a polar function is given by r = f(θ) = 3 sin(πθ)+6. as θ increases on the interval 0 < θ < 0.5, which of…

a polar function is given by r = f(θ) = 3 sin(πθ)+6. as θ increases on the interval 0 < θ < 0.5, 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 on the and are because on the the graph is result, on the polar plane the points on the graph are getting the origin. below the x - axis above the r - axis above the x - axis below the r - axis submit answer still stuck

a polar function is given by r = f(θ) = 3 sin(πθ)+6. as θ increases on the interval 0 < θ < 0.5, 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 on the and are because on the the graph is result, on the polar plane the points on the graph are getting the origin. below the x - axis above the r - axis above the x - axis below the r - axis submit answer still stuck

Answer

Explanation:

Step1: Analyze the range of $\sin(\pi\theta)$

For $0 < \theta<0.5$, let $u = \pi\theta$. Then $0<u < 0.5\pi$. The sine - function $y = \sin(u)$ is positive for $0 < u<0.5\pi$.

Step2: Analyze the value of $r = f(\theta)$

We have $r=f(\theta)=3\sin(\pi\theta)+6$. Since $\sin(\pi\theta)>0$ for $0 < \theta<0.5$, then $3\sin(\pi\theta)>0$ and $r = 3\sin(\pi\theta)+6>6>0$. In polar coordinates, $r$ represents the distance from the origin. A positive $r$ value means the points are on the ray in the direction of the angle $\theta$ from the origin. Also, for polar coordinates, when $r>0$, the points are above the $r = 0$ line (in the context of polar - coordinate orientation).

Step3: Determine the position relative to the origin

Since $r = 3\sin(\pi\theta)+6>0$, the points are getting farther from the origin as $\theta$ increases in the interval $0 < \theta<0.5$.

Answer:

The points are positive because they lie above the r - axis on the polar plane and are getting farther from the origin because on the given interval the graph has positive $r$ values.