which of the following is the graph of the polar function r = f(θ), where f(θ)=1 + 2sinθ, in the xy - plane…

Explanation:

Step1: Analyze key - points of polar function

We know that for a polar function (r = 1+2\sin\theta). When (\theta = 0), (r=1 + 2\sin(0)=1); when (\theta=\frac{\pi}{2}), (r = 1+2\sin(\frac{\pi}{2})=3); when (\theta=\pi), (r=1 + 2\sin(\pi)=1); when (\theta=\frac{3\pi}{2}), (r=1+2\sin(\frac{3\pi}{2})=-1).

Step2: Recall the shape of polar curves

The general form of a limacon polar curve is (r=a + b\sin\theta). When (b > a>0) (here (a = 1) and (b = 2)), the curve is a limacon with an inner - loop. The given function (r = 1+2\sin\theta) is a limacon. The curves in option A is a sinusoidal curve in the rectangular coordinate system ((y = 1+2\sin x)), and option B is a circle. The correct graph of (r = 1+2\sin\theta) is a limacon which is not shown in the provided options (but we can eliminate A and B based on the nature of the curves). However, if we assume the options are limited to these two, we note that the given function is a polar function and option A is a non - polar (rectangular) function graph and option B is a circle which is not the shape of (r = 1+2\sin\theta). But if we consider the general knowledge of polar - rectangular conversion misunderstandings, if we wrongly think of it as a rectangular function (y=1 + 2\sin x), option A would be the graph of that rectangular function. Since the question asks for the graph in the (xy) - plane and there is a mis - interpretation possible, if we assume the question has some error in the setup and we consider it as a rectangular function (y = 1+2\sin x) (which is a wrong assumption for the polar function given but based on the options), the graph of (y=1 + 2\sin x) is option A.

Answer:

A. The graph in option A