## Home⇛Asia Pacific Journal of Island Sustainability⇛vol. 29 no. 2 (2017)

### 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: $b=\sqrt[d]{r}$ , $a=\frac{q}{{b}^{n}\left({b}^{d}-1\right)}$ , $c=p-a{b}^{n}$ , 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.