what is the general equation of a sine function with an amplitude of 6, a period of \\( \\frac{\\pi}{4} \\)…

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) is related to the period ((T=\frac{2\pi}{B})), (C) is the horizontal shift, and (D) is the vertical shift.

Step2: Determine the value of (A)

Given the amplitude (A = 6).

Step3: Determine the value of (B)

Given the period (T=\frac{\pi}{4}). Using the formula (T=\frac{2\pi}{B}), we substitute (T) into it: (\frac{\pi}{4}=\frac{2\pi}{B}). Cross - multiply gives (B\pi = 8\pi), so (B = 8).

Step4: Determine the value of (C)

Given the horizontal shift (C=\frac{\pi}{2}).

Step5: Substitute (A), (B), and (C) into the general form

Substitute (A = 6), (B = 8), and (C=\frac{\pi}{2}) into (y = A\sin(B(x - C))) (since there is no vertical shift (D = 0)), we get (y=6\sin\left(8\left(x-\frac{\pi}{2}\right)\right))

Answer:

(y = 6\sin\left(8\left(x-\frac{\pi}{2}\right)\right)) (the third option in the given choices)