Question

5 matching 1 point find the key features of the function ( f(x)=8 cos left(\frac{pi}{6} x
ight) ). match the correct value from the list with the key feature. a. 3 b. 8 c. 12 d. ( \frac{1}{8} ) e. ( \frac{1}{12} ) f. ( \frac{pi}{3} ) g. ( x = 0 ) h. ( y = 0 ) amplitude period frequency midline previous

Explanation:

Step1: Recall the general form of a cosine function

The general form of a cosine function is \(y = A\cos(Bx - C)+D\). For the function \(f(x)=8\cos(\frac{\pi}{6}x)\), we have \(A = 8\), \(B=\frac{\pi}{6}\), \(C = 0\), \(D=0\).

Step2: Calculate the amplitude

The amplitude of a cosine function \(y = A\cos(Bx - C)+D\) is given by \(|A|\). Since \(A = 8\), the amplitude is \(|8|=8\).

Step3: Calculate the period

The period of a cosine function \(y = A\cos(Bx - C)+D\) is given by \(T=\frac{2\pi}{|B|}\). Here, \(B=\frac{\pi}{6}\), so \(T=\frac{2\pi}{\frac{\pi}{6}}= 12\).

Step4: Calculate the frequency

The frequency \(f\) of a cosine function is the reciprocal of the period. Since \(T = 12\), \(f=\frac{1}{T}=\frac{1}{12}\).

Step5: Determine the mid - line

The mid - line of a cosine function \(y = A\cos(Bx - C)+D\) is \(y = D\). Here, \(D = 0\), so the mid - line is \(y = 0\).

Answer:

amplitude: b. 8
period: c. 12
frequency: e. \(\frac{1}{12}\)
midline: h. \(y = 0\)