12 mark for review a polar function is given by r = f(θ) = -1 + sin θ. as θ increases on the interval 0 < θ…

Explanation:

Step1: Analyze the sign of (r)

For (0 <\theta<\frac{\pi}{2}), we know that (\sin\theta\in(0,1)). Then (r=- 1+\sin\theta). Since (\sin\theta\in(0,1)), (r=-1 + \sin\theta<0) when (0 <\theta<\frac{\pi}{2}). In polar - coordinates, when (r < 0), the points ((r,\theta)) are plotted in the opposite direction of the terminal side of the angle (\theta). So the points are above the (x) - axis.

Step2: Analyze the change of (|r|)

We take the derivative of (r) with respect to (\theta), (r'(\theta)=\cos\theta). For (0 <\theta<\frac{\pi}{2}), (\cos\theta>0), which means (r) is increasing on the interval ((0,\frac{\pi}{2})). Since (r=-1+\sin\theta) and (r<0) on ((0,\frac{\pi}{2})), as (r) increases (gets less negative), (|r|) is decreasing. The distance from the origin in polar - coordinates is (|r|), so the points are getting closer to the origin.

Answer:

A. The points on the graph are above the (x) - axis and are getting closer to the origin.