# How do you identify equations as exponential growth, exponential decay, linear growth or linear decay f(x) = 3 (1/2) ^2?

Aug 1, 2018

The function that you provided is just a constant: 0.75.

However, we can think about what each of those terms means:

Exponential Growth / Decay:
$f \left(x + 1\right) = a \cdot f \left(x\right)$
for some number $a$ that we call the base. If $f \left(x + 1\right) > f \left(x\right)$, it's growth. Otherwise, it's decay. This means that if $a > 1$, we have a growth and if $a < 1$, we have a decay. Functions of this type look like the following:
$f \left(x\right) = c \cdot {a}^{x}$
If the number you provided is part of an exponential decay, it may look like this
$f \left(x\right) = 3 \cdot {\left(\frac{1}{2}\right)}^{x}$

If the number you provided is part of an exponential growth, it may look like this
$f \left(x\right) = {\left(\frac{1}{2}\right)}^{2} {3}^{x}$

Linear Growth / Decay:
$f \left(x + 1\right) = a + f \left(x\right)$
for some number $a$ that we call the slope. If $f \left(x + 1\right) > f \left(x\right)$, it's growth. Otherwise, it's decay. This means that if $a > 0$, we have a growth and if $a < 0$ we have a decay. Functions of this type look like the following:
$f \left(x\right) = a x + b$

If the number you provided was part of a linear growth, it may look like this
$f \left(x\right) = {\left(\frac{1}{2}\right)}^{2} x$
If the number you provided was part of a linear decay, it may look like this
$f \left(x\right) = - x \cdot {\left(\frac{1}{2}\right)}^{2} + 1$