### 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.