Question

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

Answer 1

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(0) = 1`
`y(1) = 1/2`
`y(2) = 1/4`
`y(3) = 1/8`

It is clear that as `x -> infty`, `y -> 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`..