Given:

The population of a city risen and fallen over a 20-year interval. Its populations may be modeled by the function \(y = 12000 + 8000 \sin (0.628x)\), where domain is the year since 1980 and the range is the population of the city.

Calculation:

The given function is \(y = 12000 + 8000 \sin (0.628x).\)

Comparing, the function with the standard form of the sinusoidal function \(y = A \sin (Bx - C) + D\)

\(A = 8000, B = 0.628, C = 0, D = 12000\)

The amplitude is \(A = 8000\)

The periodis given by

\(\frac{2 \pi}{B}\)

\(\frac{2 \pi}{0.628}= \frac{3.14}{0.314}\)

\(= 10\)

And the phase shift is \(C = 0\)

Using these properties, the graph of the function in the domain [0,40] has been shown below

The population of a city risen and fallen over a 20-year interval. Its populations may be modeled by the function \(y = 12000 + 8000 \sin (0.628x)\), where domain is the year since 1980 and the range is the population of the city.

Calculation:

The given function is \(y = 12000 + 8000 \sin (0.628x).\)

Comparing, the function with the standard form of the sinusoidal function \(y = A \sin (Bx - C) + D\)

\(A = 8000, B = 0.628, C = 0, D = 12000\)

The amplitude is \(A = 8000\)

The periodis given by

\(\frac{2 \pi}{B}\)

\(\frac{2 \pi}{0.628}= \frac{3.14}{0.314}\)

\(= 10\)

And the phase shift is \(C = 0\)

Using these properties, the graph of the function in the domain [0,40] has been shown below