Graphs of Polar Equations
A radar sweep traces a curve by reporting one distance for each angle as it turns. That is exactly what a polar equation does: feed it an angle, and it hands back a radius. Plot every such point and a shape appears, often a circle, a heart-shaped curve, or a flower of petals that no y = f(x) formula draws as cleanly.
By the end of this page you can:
- Build a polar graph by reading the radius the equation gives at each angle.
- Recognize circles, limacons, cardioids, and roses from the form of the equation.
- Predict the number of rose petals from the coefficient on theta.
The floor. A point (r, theta) sits at distance r from the origin along the ray at angle theta. To graph a polar equation, sweep theta and plot the r it returns.
Quick Reference
| Field | Value |
|---|---|
| Textbook | Stewart, Redlin, Watson, Precalculus 7e |
| Chapter | 8 |
| Section | 8.2 Graphs of Polar Equations |
| Open alternate | OpenStax Precalculus 2e, 10.4 Polar Coordinates: Graphs |
| Difficulty | Core |
| Time | 30 minutes |
Before You Start
The Idea
A polar equation states the radius as a rule in terms of the angle, written r = f(theta). To graph it, you walk theta from 0 to 2 pi, compute r at convenient angles, plot each (r, theta), and connect the points in the order the angle increases. When r comes out negative, the point lands on the opposite ray, at angle theta + pi.
Three families cover most of what you meet. A circle comes from r = a (centered at the origin) or from r = a sin theta and r = a cos theta (passing through the origin). A limacon comes from r = a + b sin theta or r = a + b cos theta; when the absolute values of a and b are equal it pinches to a point at the origin and is called a cardioid for its heart shape. A rose comes from r = a sin(n theta) or r = a cos(n theta), and the coefficient n controls how many petals appear.
Key Formulas
Limacons and cardioids: \[ r = a + b\sin\theta \qquad r = a + b\cos\theta \] The curve is a cardioid exactly when \(|a| = |b|\).
Roses, with n a positive integer: \[ r = a\sin(n\theta) \qquad r = a\cos(n\theta) \] The rose has \(n\) petals when \(n\) is odd, and \(2n\) petals when \(n\) is even.
Source: Stewart 7e, Section 8.2 Graphs of Polar Equations.
Worked Example
Identify and sketch the curve r = 1 + cos theta, then state how many petals the curve r = 4 sin(3 theta) has.
For r = 1 + cos theta, read the form a + b cos theta with a = 1 and b = 1. Since the absolute values are equal, this is a cardioid. Build a short table:
| theta | cos theta | r = 1 + cos theta |
|---|---|---|
| 0 | 1 | 2 |
| pi/2 | 0 | 1 |
| pi | -1 | 0 |
| 3pi/2 | 0 | 1 |
The curve reaches farthest out (r = 2) along the positive x-axis and pinches to the origin at theta = pi, giving the heart pointing left to right.
For r = 4 sin(3 theta), the coefficient on theta is n = 3.
Show the petal count
Because n = 3 is odd, the rose has n = 3 petals (not 2n = 6). Each petal reaches length 4. A quick check: petals occur where sin(3 theta) hits 1, and over one full turn the odd coefficient retraces the same three petals rather than drawing six distinct ones.
Connections
This builds on plotting polar coordinates, since every point on a polar graph is a (r, theta) pair, and on the sine and cosine graphs, whose shape drives how r grows and shrinks as the angle turns. It leads toward complex numbers in polar form and toward area and arc length of polar curves in calculus, where these same families reappear as regions to measure.
Resources
- Stewart, Redlin, Watson, Precalculus 7e, Section 8.2 Graphs of Polar Equations (primary).
- OpenStax Precalculus 2e, Section 10.4 Polar Coordinates: Graphs (alternate).
Last updated: 2026-06-24