When drawing a polar curve, if the value of r is negative, what will happen?

Answer 1

|r| is the distance from the origin to the point on the function. If r is negative, then one should move this distance not in the direction of theta from the origin, but in the direction of - theta, which lies in the opposite quadrant, relative to the origin.

For example, let r = cos(theta), then the graph of r will be a circle or radius 1/2 centered at (1/2,0), which is only plotted in quadrants 1 and 4. This is because as theta runs from 0 to p/2, cos(theta) is non-negative, and the top part of the circle is graphed in Q1. As theta runs from pi / 2 to pi, that is, through the Q2 values, cos(theta) is negative, and so the graph of r is plotted in quadrant 4, the bottom half of the circle. Similarly, for Q3 values of theta, r is negative, and the top half of the circle is graphed (again), and for Q4 values of theta, r is positive, and the bottom half of the circle is graphed (again). Thus, as theta runs from 0 to 2pi, the circle is graphed twice, but never appears in quadrants 2 or 3..