Question

How do you determine if the equation `y=39(0.098)^t` represents exponential growth or decay?

Answer 1

I would say decay.

Explanation:

We can have a look at the base `0.098` of your exponential. Written as it is it doesn't tell us much but what if we write it as:

`98/1000`

this is very good because we see that if you try to use your exponent `t` applied to this fraction you see that:

`98^t` becomes big BUT `1000^t` becomes bigger!!!

So, the fraction `(98/1000)^t` will become very small (the denominator is always bigger) when `t` increases!

Try with `t=1` and `t=2` you'll get:

`98^1/1000^1=0.098`

and:

`98^2/1000^2=0.0096`

So our exponential function will return us values that become smaller as `t` increases!

Graphically we see this as:
graph{39(0.098)^x [-11.25, 11.25, -5.625, 5.625]}

Hope it helps!