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

1 Answer
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(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#..