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More On Quartics Architecture
Designing Quartics To Meet Your Needs
Following earlier posts promoting function design for learning and skills development, rather than simply finding roots, this post furthers my earlier posts on the concepts of developing zones or domains for polynomial generics y=Ax⁴+Cx²+Dx+E with specified numbers of Real roots and Turning Points Tp’s commencing with Genetic parents y=Ax⁴+Cx²+E.
This post assumes math at the high school level.
Note: Throughout this post, where appropriate for simplicity, we use Ax⁴ coefficient A=1.
Design Specs
Create a domain inside 2 generic Polynomials y=x⁴+Cx²+Dx+E and y=x⁴+Cx²+Mx+E shown in black and blue respectively in Graph 1 where all the functions between the boundaries have 3 Tp’s and 2 roots. We keep both constants E=0 for concept simplicity.
Concept is envisaged in Graph 1 with such a domain shaded green.
The red function is Big M a genetic y=-3Ax⁴-Cx²+E which traces the Turning Points of all generics y=x⁴+Cx²+Dx+E. It is introduced in several of my earlier posts e.g.:
Quartic Polynomials With Specified Turning Points Using Big M
Designing Quartic Polynomials — Converging Turning Points
Designing Generic Quartics Within 4 Real Roots Domains
The black function on the outer edge is a recently posted limiting function based on a 1 : 8 Ratio of its Tp(y’s) and has its Tp’s(2 & 3) merged at Big M’s Tp above which with increased Dx it has only one Tp, Tp(1) and 2 Real Roots.
Graph 1
Functions Design
Find the black and blue functions y=x⁴+Cx²+Dx+E and y=x⁴+Cx²+Mx+E given C=-7 and E=0 using the red genetic Quartic Big M, y=-3x⁴-Cx²+E providing roots and Tp’s.
Hence: Big M y=-3x⁴+7x²+0 with roots easily calculated using the Quadratic Equation; +-1.53. Root +1.53 is the Tp(3) and merger of 2 roots of blue: Hence blue: y=x⁴-7x²+Mx. Sub in x=+1.53
