AP PRECALCULUS — UNIT 3B · TRIGONOMETRIC & POLAR FUNCTIONS
3.14B — Polar Function Graphs
Notes — Behavior, Zeros, and Intersections
💡 Learning Objectives (3.14.A Part 2)
By the end of this lesson you will be able to:
- Determine intervals where r is increasing or decreasing as θ increases
- Identify the values of θ at which r = 0 (the curve crosses the pole)
- Find the angles where r reaches its maximum or minimum value
- Find points of intersection between two polar curves
1. r as a Function of θ
If you ignore the polar interpretation and just plot r vs. θ on Cartesian axes, you get the ‘rectangular trace’ of the polar function. This is a useful intermediate picture for understanding the polar shape:
- Where r is INCREASING: as θ grows, the curve moves AWAY from the pole
- Where r is DECREASING: as θ grows, the curve moves TOWARD the pole
- Where r = 0: the curve passes through the pole
- Where r is at a maximum: the curve is at its FARTHEST point on that ray
2. Worked Analysis — r = 2 + 3 sin θ (Inner Loop Limaçon)
📘 Example — Behavior over [0, 2π)
- At θ = 0: r = 2
- At θ = π/2: r = 5 (maximum)
- At θ = π: r = 2
- Near θ = 7π/6: r = 2 + 3(−1/2) = 1/2
- At θ ≈ 7π/6 (or so): r = 0 — find by 2 + 3 sin θ = 0, so sin θ = −2/3, giving θ ≈ 3.871 or 5.553
- Between those two angles, r < 0 — that is the INNER LOOP, traced ‘inside out’
- At θ = 3π/2: r = −1 (minimum)
3. Where Does the Curve Pass Through the Pole?
Setting r = 0 and solving gives the angles where the curve crosses the pole. These are important features for sketching and for finding intersections.
📘 Example — Find pole crossings of r = 1 − 2 cos θ
- Solve 1 − 2 cos θ = 0
- cos θ = 1/2, so θ = π/3 or 5π/3 in [0, 2π)
- The cardioid passes through the pole at these two angles.
4. Maximum and Minimum r Values
To find where r reaches its largest or smallest value, treat r = f(θ) as a function and find where it's maximized/minimized over the requested θ-interval:
- For r = a + b cos θ with b > 0: max is at θ = 0 (cos = 1), value a + b; min at θ = π (cos = −1), value a − b
- For r = a + b sin θ with b > 0: max at θ = π/2, value a + b; min at θ = 3π/2, value a − b
- For roses r = a cos(nθ): max |r| = |a|, occurs at the petal tips
5. Intersections of Two Polar Curves
To find the points where two polar curves r = f(θ) and r = g(θ) cross, set them equal and solve for θ:
f(θ) = g(θ)
⚠️ Important caveat
Polar coordinates are NOT unique, so two curves can cross at a point even if they don't reach that point at the SAME θ-value. Always check the POLE (origin) separately by seeing whether each curve has any θ for which r = 0.
📘 Example — Find intersections of r = 1 + cos θ and r = 1 − cos θ
- Set equal: 1 + cos θ = 1 − cos θ
- So cos θ = 0, giving θ = π/2 and 3π/2
- At θ = π/2: r = 1 + 0 = 1 — point (1, π/2). Both curves agree here.
- At θ = 3π/2: r = 1, same point. (Same intersection.)
- Pole check: 1 + cos θ = 0 ⇒ θ = π; 1 − cos θ = 0 ⇒ θ = 0. Both curves pass through the pole at different θ — but the pole IS shared, so it's an intersection point too.
6. Reading Behavior from a Graph
Given a sketch of a polar curve, you can describe its features:
- Where the curve is FARTHEST from the pole (max r) — corresponds to a peak in the rectangular trace
- Where it's CLOSEST or AT the pole (r = 0) — corresponds to a zero of the rectangular trace
- Where it's moving toward (decreasing r) or away from (increasing r) the pole as θ grows
- Symmetries about axes or the pole, identified visually
7. Tangent Lines at the Pole
If a polar curve passes through the pole at θ = θ₀ (i.e., r(θ₀) = 0), then the tangent line to the curve at the pole is θ = θ₀. So a rose r = sin(2θ) crosses the pole and has tangent lines along θ = 0, π/2, π, 3π/2, etc.
8. Summary
- As θ increases, increasing r means moving away from the pole, decreasing r means moving toward it
- Setting r = 0 finds where the curve passes through the pole; this also gives tangent directions
- Maximum/minimum r values determine the curve's farthest and closest reach
- Intersections: solve f(θ) = g(θ), and ALSO check the pole separately
- A rectangular trace of r vs. θ is a useful intermediate step before sketching