which of the following is the graph of the polar function (r = f(\theta)), where (f(\theta)=4 - 2sin\theta)…
which of the following is the graph of the polar function (r = f(\theta)), where (f(\theta)=4 - 2sin\theta), in the polar plane with the domain (0,2pi))?
Answer
Explanation:
Step1: Analyze key - points of polar function
The general form of a polar function is $r = a\pm b\sin\theta$ or $r=a\pm b\cos\theta$. For $r = 4 - 2\sin\theta$, when $\theta = 0$, $r=4-2\sin(0)=4$. When $\theta=\frac{\pi}{2}$, $r = 4-2\sin(\frac{\pi}{2})=2$. When $\theta=\pi$, $r=4 - 2\sin(\pi)=4$. When $\theta=\frac{3\pi}{2}$, $r=4-2\sin(\frac{3\pi}{2})=6$.
Step2: Recall polar - graph shapes
The function $r = a - b\sin\theta$ with $a>b>0$ represents a limacon. The limacon has an inner - loop when $\frac{a}{b}< 2$ and no inner - loop when $\frac{a}{b}\geq2$. Here $a = 4$ and $b = 2$, and $\frac{a}{b}=2$, so it is a cardioid - like limacon without an inner - loop.
Step3: Match with graphs
By comparing the calculated values of $r$ at key angles and the general shape of the limacon, we can identify the correct graph.