What is the Degree of this Curve?

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Discussion Overview

The discussion revolves around determining the degree of a polynomial curve that passes through specific points labeled A0, A3, and A6. Participants explore the implications of fitting polynomials to a set of points, including the minimum degree required for such a fit.

Discussion Character

  • Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant questions whether the degree of the curve is 6, given the endpoints A0, A3, and A6.
  • Another participant states that the minimum degree polynomial for n points is n-1, suggesting that with 7 points, the minimum fitting polynomial is of degree six.
  • A further contribution reiterates the n-1 rule but raises the possibility that a polynomial of lower degree could fit the points, questioning if degree six is indeed the maximum for the minimum degree polynomial.
  • Another participant introduces Lagrange interpolation as a method to derive a polynomial that fits the points, but seeks clarification on the term "endpoint" as it relates to A3.

Areas of Agreement / Disagreement

Participants express differing views on whether the degree of the polynomial is definitively six, with some suggesting the possibility of lower degree polynomials fitting the points. The discussion remains unresolved regarding the exact degree and the interpretation of the endpoints.

Contextual Notes

There is an assumption that the points A0, A3, and A6 are part of a larger set of 7 points, but the implications of this on the degree of the polynomial are not fully explored. The term "endpoint" is also not clearly defined in the context of the discussion.

mymachine
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I wonder how many degree of this curve where the endpoints are A0, A3, and A6?

Is it degree of 6?
 

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The minimum degree polynomial, when given n points, is of degree n-1. You are given 7 points so the minimum fitting polynomial is of degree six.
 
mathman said:
The minimum degree polynomial, when given n points, is of degree n-1. You are given 7 points so the minimum fitting polynomial is of degree six.
Isn't this the maximum of the minimum degree polynomial that exactly fits n points? It's possible that the n points could be exactly fitted with a polynomial of lower degree.
 
You can use Lagrange interpolation to write down an explicit polynomial that passes through the given points. As the Wikipedia article notes, such a polynomial will be of minimum degree. However, what do you mean by saying that A3 is an endpoint?
 

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