Question

A sine function has a maximum y value of 8 and a minimum y value of -2. The period of the sine function is 4pi, and there is no phase shift/horizontal translation. What is the equation of the sine function?

Answer 1

Answer:

$\displaystyle y = 5\sin(\frac{1}{2} x) +3$

Step-by-step explanation:

The Sine Function

The general form of the sine function (with no phase shift) is:

$y = A\sin(\omega x) +M$

Where:

A = Amplitude

ω = angular frequency

M= Midline or vertical shift

The midline can be calculated as the mean value of the maximum and minimum values of the oscillation, thus:

$M=\frac{8-2}{2}=3$

The amplitude is half the difference between the maximum and minimum values of the oscillation:

$A=\frac{8+2}{2}=5$

The angular frequency is calculated in terms of the period T as:

$\displaystyle \omega=\frac{2\pi}{T}$

Since T=4π:

$\displaystyle \omega=\frac{2\pi}{4 \pi}=\frac{1}{2}$

Substituting in the general form of the sine function:

$\boxed{\displaystyle y = 5\sin(\frac{1}{2} x) +3}$

The first choice is correct