Mathematics
Experiment with Numbers
Calculus spreadsheets allow for examining familiar functions such as second-degree polynomials. To some extent, variable transformation processes should be applied with a great deal of calculation management. Meantime, imagination is being required throughout any experiment with numbers. Lately, I manipulated the sum of coefficients in a second-degree polynomial equation, to describe relationships between the variable types. The final result is the creation of a new expression which it is renewed fundamentally. This modest article introduces the function
q(p) = Ap²+Bp+1. Hereby, I am referring to q(p) instead of the familiar p(x) because of the variable p is containing information on both the variable x and the coefficient materials. Indeed, the new equation reveals a concept such as properties in p are being found in q.
Requirements
There are three analytical features in the inputs:
- Two arbitrary zeroes;
- One arbitrary abscissa coordinate.
Methodology
To begin with, the algebraic transformation from the sum of quadratic coefficients to the variable p is requiring to use a calculus software. I solved for p in the simplified form of the sum of quadratic coefficients. As a result, I have obtained a mixed equation which it expresses p in terms of the two zeroes, the specified abscissa coordinate, and the sum of quadratic coefficients. This equation has two branches because of the calculus software is solving with using a square root. Consequently, from the arbitrary abscissa coordinate, I obtain two arbitrary abscissa coordinates: without themselves being zeroes with respect to q, the sum of p+ and p- is equal to that of the arbitrary zeroes x and y. Screenshot 1 depicts the step-formulas in the process towards my developing q(p).
First, I calculated all new values for q and p, from the genuine formulas.
Second, I compared the values are obtained with the calculation of q(p), with using the new formula, with the values are obtained throughout several calculation steps upon the use of the second-degree polynomial equation
q(p) = Ax²+Bx+1. Despite the redundant reference to the second-degree polynomial equation, still it is being fast directly to calculate q(p) with the formula q = ((x-p)*(y-p))/(x*y), for x and y are the two zeroes and p is the independent variable. Basically, this formula reads an only ordinate value from a pair of abscissas: on chart 1, the crossing line over the curve details the measures of yp given two sound abscissas p.
Third, in my choice for nomenclature, I made a distinction between q and yp because of they are originating from different formulas. Indeed, q is blind to the steps are required for performing usual iterations on the classical expression q(p)= Ap²+Bp+1. Hence confidence in the framework requirements and the calculation mechanics is enough to obtaining yp for any p, in the data series. Bitmap 1 shows an overview on the exemplar of this experiment with numbers. Moreover, I adjusted the values accuracy in table 4, for the calculus spreadsheet features would perform well to read the yp beside p.
Meta-analysis
Fourth, I made some observation, and I earned experiences with modelling a straight-forward expression for the equation of second-degree polynomials without knowing ahead of time anything of the quadratic coefficients. I had to double-check everything to ensure that the two values were equal; and, it happens that there are equal, still. However, q is a fully-fledged dependent variable on the zeroes while its domain is that of the traditional variable y. Until I performed this experiment with numbers, it was necessary to retrieving quadratic coefficients from a pair of zeroes, in order to calculate ordinates from abscissas in second-degree polynomials. In spite of the initial definition for the sum of quadratic parameters, still each coefficient set that is being retrieved, with some formulas, does not sum to any q value anymore. It seems that this property in A+B+C had been being carried over q(p) through an early stage of algebraic manipulation wherein A+B+C equaled q.
Conclusion
Finally, I am happy to describe this experiment with numbers with great deal of imagination and creativity. In handling efficiently a calculus spreadsheet, I made progress on the path to variable modelling. Thanks for reading me!
