question a polar function is given by r = f(θ) = 3 cos (1/4θ) + 6. as θ increases on the interval 2π < θ <…

question a polar function is given by r = f(θ) = 3 cos (1/4θ) + 6. as θ increases on the interval 2π < θ < 4π, which of the following is true about the points of the graph of r = f(θ) on the xy - plane? answer attempt 1 out of 2 the points are because they lie and are because on the right, the graph is negative positive submit answer

question a polar function is given by r = f(θ) = 3 cos (1/4θ) + 6. as θ increases on the interval 2π < θ < 4π, which of the following is true about the points of the graph of r = f(θ) on the xy - plane? answer attempt 1 out of 2 the points are because they lie and are because on the right, the graph is negative positive submit answer

Answer

Explanation:

Step1: Analyze the cosine - function range

Let (u=\frac{1}{4}\theta). When (\theta\in(2\pi,4\pi)), then (u\in(\frac{\pi}{2},\pi)). The cosine function (y = \cos(u)) has the range (- 1<\cos(u)<0) for (u\in(\frac{\pi}{2},\pi)).

Step2: Find the range of (r = f(\theta))

We have (r = 3\cos(\frac{1}{4}\theta)+6). Since (-1 <\cos(\frac{1}{4}\theta)<0) when (\theta\in(2\pi,4\pi)), then (3\times(- 1)+6<3\cos(\frac{1}{4}\theta)+6<3\times0 + 6), which simplifies to (3<r<6).

Answer:

The points are positive because they lie in the (xy -)plane and are positive because on the right, the graph is positive.