a polar function is given by r = f(θ) = 2 sin (1/2 θ) + 5. as θ increases on the interval -2π < θ < -π…

a polar function is given by r = f(θ) = 2 sin (1/2 θ) + 5. as θ increases on the interval -2π < θ < -π, which of the following is true about the points of the graph of r = f(θ)? answer attempt 1 out of 2 the points are positive because they lie above the r - axis on the polar plane and are decreasing because on the polar plane from left to right the graph is going down. as a result, on the polar plane the distance from f(θ) to the origin is increasing decreasing submit answer

a polar function is given by r = f(θ) = 2 sin (1/2 θ) + 5. as θ increases on the interval -2π < θ < -π, which of the following is true about the points of the graph of r = f(θ)? answer attempt 1 out of 2 the points are positive because they lie above the r - axis on the polar plane and are decreasing because on the polar plane from left to right the graph is going down. as a result, on the polar plane the distance from f(θ) to the origin is increasing decreasing submit answer

Answer

Explanation:

Step1: Analyze the polar - function

We have (r = f(\theta)=2\sin(\frac{1}{2}\theta)+5).

Step2: Consider the interval (-2\pi<\theta<-\pi)

Let (u = \frac{1}{2}\theta), then when (\theta=-2\pi), (u =-\pi) and when (\theta =-\pi), (u=-\frac{\pi}{2}). The function becomes (r = 2\sin(u)+5). The sine - function (y = \sin(u)) is decreasing on the interval (-\pi<u<-\frac{\pi}{2}).

Step3: Analyze the behavior of (r)

Since (r = 2\sin(u)+5) and (\sin(u)) is decreasing on (-\pi<u<-\frac{\pi}{2}), and the coefficient of (\sin(u)) is positive ((2>0)), the function (r = 2\sin(u)+5) is also decreasing on the corresponding (\theta) - interval (-2\pi<\theta<-\pi). In polar coordinates, (r) represents the distance from the point ((r,\theta)) to the origin.

Answer:

decreasing