How do you identify equations as exponential growth, exponential decay, linear growth or linear decay #f(x) = 3 (1/2) ^2#?

1 Answer
Aug 1, 2018

The function that you provided is just a constant: 0.75.

However, we can think about what each of those terms means:

Exponential Growth / Decay:
#f(x+1) = a * f(x) #
for some number #a# that we call the base. If #f(x+1)>f(x)#, it's growth. Otherwise, it's decay. This means that if #a > 1#, we have a growth and if #a < 1#, we have a decay. Functions of this type look like the following:
#f(x) = c * a^x#
If the number you provided is part of an exponential decay, it may look like this
#f(x) = 3 * (1/2)^x#

If the number you provided is part of an exponential growth, it may look like this
#f(x) = (1/2)^2 3^x#

Linear Growth / Decay:
#f(x+1) = a + f(x)#
for some number #a# that we call the slope. If #f(x+1) > f(x)#, it's growth. Otherwise, it's decay. This means that if #a >0#, we have a growth and if # a< 0# we have a decay. Functions of this type look like the following:
#f(x) = a x + b#

If the number you provided was part of a linear growth, it may look like this
#f(x) = (1/2)^2 x #
If the number you provided was part of a linear decay, it may look like this
#f(x) = - x * (1/2)^2 + 1 #