Unit 6 · Topic 3.14
Polar Function Graphs
Interpret a polar function as a changing angle paired with a signed radial displacement, then use tables, key features, and restricted domains to trace its graph.
Learning Goals
- Interpret the input and output of \(r=f(\theta)\).
- Plot positive, zero, and negative radius values in input order.
- Use a restricted \(\theta\)-domain to identify a selected portion of a polar graph.
- Locate zeros, greatest radial distances, and possible symmetries.
- Recognize and analyze common polar curve families.
1. Read a Polar Function
In a polar function
the input \(\theta\) is an angle measured from the polar axis, and the output \(r\) is a signed radial displacement from the pole. Each input produces the polar point \((f(\theta),\theta)\).
This is different from \(y=f(x)\): \(r\) is not a vertical coordinate, and increasing \(\theta\) rotates the directed ray rather than moving horizontally.
2. Trace the Curve in Input Order
- Choose an input angle \(\theta\).
- Evaluate \(r=f(\theta)\).
- Rotate to the directed angle \(\theta\).
- Move \(r\) units along that ray, reversing direction when \(r<0\).
- Repeat for increasing inputs and connect the points in that order.
3. Interpret the Sign of the Output
| Output | Location |
|---|---|
| \(r>0\) | \(r\) units along the \(\theta\)-ray |
| \(r=0\) | At the pole |
| \(r<0\) | \(|r|\) units opposite the \(\theta\)-ray |
4. A Domain Selects the Part That Is Traced
A restriction \(a\le\theta\le b\) tells the graph where to begin, how the point moves, and where to stop. Two endpoint coordinates can name the same point while the values between them trace an entire arc or loop.
For \(r=2\cos\theta\), the interval \([-\pi/2,\pi/2]\) traces the circle exactly once. Extending the domain to \([0,2\pi]\) traces the same circle twice because negative radii reproduce points already visited.
5. Build a Key-Angle Table
Start with angles where the trigonometric expression has exact values. Add inputs where \(r=0\), where \(|r|\) is greatest, or where its sign changes.
| \(\theta\) | \(r=2\cos\theta\) |
|---|---|
| \(-\pi/2\) | \(0\) |
| \(-\pi/3\) | \(1\) |
| \(0\) | \(2\) |
| \(\pi/3\) | \(1\) |
| \(\pi/2\) | \(0\) |
6. Example: \(r=2\cos\theta\)
The point leaves the pole, reaches \((2,0)\), and returns to the pole. The graph is a circle to the right of the pole.
7. Confirm the Shape Algebraically
Multiply by \(r\), then use \(r^2=x^2+y^2\) and \(r\cos\theta=x\):
The rectangular form confirms a circle centered at \((1,0)\) with radius 1.
8. Negative Radius Can Retrace the Curve
At \(\theta=2\pi/3\),
The point \((-1,2\pi/3)\) lies opposite the \(2\pi/3\)-ray, so it is equivalent to \((1,5\pi/3)\). Its rectangular coordinates are
a point on the lower half of the same circle. Ignoring the sign would place it incorrectly.
9. Find Zeros and Greatest Distance
Zeros occur where \(f(\theta)=0\), so the curve passes through the pole. Distance from the pole is \(|r|\), not \(r\).
For \(r=2\cos\theta\), the greatest distance is 2. The inputs \(0\) and \(\pi\) produce opposite signed radii but the same point \((2,0)\).
10. Test for Possible Symmetry
| Possible symmetry | Substitution test |
|---|---|
| Polar axis | Replace \(\theta\) with \(-\theta\) |
| Line \(\theta=\pi/2\) | Replace \(\theta\) with \(\pi-\theta\) |
| Pole | Replace \(\theta\) with \(\theta+\pi\), or \(r\) with \(-r\) |
An unchanged equation confirms the indicated symmetry. A failed substitution does not by itself prove that the graph lacks symmetry because polar points have multiple representations.
11. Recognize Common Polar Families
| Typical equation | Common shape | Feature to inspect |
|---|---|---|
| \(r=a\cos\theta\) or \(r=a\sin\theta\) | Circle through the pole | Orientation and diameter \(|a|\) |
| \(r=a\pm b\cos\theta\) or \(r=a\pm b\sin\theta\) | Limaçon or cardioid | Zeros and inner loop |
| \(r=a\cos(n\theta)\) or \(r=a\sin(n\theta)\) | Rose | Petal count and length |
| \(r^2=a^2\cos(2\theta)\) or \(r^2=a^2\sin(2\theta)\) | Lemniscate | Where \(r^2\ge0\) |
| \(r=a+b\theta\) | Spiral | Radial change per revolution |
These patterns help form a prediction, but a table and the equation's behavior should justify the graph.
12. Cardioid Example: \(r=2+2\cos\theta\)
| \(\theta\) | \(r\) | Feature |
|---|---|---|
| \(0\) | \(4\) | Farthest point on the polar axis |
| \(\pi/2\) | \(2\) | Upper point |
| \(\pi\) | \(0\) | Cusp at the pole |
| \(3\pi/2\) | \(2\) | Lower point |
| \(2\pi\) | \(4\) | Returns to the start |
Because replacing \(\theta\) by \(-\theta\) leaves the equation unchanged, the cardioid is symmetric about the polar axis and faces right.
13. Rose Example: \(r=3\sin(2\theta)\)
The curve reaches the pole when
Petal tips occur when \(|r|=3\), at \(\theta=\pi/4+k\pi/2\). Negative radii generate petals opposite their directed rays, producing four petals of length 3.
On \(0\le\theta\le\pi/2\), the point moves from the pole to the tip at \((3,\pi/4)\) and back to the pole, tracing one complete petal.
14. Use Technology Purposefully
- Select polar mode and radians.
- Enter \(r\) as a function of \(\theta\).
- Set the requested \(\theta\)-minimum and \(\theta\)-maximum.
- Choose a small enough \(\theta\)-step to show loops and cusps.
- Set the rectangular viewing window using the expected maximum \(|r|\).
- Compare the display with a key-angle table.
15. AP Workflow and Common Errors
- State the domain for \(\theta\).
- Find zeros and inputs where \(|r|\) is greatest.
- Build a table that includes sign changes.
- Plot each signed radius correctly.
- Connect points in increasing-\(\theta\) order.
- Use symmetry or rectangular conversion as a check.
- Describe the traced portion, direction, and endpoints.
- Do not treat \(r\) as a y-coordinate.
- Do not replace a negative radius with its absolute value.
- Do not confuse maximum \(r\) with maximum distance \(|r|\).
- Do not assume \([0,2\pi]\) is always the shortest interval that traces a curve once.
- Do not connect table points in spatial order instead of input order.
Key Takeaways
- Graph polar equations using symmetry, key angles, zeros, and maximum radii.
- Core relationship: \(r=2\cos\theta\)
- Error check: Do not treat r as a vertical coordinate.
- Identify the center and radius of \(r=4\sin\theta\).
- Make a five-angle table for \(r=1-2\cos\theta\) and identify every negative output.
- Explain why \((-2,\pi/6)\) and \((2,7\pi/6)\) name the same point.
- Describe the single petal traced by \(r=3\sin(2\theta)\) on \(0\le\theta\le\pi/2\).
- For \(r=2+2\sin\theta\), find the pole input, greatest radial distance, and axis of symmetry.