4. write the equation of a cosine function with an amplitude of 4 and period of 4.

Explanation:

Step1: Recall the general form of cosine function

The general form of a cosine function is (y = A\cos(Bx + 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. Given (A = 4) (amplitude) and (T = 4) (period).

Step2: Find the value of (B)

Since (T=\frac{2\pi}{|B|}), and (T = 4). Then (4=\frac{2\pi}{|B|}), solving for (B): [ \begin{align*} |B|&=\frac{2\pi}{4}\ |B|&=\frac{\pi}{2} \end{align*} ] We can take (B=\frac{\pi}{2}) (assuming no phase - shift (C = 0) and no vertical shift (D = 0))

Answer:

(y = 4\cos\left(\frac{\pi}{2}x\right))