# How do you determine if the equation y=(1/2)^x represents exponential growth or decay?

Jul 12, 2016

The function decays exponentially.

#### Explanation:

Intuitively, you can determine if a function is exponentially growing (heading toward infinity) or decaying (heading toward zero) by graphing it or simply evaluating it at a few increasing points.

Using your function as an example:
$y \left(0\right) = 1$
$y \left(1\right) = \frac{1}{2}$
$y \left(2\right) = \frac{1}{4}$
$y \left(3\right) = \frac{1}{8}$

It is clear that as $x \to \infty$, $y \to 0$. Graphing the function will also make this result more intuitive:
graph{(1/2)^x [-2.625, 7.375, -0.64, 4.36]}

You can see that the function rapidly approaches zero as $x$ increases, that is, it decays

The rule to work by is that for $y = {r}^{x}$, the function is exponential growth if $| r | > 1$, and exponential decay if $| r | < 1$..