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

Explanation:

Step1: Recall the general form of a 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 phase - shift, and (D) is the vertical shift.

Step2: Determine the value of (A)

Given that the amplitude (A = 5).

Step3: Determine the value of (B)

Since the period (T = 2\pi), and (T=\frac{2\pi}{B}). Substituting (T = 2\pi) into the formula (T=\frac{2\pi}{B}), we get (2\pi=\frac{2\pi}{B}), then (B = 1).

Step4: Determine the value of (C)

Given that the phase - shift (C=\frac{1}{5}).

Step5: Determine the value of (D)

Since there is no vertical shift, (D = 0).

Step6: Write the equation of the sine function

Substitute (A = 5), (B = 1), (C=\frac{1}{5}), and (D = 0) into the general form (y=A\sin(B(x - C))+D). We get (y = 5\sin\left(1\times\left(x-\frac{1}{5}\right)\right)+0), which simplifies to (y = 5\sin\left(x-\frac{1}{5}\right)).

Answer:

(y = 5\sin\left(x-\frac{1}{5}\right))