Question

1. graph one complete primary period of the function ( y = 100 cos ( \frac { pi } { 4 } t + pi ) - 50 ) and provide the requested information (use interval notation as appropriate).

amplitude:
average value:
maximum value:
minimum value:
period:
vertical shift:
horizontal shift:
endpoints of the primary interval:
domain:
range:

Explanation:

Step1: Recall the general form of cosine function

The general form of a cosine function is \(y = A\cos(B(t - C))+D\). For the function \(y = 100\cos(\frac{\pi}{4}t+\pi)-50\), we can rewrite it as \(y = 100\cos(\frac{\pi}{4}(t + 4))-50\). Here, \(A = 100\), \(B=\frac{\pi}{4}\), \(C=- 4\), \(D=-50\).

Step2: Calculate the amplitude

The amplitude of a cosine function \(y = A\cos(B(t - C))+D\) is given by \(|A|\). So, the amplitude \(=\vert100\vert = 100\).

Step3: Calculate the period

The period of a cosine function \(y = A\cos(B(t - C))+D\) is \(T=\frac{2\pi}{|B|}\). Substituting \(B = \frac{\pi}{4}\), we get \(T=\frac{2\pi}{\frac{\pi}{4}}=8\).

Step4: Calculate the vertical shift

The vertical shift of a cosine function \(y = A\cos(B(t - C))+D\) is \(D\). So, the vertical shift is \(-50\).

Step5: Calculate the horizontal shift

The horizontal shift of a cosine function \(y = A\cos(B(t - C))+D\) is \(C\). So, the horizontal shift is \(-4\) (4 units to the left).

Step6: Calculate the maximum and minimum values

The maximum value of \(y = A\cos(B(t - C))+D\) is \(A + D\). Substituting \(A = 100\) and \(D=-50\), we get \(100-50=50\). The minimum value is \(-A + D\). Substituting \(A = 100\) and \(D=-50\), we get \(-100-50=-150\).

Step7: 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 \(-50\).

Step8: Determine the domain and range

For the primary interval (one - period interval), we consider the standard interval for the cosine function. Since the period \(T = 8\) and the horizontal shift is \(-4\), the endpoints of the primary interval are found by solving \(0\leq\frac{\pi}{4}(t + 4)\leq2\pi\). First, \(0\leq\frac{\pi}{4}(t + 4)\) gives \(t\geq - 4\), and \(\frac{\pi}{4}(t + 4)\leq2\pi\) gives \(t+4\leq8\) or \(t\leq4\). So, the endpoints of the primary interval are \(t=-4\) and \(t = 4\). The domain of the primary interval is \([-4,4]\). The range is \([-150,50]\) (from the minimum and maximum values).

Answer:

Amplitude: \(100\)
Period: \(8\)
Vertical Shift: \(-50\)
Horizontal Shift: \(-4\)
Maximum Value: \(50\)
Minimum Value: \(-150\)
Average Value: \(-50\)
Endpoints of the Primary Interval: \(-4\) and \(4\)
Domain (Primary Interval): \([-4,4]\)
Range: \([-150,50]\)