Philippine E-Journals

Patterns of linear, quadratic and exponential function

Rene V. Torres

 

Abstract:

Generally, when independent variable of a function is used as an exponent, the function is exponential. Hence, the following can be examples of exponential functions: $f\left(x\right)=a{b}^x+c$ , $f\left(x\right)=a{e}^{bx}+c$ , or $f\left(x\right)=e^{a{x}^2+bx+c}$ .Deriving functions of these types given the set of ordered pairs is difficult. This study was conducted to derive formulas for the arbitrary constants ${}^b$ , and ${}^c$ of the exponential function $f\left(x\right)=a{b}^x+c$ . It applied the inductive method by using definitions of functions to derive the arbitrary constants from the patterns produced. The findings of the study were: a) For linear, given the table of ordered pairs, equal differences in x produce equal first differences in y; b) for quadratic, given the table of ordered pairs, equal differences in x produce equal second differences in y; and c) for an exponential function, given a table of ordered pairs, equal differences in x produce a common ratio in the first differences in y. The study obtained the following forms: $b=\sqrt[d]{r}$ , $a=\frac{q}{b^n(b^d-1)}$ , $c=p-a{b}^n$ , Since most models developed used the concept of linear and multiple regressions, it is recommended that pattern analysis be used specifically when data are expressed in terms of ordered pairs.