Question

What is the relationship between the graphs of r = f(θ) and r = f(θ − k) in the θr-plane for k > 0? What is the relationship between the two graphs in polar coordinates?

Short answer

In $\theta r$-plane, the graph of $r=f(\theta -k)$is shifted version of the graph of $r=f\left(\theta \right)$by k units towards right.

In polar coordinates the graph of $r=f(\theta -k)$is rotated version of the graph of $r=f\left(\theta \right)$by angle $k$degree in couterclockwise direction.

Step-by-step solution

Step 1. Given information

There are two planes.

The functions are :$$\begin{gathered} r=f\left(\theta \right) \\ r=f(\theta -k) \end{gathered}$$

Step 2. Relation between graphs in both plane. 

In $\theta r$-plane, the graph of $r=f(\theta -k)$is obtained by shifting the graph of $r=f\left(\theta \right)$by k units in horizontal direction towards right.

In polar coordinates the graph of $r=f(\theta -k)$is obtained by rotating the graph of $r=f\left(\theta \right)$by angle $k$degree in anticlockwise direction.