To explore Golden Ratio and identify its existence in the graph of a Symmetric and an Asymmetric Quartic Curve.
Introduction:
When I was introduced to the Topic of Quadratic Equations and Graphs, I was told my Teacher about the connection of Golden Ratio with the Quadratic Equation Roots. Later, he showed us a short documentary movie on the Golden Ratio and the mystery of Golden Ratio in the branch of Mathematics. This intrigued me into exploring this fascinating ratio. Since childhood I have a passion to explore more in depth of the interesting facts which my teacher explains in class for few specific topics of Math. I have the tendency of getting more involved into those kind of topics which my teacher teachers and then I try to explore more of it in my free time through online research. I remember in my Grade.8 when my teacher introduced to the concept of how pi value has been a mystery in Mathematics and how it actually originated in Mathematics. I took this in great passion and then started exploring more into the pi and how the digits of pi are being approximated till date using super computers. I found Golden Ratio in the similar interests of my deep exploration and then decided to make this research as my topic of IA since on exploring through various websites, I found that this ratio is also called God’s favorite ratio due to its existence in many historical facts and specimens of nature. Initially I will understand what the Golden Ratio and its historical facts. I will see in what fields is this ratio connected in real-life and how dominant is it in nature and other interesting facts.
Aim:
The aim of my investigation is to explore Golden Ratio in Quartic Curves.
Plan:
In order to proceed with my investigation, I will initially understand the definition of Golden Ratio and its mystery in the branch of Mathematics and in various fields. I will then explore the Quartic Function and its graphical properties. I will take the case of both a Symmetric and an Asymmetric Quartic curve to explore for possible Golden Ratios in these 4th degree polynomial graphs.
GOLDEN RATIO:
Definition:
Since the whole of my Internal Assessment is based on exploring Golden Ratio in the different types of Quartic Functions, I initially tried to thoroughly understand the Golden Ratio and various special geometrical properties connected with it. Also, I will explore on why the Golden Ratio is so special in the branch of Mathematics and how it is significant in real-life.
History of Golden Ratio: Golden Ratio is a very special number in the branch of Mathematics which is found in many of the things in nature and real-life. The mystery behind why it exists so perfectly in many natural specimens is still a mystery surrounding the great Mathematicians and Researches of history. The value of Golden Ratio is 1.618…. and its reciprocal is 1/(1.618….)=0.618… One more interesting fact is that the reciprocal of the Golden Ratio has the same value as that of the original Golden Ratio. Such is the fascination of the Golden Ratio that many people have used this ratio in the construction of various Historical Monuments, in famous artistic paintings etc.
The world famous Monalisa Painting by Leonardo Da Vinci also has the Golden Ratio in it.
Symmetric Quartic Function:
The Quartic Function is symmetric if it has even power of ‘x’. We call it as an even function since its degree or the higher power of ‘x’ has an even power. The reason why the functions with even power of x will be symmetric is that for every x value there are two y values, one with y value and one with –y value differing by a sign but having the same value for ‘y’. This happens for every x value of the curve thus demonstrating the symmetricity along certain vertical line.
Let me take an example Symmetric function to demonstrate this:
The Symmetric function
is graphed on the axes and I obtained the following graph:
Graph:
The above graph is plotted with key points of intersection using the paint as follows:
