Question

How do you determine if the equation `y = 4^x` represents exponential growth or decay?

Answer 1

Since `4>1`, this represents exponential growth.

Explanation:

The term that the exponent is attached to (in this case, `4`) can tell you whether or not the exponential equation will grow or decay.

Since `4>1`, increasing the exponent will make function increase (for example, `4^2=16`, and `4^3=64`, so the function is increasing rapidly), so `y=4^x` is a growth function.

This can be turned into a rule:

For the general exponential function `y=b^x`,

• the function represents growth if `b>1`
• the function represents decay if `0 < b < 1`

We could imagine a decay function, say, `y=(1/4)^x`.

Here, as the exponent increases, the function value will decrease. Think along the lines of `(1/4)^2=1/16`, but `(1/4)^3=1/64` which is much closer to `0`.

`y=4^x` graphed looks like:

graph{4^x [-12.52, 15.96, -3.28, 10.96]}

The function grows rapidly.

Whereas, the graph of `y=(1/4)^x` shrinks (decays).

graph{(1/4)^x [-12.52, 15.96, -3.28, 10.96]}