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

##### 1 Answer

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:

where

Now, there are two type of exponential behaviours:

**Exponential growth**: the value of#y(x)# 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#cx# 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:

#1/2 cdot 1/2 = 1/4 " , " 1/2 cdot 1/2 cdot 1/2 = 1/8 < 1/4 ...# - These explanations are right if
#a > 0# . If#a<0# , then the results are the opposites.

So, to sum up:

- If
#b > 1# , then:

#color(blue) ((1))# If#a,c# are both possitive or negative, we find exponential growth.

#color(blue) ((2))# If#a,c# have different signs, we find exponential decay. - If
#b < 1# , then:

#color(blue) ((3))# If#a, c# are both possitive or negative, we find exponential decay.

#color(blue) ((4))# If#a, c# have different sign, we find exponential growth.

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

So, finally,