Question

A population of coyotes is known to show periodic behavior over time. The table shows the population of coyotes in a specific area over time.

Answer 1

1. B

2. A, D

Explanation:

Answer 2

second option
`f(x) = -25cos((\pi)/(3)x) + 50`

Explanation:

We have a function of the form
`Acos(bx + c) + h`

We know that the cos(x) function is periodic.

That's why
`Acos(x) = A` when
`x = k\pi`

Where k is an even number.

Also
`Acos(x) = -A` when
`x = k\pi`

Where k is an odd integer.

Finally
`Acos(x) = 0` when
`x = k((\pi)/(2))` and k is an odd integer.

With this information we can evaluate the options given for the function f(x) with the values presented in the attached table and see which one is more similar.

For example, for the point (0, 25) we have Acos(0) = A.

Then A + h = 25

Of the options presented, the one that best approximates this result is:

`f(x) = -25cos((\pi)/(3)x) + 50`

Because:

`f(0) = -25cos((\pi)/(3)(0)) + 50`

`f(0) -25 +50 = 25`

If we try another point, for example (3, 75) we have:

`f(3) = -25cos((\pi)/(3)(3)) + 50`

We know that
`cos(\pi) = -1`

So:

`f(3) = 25 + 50 = 75`

In point (6, 26) we have:

`f(6) = -25cos((\pi)/(3)(6)) + 50`

`cos(2\pi) = 1`

`f(6) = -25 + 50 = 25`

Finally the answer is the second option