Question

find the amplitude and the period for ( y = - 3 cos \frac { 1 } { 4 } x ). write the period as an exact value involving ( pi ).

Explanation:

Step1: Recall the formula for the period of a cosine function

The general form of a cosine function is \(y = A\cos(Bx + C)+D\). The period \(T\) is given by \(T=\frac{2\pi}{|B|}\).

Step2: Identify the value of \(B\) in the given function

For the function \(y=-3\cos(\frac{1}{4}x)\), we have \(B = \frac{1}{4}\).

Step3: Calculate the period

Substitute \(B=\frac{1}{4}\) into the period formula \(T=\frac{2\pi}{|B|}\). Since \(|B|=\frac{1}{4}\), then \(T=\frac{2\pi}{\frac{1}{4}}\). Using the rule for dividing by a fraction (\(\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}\)), we get \(T = 2\pi\times4=8\pi\).

Answer:

Amplitude: \(3\)
Period: \(8\pi\)