Question

the graph of a periodic function ( f ) is shown below.
a. what is the period of this function?
b. what is the amplitude of this function?
c. write a function formula for ( f ). (enter \theta\ for ( \theta ).)
( f(\theta)=)

Explanation:

Step1: Determine the general form of the function

The general form of a periodic function (cosine - type) is \(f(\theta)=A\cos(B\theta)+k\). The amplitude \(A\) is the distance from the mid - line to the maximum (or minimum) value. The period \(T=\frac{2\pi}{B}\). The mid - line \(k\) is the average of the maximum and minimum values.

Step2: Find the mid - line \(k\)

The maximum value \(y_{max} = 3\) and the minimum value \(y_{min}=-3\). Then \(k=\frac{y_{max}+y_{min}}{2}=\frac{3+( - 3)}{2}=0\)

Step3: Use the period to find \(B\)

We know the period \(T=\frac{\pi}{2}\). Since \(T = \frac{2\pi}{B}\), then \(\frac{\pi}{2}=\frac{2\pi}{B}\). Cross - multiply gives \(B\pi=4\pi\), so \(B = 4\)

Step4: Write the function formula

Since the function has a maximum at \(\theta = 0\) (cosine - type function), and \(A = 3\), \(k = 0\), \(B=4\), the function formula is \(f(\theta)=3\cos(4\theta)\)

Answer:

\(f(\theta)=3\cos(4\theta)\)