Polar function graphs

To trace $r = f(\theta)$, march θ through its sweep and, at each angle, plot the point at distance r along that direction. Where $f(\theta) = 0$ the pen touches the pole (the origin); where f is large the curve bulges away from it. The polar graph is the trail the pen leaves.

It helps to first sketch r against θ on ordinary rectangular axes, a plain wave, and then read it as driving instructions: wave high means far from the pole, wave at zero means at the pole, wave dipping negative means the pen crosses to the opposite side. But keep the two pictures straight. The rectangular wave is the instruction sheet; the polar curve is what those instructions draw. Handing in the wave as "the graph of r = cos 2θ" is the single most common polar error.

Three equation shapes all draw circles:

1. r = a: every direction, same distance. A circle of radius |a| centered at the pole.
2. r = a cos θ: a circle THROUGH the pole, centered at (a/2, 0) on the x-axis, with diameter |a|. At θ = 0 the pen is at distance a; by θ = π/2 it is back at the pole. The full circle is traced once as θ runs from 0 to π.
3. r = a sin θ: the same circle rotated to sit on the y-axis, centered at (0, a/2), diameter |a|.

The trap in the second and third shapes is reading a as the radius and the pole as the center. Check with two points: r = 6 cos θ gives r = 6 at θ = 0 and r = 0 at θ = π/2, so the circle runs from (6, 0) through the origin: center (3, 0), radius 3.

Equations $r = a\sin(n\theta)$ and $r = a\cos(n\theta)$ (n a whole number, n ≥ 2) draw roses: petal loops radiating from the pole, each petal reaching out to distance |a|.

The petal count follows a parity rule, and it is worth knowing why rather than just memorizing it. When n is odd, the rose has exactly n petals: the sweep from π to 2π retraces the same petals because the negative-r arcs fold onto loops already drawn. When n is even, the negative-r arcs fold onto NEW positions, so the rose has 2n petals. So r = sin 3θ has 3 petals, while r = cos 2θ has 4. The n in the argument is not the petal count itself and it is certainly not multiplied by a; a controls only how long the petals are.

Equations $r = a \pm b\sin\theta$ or $r = a \pm b\cos\theta$ (a, b > 0) draw limaçons, and comparing a to b predicts the shape:

1. a < b: r dips negative for part of the sweep, and those points fold across the pole, drawing a small inner loop.
2. a = b: r just touches 0 once, producing the heart-shaped cardioid with a cusp at the pole.
3. a > b: r never reaches 0; the curve misses the pole entirely (dimpled, or fully convex when a ≥ 2b).

The folding rule is the key mechanic: a point with negative r at angle θ is plotted at distance |r| in the OPPOSITE direction, angle θ + π. Negative r values are not skipped, not undefined, and not "the same point labeled differently": they are real points on the far side of the pole, and they are exactly where inner loops and the extra even-n petals come from.