Question

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

Explanation:

Step1: Recall the general form of cosine function

The general form of a cosine function is \(y = A\cos(B(x - C))+D\), where \(A\) is the amplitude, \(B\) affects the period (\(T=\frac{2\pi}{|B|}\)), \(C\) is the phase - shift, and \(D\) is the vertical shift.
For the function \(y = 3\cos(x)\), we can rewrite it as \(y=3\cos(1\times(x - 0))+0\).

Step2: Calculate the amplitude

The amplitude \(A\) is given by \(|A|\). Here, \(A = 3\), so the amplitude is \(|3|=3\).

Step3: Calculate the period

Since \(B = 1\), using the formula \(T=\frac{2\pi}{|B|}\), we have \(T=\frac{2\pi}{|1|}=2\pi\).

Step4: Calculate the phase - shift

The phase - shift is given by \(C\). Here, \(C = 0\), so the phase - shift is \(0\).

Step5: Calculate the vertical shift

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

Answer:

Period: \(2\pi\)
Amplitude: \(3\)
Phase Shift: \(0\)
Vertical Shift: \(0\)