2. graph one complete primary period of the function $y = 125\\sin(\\frac{1}{8}t + \\frac{\\pi}{4}) + 175$…

2. graph one complete primary period of the function $y = 125\\sin(\\frac{1}{8}t + \\frac{\\pi}{4}) + 175$ and provide the requested information (use interval notation as appropriate).\namplitude:\nvertical shift:\naverage value:\nhorizontal shift:\nmaximum value:\nendpoints of the primary interval:\nminimum value:\ndomain:\nperiod:\nrange:

2. graph one complete primary period of the function $y = 125\\sin(\\frac{1}{8}t + \\frac{\\pi}{4}) + 175$ and provide the requested information (use interval notation as appropriate).\namplitude:\nvertical shift:\naverage value:\nhorizontal shift:\nmaximum value:\nendpoints of the primary interval:\nminimum value:\ndomain:\nperiod:\nrange:

Answer

Explanation:

Step1: Find the amplitude

For a sine function (y = A\sin(Bt - C)+D), the amplitude is (|A|). Here (A = 125), so the amplitude is (|125|=125).

Step2: Find the vertical shift

The vertical shift is (D). Here (D = 175), so the vertical shift is (175).

Step3: Find the average value

The average value of a sine function (y = A\sin(Bt - C)+D) is (D). So the average value is (175).

Step4: Find the horizontal shift

For (y = A\sin(Bt - C)+D), the horizontal shift is (\frac{C}{B}). Here (B=\frac{1}{8}) and (C =-\frac{\pi}{4}) (since (y=125\sin(\frac{1}{8}t+\frac{\pi}{4})+175=125\sin(\frac{1}{8}(t + 2\pi))+175)), so the horizontal shift is (- 2\pi).

Step5: Find the maximum value

The maximum value of (y = A\sin(Bt - C)+D) is (A + D). So (125+175 = 300).

Step6: Find the minimum value

The minimum value of (y = A\sin(Bt - C)+D) is (-A+D). So (- 125 + 175=50).

Step7: Find the period

The period of (y = A\sin(Bt - C)+D) is (T=\frac{2\pi}{|B|}). Here (B=\frac{1}{8}), so (T=\frac{2\pi}{\frac{1}{8}}=16\pi).

Step8: Find the endpoints of the primary interval

For (y=\sin t), the primary interval is ([0,2\pi]). For (y = 125\sin(\frac{1}{8}t+\frac{\pi}{4})+175), we set (\frac{1}{8}t+\frac{\pi}{4}=0) and (\frac{1}{8}t+\frac{\pi}{4}=2\pi). Solving (\frac{1}{8}t+\frac{\pi}{4}=0) gives (t=-2\pi), and solving (\frac{1}{8}t+\frac{\pi}{4}=2\pi) gives (t = 14\pi). So the endpoints are ([-2\pi,14\pi]).

Step9: Find the domain

The domain of a sine function (y = 125\sin(\frac{1}{8}t+\frac{\pi}{4})+175) is all real numbers, ((-\infty,\infty)).

Step10: Find the range

Since the minimum value is (50) and the maximum value is (300), the range is ([50,300]).

Answer:

Amplitude: (125) Vertical Shift: (175) Average Value: (175) Horizontal Shift: (-2\pi) Maximum Value: (300) Minimum Value: (50) Period: (16\pi) Endpoints of the Primary Interval: ([-2\pi,14\pi]) Domain: ((-\infty,\infty)) Range: ([50,300])