4. $y=-2+sin(x)$\nphase shift:______ period:______\nvertical shift:______ amplitude:______

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) determines the period ((T=\frac{2\pi}{|B|})), (C) is the phase - shift, and (D) is the vertical shift. For the function (y=-2+\sin(x)), we can rewrite it as (y = 1\sin(1(x - 0))- 2).

Step2: Calculate the amplitude

The amplitude (A) is given by (|A|). Here, (A = 1), so the amplitude is (|1|=1).

Step3: Calculate the period

Using the formula (T=\frac{2\pi}{|B|}), with (B = 1), we have (T=\frac{2\pi}{|1|}=2\pi).

Step4: Determine the phase - shift

The phase - shift is given by (C). Since (C = 0), the phase - shift is (0).

Step5: Determine the vertical shift

The vertical shift is given by (D). Here, (D=-2), so the vertical shift is (-2).

Answer:

Phase shift: (0) Period: (2\pi) Vertical shift: (-2) Amplitude: (1)