write the equation of a sine function that has the following characteristics. assume that there is no…

Explanation:

Step1: Recall the general form of sine function

The general form of a sine function is (y = A\sin(B(x - C))+D), where (A) is the amplitude, (B=\frac{2\pi}{Period}), (C) is the phase - shift, and (D) is the vertical shift.

Step2: Determine the value of (A)

Given amplitude (A = 4).

Step3: Determine the value of (B)

Given period (T=4\pi). Since (B=\frac{2\pi}{T}), substituting (T = 4\pi) gives (B=\frac{2\pi}{4\pi}=\frac{1}{2}).

Step4: Determine the value of (C)

Given phase - shift (C=\frac{1}{6}).

Step5: Determine the value of (D)

Given no vertical shift, so (D = 0).

Step6: Substitute (A), (B), (C), and (D) into the general form

Substituting (A = 4), (B=\frac{1}{2}), (C=\frac{1}{6}), and (D = 0) into (y = A\sin(B(x - C))+D), we get (y=4\sin\left(\frac{1}{2}\left(x-\frac{1}{6}\right)\right)).

Answer:

(y = 4\sin\left(\frac{1}{2}x-\frac{1}{12}\right))