Lesson 3: Graphing Polar Equations

Estimated time: 35-40 minutes

Classic Polar Curves

Circles: r = a, r = a cosθ, r = a sinθ

Cardioids: r = a ± a cosθ or r = a ± a sinθ (heart shape)

Limacons: r = a ± b cosθ or r = a ± b sinθ (with/without inner loop depending on a/b)

Rose curves: r = a cos(nθ) or r = a sin(nθ) — n petals if n is odd, 2n petals if n is even

Lemniscates: r² = a² cos(2θ) or r² = a² sin(2θ) — figure-eight shape

This is r = a + a cosθ with a = 2. Symmetric about polar axis.

θ=0: r=4. θ=π/2: r=2. θ=π: r=0. θ=3π/2: r=2.

Heart-shaped curve passing through pole at θ=π, maximum r=4 at θ=0.

n=2 (even), so 2n = 4 petals. Maximum r = 3.

Petals at θ=π/4, 3π/4, 5π/4, 7π/4. Graph from 0 to 2π.

n=3 (odd), so 3 petals. Maximum r = 4. Graph from 0 to π.

Since b > a (2 > 1), this has an inner loop. The inner loop occurs where r < 0.

Polar axis (x-axis): Replace θ with −θ. If equation unchanged, symmetric.

Line θ=π/2 (y-axis): Replace θ with π−θ. If unchanged, symmetric.

Pole (origin): Replace r with −r. If unchanged, symmetric.

1. How many petals does r = 5 sin(4θ) have?

n=4 (even), so 8 petals.

2. What type of curve is r = 3 − 3sinθ?

Cardioid (a = b = 3).

3. Does r = 2cosθ have an inner loop?

No. r = 2cosθ is a circle, not a limacon.