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

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

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

Answer

Explanation:

Step1: Analyze the cosine - function behavior

We know that the function (y = \cos\theta) is decreasing on the interval (\frac{\pi}{2}<\theta<\pi). The given polar function is (r=\cos\theta - 6).

Step2: Determine the sign of (r)

When (\theta=\frac{\pi}{2}), (r=\cos\frac{\pi}{2}-6=0 - 6=-6). When (\theta=\pi), (r=\cos\pi-6=-1 - 6=-7). Since (\cos\theta) is decreasing on (\frac{\pi}{2}<\theta<\pi), (r = \cos\theta-6) is also decreasing on this interval and (r<0) for (\frac{\pi}{2}<\theta<\pi). In polar coordinates, when (r < 0), the points ((r,\theta)) are plotted in the opposite - direction of the angle (\theta). As (\theta) increases from (\frac{\pi}{2}) to (\pi) (moving counter - clockwise in the second quadrant), since (r<0), the points of the graph in the (xy) - plane are moving closer to the origin (because (|r|) is getting larger as (\cos\theta) decreases from (0) to (- 1) and (r=\cos\theta - 6) is getting more negative) and are moving from right to left.

The points are moving closer to the origin because they lie in the second quadrant (due to (\frac{\pi}{2}<\theta<\pi)) and are moving from right to left because on the interval (\frac{\pi}{2}<\theta<\pi) from left to right, the graph is moving closer to the origin.

Answer:

The points are moving closer to the origin because they lie in the second quadrant and are moving from right to left because on the interval (\frac{\pi}{2}<\theta<\pi) from left to right, the graph is moving closer to the origin.