3. graph one complete primary period of the function ( g(t)=15 cos left(\frac{pi}{6} t+\frac{2…

3. graph one complete primary period of the function ( g(t)=15 cos left(\frac{pi}{6} t+\frac{2 pi}{3}\right)-5 ) 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:

3. graph one complete primary period of the function ( g(t)=15 cos left(\frac{pi}{6} t+\frac{2 pi}{3}\right)-5 ) 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 cosine function

The general form of a cosine function is (y = A\cos(B(t - C))+D). For the function (g(t)=15\cos(\frac{\pi}{6}t+\frac{2\pi}{3}) - 5), we can rewrite it as (g(t)=15\cos(\frac{\pi}{6}(t + 4))-5).

Step2: Calculate the amplitude

The amplitude (A) is given by (|A|). Here, (A = 15), so the amplitude is (15).

Step3: Calculate the period

The period (T) of a cosine function (y = A\cos(B(t - C))+D) is (T=\frac{2\pi}{|B|}). Since (B=\frac{\pi}{6}), then (T=\frac{2\pi}{\frac{\pi}{6}}=12).

Step4: Calculate the vertical shift

The vertical shift (D=- 5).

Step5: Calculate the average value

The average value of a cosine function (y = A\cos(B(t - C))+D) is (D). So the average value is (-5).

Step6: Calculate the maximum value

The maximum value of (y = A\cos(B(t - C))+D) is (A + D). Substituting (A = 15) and (D=-5), we get (15+( - 5)=10).

Step7: Calculate the minimum value

The minimum value of (y = A\cos(B(t - C))+D) is (-A + D). Substituting (A = 15) and (D=-5), we get (-15+( - 5)=-20).

Step8: Calculate the horizontal shift

For (y = A\cos(B(t - C))+D), the horizontal shift is (C). Since (g(t)=15\cos(\frac{\pi}{6}(t + 4))-5), the horizontal shift is (-4).

Step9: Find the endpoints of the primary interval

For the standard cosine function (y=\cos(t)), the primary interval is ([0,2\pi]). For (y = A\cos(B(t - C))+D), we set (B(t - C)=0) and (B(t - C)=2\pi). If (B=\frac{\pi}{6}) and (C=-4), then (\frac{\pi}{6}(t + 4)=0) gives (t=-4), and (\frac{\pi}{6}(t + 4)=2\pi) gives (t = 8). So the endpoints of the primary interval are ([-4,8]).

Step10: Determine the domain

The domain of a cosine function (y = A\cos(B(t - C))+D) is all real numbers, so in interval notation, the domain is ((-\infty,\infty)).

Step11: Determine the range

Since the maximum value is (10) and the minimum value is (-20), the range is ([-20,10]).

Answer:

  • Amplitude: (15)
  • Vertical Shift: (-5)
  • Average Value: (-5)
  • Horizontal Shift: (-4)
  • Maximum Value: (10)
  • Minimum Value: (-20)
  • Period: (12)
  • Endpoints of the Primary Interval: ([-4,8])
  • Domain: ((-\infty,\infty))
  • Range: ([-20,10])