Question

A population of 240 birds increases at a rate of 16% annually. Jemel writes an exponential function of the form f(x) = abx to represent the number of birds after x years. Which values should she use for a and b?

Answer 1

The value of a is 240 and b is 1.16.

Explanation:

Given,

The initial population of the birds = 240,

Annual rate of increasing = 16 %,

Hence, the population of birds after x years

`P=240(1+(16)/(100))^x`

`=240(1+0.16)^x`

`=240(1.16)^x`

We can put P = f(x), ( because, f(x) also shows the population of birds after x years )

`\implies f(x) = 240(1.16)^x` --------(1)

According to the question,

`f(x) =ab^x` --------(2),

From equation (1) and (2),

a = 240 and b = 1.16

Answer 2

`\boxed{\boxed{a = 240, b = 1.16}}`

Explanation:

General exponential function for growth or decay is,

`y=a(1\pm r)^(x)`

Where,

a = initial value,

r = rate of change

+ is used for growth and - is used for decay.

As here, the number of birds increasing so, the exponential function is,

`y=a(1+r)^(x)`

And initial value a = 240, r = rate of change = 16% = 0.16

Putting the values,

`y=240(1+0.16)^(x)\\\\y=240(1.16)^(x)`

Comparing this with the given equation
`y=ab^(x)`

Hence, a = 240, b = 1.16