Question

state the period, amplitude, phase shift, and vertical shift of the function.
$y = \cos(3x - \pi)$
period:
amplitude:
phase shift:
vertical shift:

Explanation:

Step1: Recall the general form of cosine function

The general form of a cosine function is \(y = A\cos(Bx - C)+D\). For the function \(y=\cos(3x-\pi)\), we have \(A = 1\), \(B = 3\), \(C=\pi\), \(D = 0\).

Step2: Calculate the period

The formula for the period of \(y = A\cos(Bx - C)+D\) is \(T=\frac{2\pi}{|B|}\). Substituting \(B = 3\) into the formula, we get \(T=\frac{2\pi}{3}\).

Step3: Determine the amplitude

The amplitude is given by \(|A|\). Since \(A = 1\), the amplitude \(|A|=1\).

Step4: Calculate the phase - shift

The formula for the phase - shift is \(\frac{C}{B}\). Substituting \(C=\pi\) and \(B = 3\) into the formula, we get \(\frac{\pi}{3}\).

Step5: Determine the vertical shift

The vertical shift is given by \(D\). Since \(D = 0\), the vertical shift is \(0\).

Answer:

Period: \(\frac{2\pi}{3}\)
Amplitude: \(1\)
Phase Shift: \(\frac{\pi}{3}\)
Vertical Shift: \(0\)