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

question a polar function is given by r = f(θ) = 3 cos(1/4 θ)+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 positive because they lie and are because on the from left to right, the graph above the r - axis above the x - axis below the x - axis below the r - axis submit answer

question a polar function is given by r = f(θ) = 3 cos(1/4 θ)+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 positive because they lie and are because on the from left to right, the graph above the r - axis above the x - axis below the x - axis below the r - axis submit answer

Answer

Explanation:

Step1: Analyze the polar - function range

Let (y = r\sin\theta=(3\cos(\frac{1}{4}\theta)+6)\sin\theta). When (2\pi<\theta < 4\pi), first consider the range of (\cos(\frac{1}{4}\theta)). Let (t=\frac{1}{4}\theta), then (\frac{\pi}{2}<t<\pi). The function (y = 3\cos t+6). Since (- 1\leqslant\cos t\leqslant0) for (\frac{\pi}{2}<t<\pi), we have (3\leqslant3\cos t + 6\leqslant6), so (r = 3\cos(\frac{1}{4}\theta)+6>0).

Step2: Determine the position relative to the x - axis

We know that in polar coordinates, (x = r\cos\theta) and (y = r\sin\theta). To determine the position of the points relative to the (x) - axis, we consider the sign of (y). Let's take some values in the interval (2\pi<\theta<4\pi). For example, when (\theta=\frac{5\pi}{2}), (r = 3\cos(\frac{5\pi}{8})+6) and (y=r\sin(\frac{5\pi}{2})=r>0). In general, for (2\pi<\theta<4\pi), we can analyze the product (y=(3\cos(\frac{1}{4}\theta)+6)\sin\theta). The function (r = 3\cos(\frac{1}{4}\theta)+6>0) in the given interval. And (\sin\theta) has both positive and negative values in the interval ((2\pi,4\pi)), but when considering the overall behavior, we note that the points of the graph of (r = f(\theta)) are above the (x) - axis.

Answer:

The points are positive because they lie above the x - axis and are moving from left to right, the graph is above the x - axis.