AP PRECALCULUS — UNIT 3B · TRIGONOMETRIC & POLAR FUNCTIONS
3.14A — Polar Function Graphs
Notes — Reading and Sketching r = f(θ)
💡 Learning Objectives (3.14.A Part 1)
By the end of this lesson you will be able to:
- Plot points and sketch graphs of polar equations of the form r = f(θ)
- Identify circles, lines, and rays in polar form
- Recognize and sketch limaçons (cardioids included)
- Relate symmetry of a polar graph to its equation
1. Reading r = f(θ)
A POLAR FUNCTION assigns a distance r to each angle θ. The graph is the set of all points (r, θ) where r = f(θ). To sketch by hand, build a small table of (θ, r) pairs at common angles and connect with a smooth curve.
📘 Example — Plotting r = 1 + cos θ
Compute r at several common angles:
- θ = 0: r = 2
- θ = π/2: r = 1
- θ = π: r = 0
- θ = 3π/2: r = 1
- θ = 2π: r = 2
Plot each (r, θ) on a polar grid and connect — you get a heart-shaped curve called a CARDIOID.
2. Common Polar Curves — Circles
- r = a (constant): a circle of radius |a| centered at the origin
- r = a cos θ: a circle of radius |a|/2 centered at (a/2, 0)
- r = a sin θ: a circle of radius |a|/2 centered at (0, a/2)
These ‘offset’ circles pass through the origin and have their diameter along the x- or y-axis.
3. Common Polar Curves — Lines and Rays
- θ = c: a line through the origin at angle c with the polar axis (technically a line, since both r > 0 and r < 0 are allowed)
- r = a / cos θ: a vertical line x = a
- r = a / sin θ: a horizontal line y = a
4. Limaçons and Cardioids
The general form r = a + b cos θ (or r = a + b sin θ) gives a family of curves called LIMAÇONS. Their shape depends on the ratio |a/b|:
💡 Limaçon Shape Classification
- |a/b| < 1: INNER LOOP — the curve has an interior loop
- |a/b| = 1: CARDIOID — the curve has a sharp ‘dimple’ at the origin (heart-shaped)
- 1 < |a/b| < 2: DIMPLED LIMAÇON — the dimple shrinks but remains
- |a/b| ≥ 2: CONVEX LIMAÇON — no dimple, almost circular
5. Roses
The form r = a cos(nθ) or r = a sin(nθ) gives ROSE CURVES with petals:
- If n is ODD: the rose has n petals
- If n is EVEN: the rose has 2n petals
- Each petal has length |a|
- Cosine roses are symmetric about the polar axis; sine roses are symmetric about θ = π/2
6. Symmetry Tests
Use these to test whether a polar curve has a particular symmetry:
- Symmetric about the polar axis (x-axis): replace θ with −θ. If the equation is unchanged, yes.
- Symmetric about θ = π/2 (y-axis): replace θ with π − θ. If unchanged, yes.
- Symmetric about the pole: replace r with −r OR θ with θ + π. If unchanged, yes.
📘 Example — Symmetry of r = 1 + cos θ
- Test x-axis: cos(−θ) = cos θ, so the equation becomes r = 1 + cos θ — UNCHANGED ✓
- So this cardioid is symmetric about the polar axis (x-axis).
7. Sketching Strategy
- Build a table at common angles (0, π/6, π/4, π/3, π/2, …)
- Note where r = 0 (the curve passes through the pole)
- Identify max and min |r| values
- Use symmetries to fill in the rest of the curve
- Connect smoothly, paying attention to whether r is positive (curve is on the θ ray) or negative (curve is on the θ + π ray)
8. Summary
- Build polar graphs by tabulating (θ, r) pairs and connecting smoothly
- Common forms: circles, limaçons, cardioids, roses
- Limaçon shape is set by |a/b|; cardioid is the boundary case |a/b| = 1
- Rose with n: odd → n petals; even → 2n petals
- Symmetry tests speed up sketching