Question

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

Answer 1

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]}

Answer 2

Exponential decay...

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

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...