3.12 Polar Function Graphs

Polar coordinates basics

A point in the polar coordinate system is written as (r, θ), where r is the distance from the origin (pole) and θ is the angle from the positive x-axis. Points can have multiple representations since adding 2π to θ or using -r with θ+π gives the same location.

Graphing common polar curves

Circles like r = a cos θ or r = a sin θ pass through the pole. Roses r = a cos(nθ) have petals whose count depends on n (odd n → n petals; even n → 2n petals). Limaçons r = a + b cos θ vary shape (with inner loop, cardioid, dimpled, or convex) based on the ratio a/b.

Analyzing polar graphs

Symmetry can be checked: replacing θ with -θ tests x-axis symmetry; replacing θ with π−θ tests y-axis symmetry. Zeros of r (where the graph passes through the pole) occur when r = 0. As θ increases, tracking how r increases/decreases reveals the direction and shape of the curve's tracing.

Key terms

Polar coordinates (r, θ)

r = distance from the pole (origin), θ = angle from the positive x-axis.

Rose curve petal count

r = a cos(nθ) or a sin(nθ): odd n gives n petals; even n gives 2n petals.

Cardioid

A heart-shaped limaçon formed when |a| = |b| in r = a + b cos θ (or sin θ).

Limaçon with inner loop

Occurs when |a| < |b| in r = a + b cos θ.

Finding zeros of r

Set r = 0 and solve for θ to find where the polar graph passes through the pole.

Try an AP-style question

What is the graph of r = 2 sin θ?

1. A circle of diameter 2 centered at (0, 1)
2. A 2-petal rose
3. A cardioid
4. A circle of radius 2 centered at the origin