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 as it doesn’t change the roots or x coordinates of Inflection and turning points.
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
