How do you determine whether each function represents exponential growth or decay #y=0.4(1/3)^x#?

2 Answers
Dec 31, 2017

Answer:

it's decay. see explanation

Explanation:

Exponential function:
Formula: #y=a*b^x+c#
where:
-a is multyplier of #b^x#;
-c moves function on y axis
-b is a base of exponential function. b must be greater than 0 and can't be 1 (everything on the first is equal one, so it makes no sense thinking about it as exponential function)
#b in (0,1)uuu(1,oo)#
if b>1 then it is growing; if #b in (0,1)# then it's decreasing.

Given: #y=0.4(1/3)^x#
a=0.4;
#b=1/3 in (0,1)quad=>quad# function represents exponential decay

graph{y=0.4(1/3)^x [-10, 10, -5, 5]}

Dec 31, 2017

Answer:

Explanation:

This is an interesting problem, but one way that you can determine weather a function is undergoing exponential decay or growth, is by considering its behavour as #x # gets large, as a comparison to when #x# is smaller...

So lets consider your function:

#y = 0.4 * (1/3)^x #

As #x to oo# or gorws to a very large number, we see it tends towards 0, this function is undergoing exponential decay, we can also see this if we sketch this function

enter image source here

But we can generalise this a bit further:

If we have # y = beta^x #

If # 0 < beta < 1 # The function represents exponential decay

If # beta > 1 # This is exponential growth

If #beta = 1 # The equation becomes #y = 1^x = 1 # This is just a straight line...

So for example:

#y = 24^(-3x) # We see ' #beta# ' Is greater than one, but...

#24^(-3x) -= (24^(-3))^x -= (1/24^3 ) ^x #

Hence this function undergoes exponential decay...