Symmetry and Asymmetry for <i>n</i>th-degree Algebraic Functions and the Tangent Lines

Norihiro Someyama

doi:doi:10.11648/j.ml.20210701.11

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Symmetry and Asymmetry for nth-degree Algebraic Functions and the Tangent Lines

Norihiro Someyama

Published in Mathematics Letters (Volume 7, Issue 1)

Received: 21 November 2020 Accepted: 13 January 2021 Published: 10 March 2021

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Abstract

We reveal one relationship between each degree algebraic function and its tangent line, via its derivative. In particular, it is easy to see and well known that asymmetry (resp. symmetry) of tangent lines of a quadratic (resp. cubic) function at its minimum and maximum zero points, but it is not easy to investigate symmetry and asymmetry of them of nth-degree functions if n is 4 or more. We thus investigate the relationship between the slopes of the tangent lines at minimum and maximum zero points of the nth-degree function. We will in this note be able to know some sufficient conditions for the ratio of their slopes to be 1 or -1. By these, we can understand that tangent lines at minimum and maximum zero points have a symmetrical (resp. asymmetrical) relationship if the ratio of their slopes is -1 (resp. 1). In other words, these properties give us symmetry and asymmetry of the functions. Furthermore, we also mention the property of the discriminant of a quadratic function.

Published in	Mathematics Letters (Volume 7, Issue 1)
DOI	10.11648/j.ml.20210701.11
Page(s)	1-6
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Algebraic Equation, Algebraic Function, Zero Point, Tangent Line, Discriminant, Inflection Point

References

[1]	Aude, H. T. R.: Notes on Quartic Curves, American Mathematical Monthly, 56 (3) (1949), 165¨C170.
[2]	Bronshtein, I. N., Semendyayev, K. A., Musiol, G., Mühlig, H.: Handbook of Mathematics; 6th Edition (Springer, 2015).
[3]	Bhargava, M., Shankar, A. and Wang, X.: Squarefree Values of Polynomial Discriminants I, arXiv preprint arXiv: 1611.09806v2 [math.NT], 2016.
[4]	Coolidge, J. L.: The Story of Tangents, The American Mathematical Monthly, Vol. 58, No. 7 (1951), pp. 449-462.
[5]	Gootman, E. C.: Calculus; Barron's College Review Series: Mathematics (Barron's, 1997).
[6]	Gowers, T. ed.: The Princeton Companion to Mathematics (Princeton University Press, 2008).
[7]	Hazewinkel, M. ed.: Encyclopaedia of Mathematics (Springer, 2001).
[8]	Larson, R. and Edwards, B. H.: Calculus; 9th Edition (Brooks/Cole, 2009).
[9]	Miyahara, S.: On the Properties Related to Tangent Lines and Inflection Points of Graphs of Quartic Functions (in Japanese), Suken-Tsushin, Volume 74 (2012).
[10]	Miyahara, S.: On the Properties Related to Tangent Lines and Inflection Points of Quartic Functions II (in Japanese), Suken-Tsushin, Volume 77 (2013).
[11]	Rinvold, R. A.: Fourth Degree Polynomials and the Golden Ratio, Volume 93, Issue 527 (2009), pp. 292-295.
[12]	Sakai, T.: Graphs and Trackings (Baifukan, in Japanese, 1963).
[13]	Stewart, J.: Calculus; Early Transcendentals; 6th Edition (Brooks/Cole, 2008).
[14]	Takagi, T.: Algebra Lecture; Revised New Edition (Kyoritsu Publication, in Japanese, 2007).
[15]	Weisstein, E. W.: CRC Concise Encyclopedia of Mathematics; English Edition; 2nd Edition (CRC Press, Kindle Version, 1998).

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Norihiro Someyama. (2021). Symmetry and Asymmetry for nth-degree Algebraic Functions and the Tangent Lines. Mathematics Letters, 7(1), 1-6. https://doi.org/10.11648/j.ml.20210701.11

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Norihiro Someyama. Symmetry and Asymmetry for nth-degree Algebraic Functions and the Tangent Lines. Math. Lett. 2021, 7(1), 1-6. doi: 10.11648/j.ml.20210701.11

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Norihiro Someyama. Symmetry and Asymmetry for nth-degree Algebraic Functions and the Tangent Lines. Math Lett. 2021;7(1):1-6. doi: 10.11648/j.ml.20210701.11

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@article{10.11648/j.ml.20210701.11,
  author = {Norihiro Someyama},
  title = {Symmetry and Asymmetry for nth-degree Algebraic Functions and the Tangent Lines},
  journal = {Mathematics Letters},
  volume = {7},
  number = {1},
  pages = {1-6},
  doi = {10.11648/j.ml.20210701.11},
  url = {https://doi.org/10.11648/j.ml.20210701.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20210701.11},
  abstract = {We reveal one relationship between each degree algebraic function and its tangent line, via its derivative. In particular, it is easy to see and well known that asymmetry (resp. symmetry) of tangent lines of a quadratic (resp. cubic) function at its minimum and maximum zero points, but it is not easy to investigate symmetry and asymmetry of them of nth-degree functions if n is 4 or more. We thus investigate the relationship between the slopes of the tangent lines at minimum and maximum zero points of the nth-degree function. We will in this note be able to know some sufficient conditions for the ratio of their slopes to be 1 or -1. By these, we can understand that tangent lines at minimum and maximum zero points have a symmetrical (resp. asymmetrical) relationship if the ratio of their slopes is -1 (resp. 1). In other words, these properties give us symmetry and asymmetry of the functions. Furthermore, we also mention the property of the discriminant of a quadratic function.},
 year = {2021}
}

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TY  - JOUR
T1  - Symmetry and Asymmetry for nth-degree Algebraic Functions and the Tangent Lines
AU  - Norihiro Someyama
Y1  - 2021/03/10
PY  - 2021
N1  - https://doi.org/10.11648/j.ml.20210701.11
DO  - 10.11648/j.ml.20210701.11
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 1
EP  - 6
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20210701.11
AB  - We reveal one relationship between each degree algebraic function and its tangent line, via its derivative. In particular, it is easy to see and well known that asymmetry (resp. symmetry) of tangent lines of a quadratic (resp. cubic) function at its minimum and maximum zero points, but it is not easy to investigate symmetry and asymmetry of them of nth-degree functions if n is 4 or more. We thus investigate the relationship between the slopes of the tangent lines at minimum and maximum zero points of the nth-degree function. We will in this note be able to know some sufficient conditions for the ratio of their slopes to be 1 or -1. By these, we can understand that tangent lines at minimum and maximum zero points have a symmetrical (resp. asymmetrical) relationship if the ratio of their slopes is -1 (resp. 1). In other words, these properties give us symmetry and asymmetry of the functions. Furthermore, we also mention the property of the discriminant of a quadratic function.
VL  - 7
IS  - 1
ER  -

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Author Information

Norihiro Someyama

Shin-yo-ji Buddhist Temple, Tokyo, Japan

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APA Style

Norihiro Someyama. (2021). Symmetry and Asymmetry for nth-degree Algebraic Functions and the Tangent Lines. Mathematics Letters, 7(1), 1-6. https://doi.org/10.11648/j.ml.20210701.11

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ACS Style

Norihiro Someyama. Symmetry and Asymmetry for nth-degree Algebraic Functions and the Tangent Lines. Math. Lett. 2021, 7(1), 1-6. doi: 10.11648/j.ml.20210701.11

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AMA Style

Norihiro Someyama. Symmetry and Asymmetry for nth-degree Algebraic Functions and the Tangent Lines. Math Lett. 2021;7(1):1-6. doi: 10.11648/j.ml.20210701.11

Copy | Download

@article{10.11648/j.ml.20210701.11,
  author = {Norihiro Someyama},
  title = {Symmetry and Asymmetry for nth-degree Algebraic Functions and the Tangent Lines},
  journal = {Mathematics Letters},
  volume = {7},
  number = {1},
  pages = {1-6},
  doi = {10.11648/j.ml.20210701.11},
  url = {https://doi.org/10.11648/j.ml.20210701.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20210701.11},
  abstract = {We reveal one relationship between each degree algebraic function and its tangent line, via its derivative. In particular, it is easy to see and well known that asymmetry (resp. symmetry) of tangent lines of a quadratic (resp. cubic) function at its minimum and maximum zero points, but it is not easy to investigate symmetry and asymmetry of them of nth-degree functions if n is 4 or more. We thus investigate the relationship between the slopes of the tangent lines at minimum and maximum zero points of the nth-degree function. We will in this note be able to know some sufficient conditions for the ratio of their slopes to be 1 or -1. By these, we can understand that tangent lines at minimum and maximum zero points have a symmetrical (resp. asymmetrical) relationship if the ratio of their slopes is -1 (resp. 1). In other words, these properties give us symmetry and asymmetry of the functions. Furthermore, we also mention the property of the discriminant of a quadratic function.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Symmetry and Asymmetry for nth-degree Algebraic Functions and the Tangent Lines
AU  - Norihiro Someyama
Y1  - 2021/03/10
PY  - 2021
N1  - https://doi.org/10.11648/j.ml.20210701.11
DO  - 10.11648/j.ml.20210701.11
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 1
EP  - 6
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20210701.11
AB  - We reveal one relationship between each degree algebraic function and its tangent line, via its derivative. In particular, it is easy to see and well known that asymmetry (resp. symmetry) of tangent lines of a quadratic (resp. cubic) function at its minimum and maximum zero points, but it is not easy to investigate symmetry and asymmetry of them of nth-degree functions if n is 4 or more. We thus investigate the relationship between the slopes of the tangent lines at minimum and maximum zero points of the nth-degree function. We will in this note be able to know some sufficient conditions for the ratio of their slopes to be 1 or -1. By these, we can understand that tangent lines at minimum and maximum zero points have a symmetrical (resp. asymmetrical) relationship if the ratio of their slopes is -1 (resp. 1). In other words, these properties give us symmetry and asymmetry of the functions. Furthermore, we also mention the property of the discriminant of a quadratic function.
VL  - 7
IS  - 1
ER  -

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