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

An exponential function is any function in the form:

#y(x) = a cdot b^{cx}#

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(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, #y(x) = -5 cdot (1/3)^{-x}# represents exponential decay (case #color(blue) ((3))#).