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

Jan 29, 2016

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, $y = {\left(\frac{1}{4}\right)}^{x}$.

Here, as the exponent increases, the function value will decrease. Think along the lines of ${\left(\frac{1}{4}\right)}^{2} = \frac{1}{16}$, but ${\left(\frac{1}{4}\right)}^{3} = \frac{1}{64}$ 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 $y = {\left(\frac{1}{4}\right)}^{x}$ shrinks (decays).

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