find a quintic function that can find the results of X values in between those coordinates. The coordinates are (0;0), (50;50), (75;75), (79,6875;100), (81,25; 75), and (100; 54,1666). A quartic function (which I calculated with my TI-84 Plus) only gave a R2 of about 0,9355 (which I think is too low).
I used http://2000clicks.com/mathhelp/GeometryPointsDeterminePolynomial.aspx
to turn f(x)=a(x^5)+b(x^4)+c(x^3)+d(x^2)+e(x)+f into f(x)=-0,000000003278645835x^5+0,000001684418146x^4+0,0001976193169x^3+0,01344428008x^2+1,02578125x+0 (I created matrices and I found the determinants; the first matrix gave 330302926E19, the second matrix gave 6202354943E19, the determinant of the coefficient matrix was -5872052017E19). When I tested the equation I found that something had gone wrong (the calculator probably did some rounding that I didn't notice), x=50 gives y=119,1052139 (instead of y=50) and the graph also doesn't look right (it is partly visible in the window, so I might not be far off).
Could you please find the correct equation? I would really appreciate it!
Sorry if something isn't clear; I don't study Mathematics and my first language is Dutch.
Thank you in advance!
ANSWER: 2137.8 x - 118.931 x^2 + 2.42318 x^3 - 0.0214954 x^4 + 0.0000701937 x^5
You are doing the right thing, you've just messed up the arithmetic. Notice that due to the extreme difference in the magnitude of some of these coefficients, it is possible that a computer round-off error is also to blame.
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QUESTION: Dear Clyde,
Thank you very much for the answer!
Is it also possible to find the equation (using the same coordinates) if the maximum y has to be 100 (here an x value of 78.118695 gives a y value of 105.17694; the x value 79,6875 should give the maximum y value of 100) and the line should go up from 0 to 75 x, go up more quickly from 75 to 79,6875 x, go down quickly from 79,6875 to 81,25 x and also go down from 81,25 to 100 x (I expect that the equation has to be +ax +bx^2 +cx^3 -dx^4 -ex^5)?
In the system I use the number 75 is the minimum acceptable x level and the number 81,25 is the maximum acceptable x level. The number 50 is random (which is unacceptable, but not as bad as 0), the number 100 is too much to ask (so it is also unacceptable, but it is a little bit better than 50 x, so the y is 54,1666 instead of 50). Scoring 79,6875 x is ideal/perfect, so that's a 100% y score (and can't be beaten), scoring at least a 75 y score is acceptable.
Is this possible and can you help me by fixing this puzzle? Thank you very much in advance.
Unfortunately not -- the polynomial given is the only one. What you are doing is called "polynomial interpolation" and is a fairly standard mathematical tool. Because polynomial interpolation can be reduced to solving a (non-degenerate) system of linear equations, its solution will be unique (assuming a solution is determined by the system), in which case you can't hope to find a different polynomial with different properties like you are asking.