4. graph the primary period of the function $f(t)=-90\\sin(\\frac{\\pi}{10}t+\\frac{3\\pi}{5})-120$ and…

4. graph the primary period of the function $f(t)=-90\\sin(\\frac{\\pi}{10}t+\\frac{3\\pi}{5})-120$ 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:

4. graph the primary period of the function $f(t)=-90\\sin(\\frac{\\pi}{10}t+\\frac{3\\pi}{5})-120$ 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: Identify the general form of the sine function

The general form of a sine function is (y = A\sin(Bt - C)+D). For the given function (f(t)=- 90\sin(\frac{\pi}{10}t+\frac{3\pi}{5})-120), we can rewrite it as (f(t)=-90\sin(\frac{\pi}{10}(t + 6))-120)

Step2: Calculate the amplitude

The amplitude (|A|). Here (A=-90), so (|A| = 90)

Step3: Calculate the period

The period (T=\frac{2\pi}{|B|}). Since (B = \frac{\pi}{10}), then (T=\frac{2\pi}{\frac{\pi}{10}}=20)

Step4: Calculate the vertical shift

The vertical shift is (D). Here (D=-120)

Step5: Calculate the horizontal shift

The horizontal shift is (-\frac{C}{B}). From (y=-90\sin(\frac{\pi}{10}(t + 6))-120), the horizontal shift is (-6)

Step6: Calculate the average value

The average value is the vertical shift, which is (-120)

Step7: Calculate the maximum and minimum values

The maximum value (=D + |A|=-120 + 90=-30) The minimum value (=D-|A|=-120-90=-210)

Step8: Find the endpoints of the primary interval

For the sine function (y = A\sin(Bt - C)+D), the primary interval for (\sin u) (where (u = Bt - C)) is (0\leq u\leq2\pi). Set (\frac{\pi}{10}(t + 6)=0), then (t=-6) Set (\frac{\pi}{10}(t + 6)=2\pi), then (t = 14)

Step9: Determine the domain and range

The domain of the primary period is ([-6,14]) The range is ([-210,-30])

Answer:

Amplitude: (90) Vertical Shift: (-120) Average Value: (-120) Horizontal Shift: (-6) Maximum Value: (-30) Minimum Value: (-210) Period: (20) Endpoints of the Primary Interval: (-6) and (14) Domain: ([-6,14]) Range: ([-210,-30])