the figure shows the graph of the polar function r = f(θ), where f(θ)=4cos(2θ), in the polar coordinate…

Explanation:

Step1: Analyze polar function symmetry

The polar function $r = 4\cos(2\theta)$ is symmetric about the polar - axis ($\theta = 0$) and the lines $\theta=\frac{\pi}{2},\theta=\pi,\theta=\frac{3\pi}{2}$. When the domain is restricted to $0\leq\theta\leq\frac{\pi}{2}$, we know that the graph of $r = 4\cos(2\theta)$ has symmetry properties. The function $y = \cos(2\theta)$ has a period of $\pi$. In polar coordinates, for $r = 4\cos(2\theta)$, when $\theta$ ranges from $0$ to $\frac{\pi}{2}$, the graph is symmetric about the line $\theta=\frac{\pi}{4}$.

Step2: Determine the remaining piece

Since one piece is in Quadrant I from $C$ to $E$, due to the symmetry of the four - petal rose curve $r = 4\cos(2\theta)$ about the line $\theta=\frac{\pi}{4}$ in the domain $0\leq\theta\leq\frac{\pi}{2}$, the other piece must be in Quadrant II from $E$ to $A$.

Answer:

B. The portion of the graph in Quadrant II from $E$ to $A$