# How do you identify equations as exponential growth, exponential decay, linear growth or linear decay M(t)=8(2)^(1/6t)?

Jun 3, 2015

A linear (decay or growth) has the variable 'in line" with the other terms as in:
$M \left(t\right) = a t + c$
The constant $a$ determines whether it is a decay ($a < 0$) or growth ($a > 0$).
For example $M \left(t\right) = 4 t + 7$ where $a = 4 > 0$ is a linear growth.

An exponential has the variable up on the "first floor" as in:
$M \left(t\right) = k {a}^{b t}$ where $k , a , b$ are all constants (with $k > 0$).
The constant $b$ determines whether it is a decay ($b < 0$) or growth ($b > 0$).
In your case you have $M \left(t\right) = 8 \cdot \left({2}^{\frac{1}{6} t}\right)$ so that it is an exponential ($t$ on the first floor) and $b = \frac{1}{6} > 0$ is an exponential growth.

Graphically:
graph{8(2^(x/6)) [-25.66, 25.66, -12.83, 12.82]}