# How do you determine if the equation  y = -5(1/3) ^ -x represents exponential growth or decay?

May 2, 2016

Let us define what is an exponential function, and when it grows or decays.

#### Explanation:

An exponential function is any function in the form:

$y \left(x\right) = a \cdot {b}^{c x}$

where $a$, $b$ and $c$ are constants, $a , c \ne 0$ and $b \ne 1$.

Now, there are two type of exponential behaviours:

• Exponential growth: the value of $y \left(x\right)$ tends to $\infty$ when $x \to \infty$.
• Exponential decay: the inverse to exponential growth.

Note: we shall note that $b$ will always be taken as a positive number, given that if $b$ is negative, some solutions of the function will be complex numbers, and we are just taking in account real functions.

Let us distinguish several cases, according to this:

• If $b > 1$, multiplying $b$ many times will increase its value. However, if exponent $c x$ is negative (because $c < 0$), then we will have something like
b^{-x} ~ 1/b^x
and this does not grow, but degrows when $x \to \infty$.
• The inverse happens when $b < 1$: it grows if $c > 0$, and degrows if $c < 0$. In this case, multiplying $b$ many times decreases its final value:
$\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \text{ , } \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{8} < \frac{1}{4} \ldots$
• These explanations are right if $a > 0$. If $a < 0$, then the results are the opposites.

So, to sum up:

• If $b > 1$, then:
$\textcolor{b l u e}{\left(1\right)}$ If $a , c$ are both possitive or negative, we find exponential growth.
$\textcolor{b l u e}{\left(2\right)}$ If $a , c$ have different signs, we find exponential decay.
• If $b < 1$, then:
$\textcolor{b l u e}{\left(3\right)}$ If $a , c$ are both possitive or negative, we find exponential decay.
$\textcolor{b l u e}{\left(4\right)}$ If $a , c$ have different sign, we find exponential growth.

On this link you cand find an example of each case.

So, finally, $y \left(x\right) = - 5 \cdot {\left(\frac{1}{3}\right)}^{- x}$ represents exponential decay (case $\textcolor{b l u e}{\left(3\right)}$).