Question

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

ight)-5 ) and provide the requested information (use interval notation as appropriate).
amplitude:
vertical shift:
average value:
horizontal shift:
maximum value:
endpoints of the primary interval:
minimum value:
domain:
period:
range:

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: \([-20,10]\)