Step-by-step examples of basic polar equation graphs

📚 Quick Study Guide

• 🧭 Polar Coordinates: A point in the polar coordinate system is represented by $(r, \theta)$, where $r$ is the distance from the origin (pole) and $\theta$ is the angle from the positive x-axis (polar axis).
• 📈 Basic Polar Equations:
  • $r = a$: Circle centered at the pole with radius $|a|$.
  • $\theta = a$: Line passing through the pole, making an angle $a$ with the polar axis.
• ❤️ Cardioids: $r = a \pm a\cos(\theta)$ or $r = a \pm a\sin(\theta)$. The shape resembles a heart.
• 🌹 Roses: $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$.
  • If $n$ is even, the rose has $2n$ petals.
  • If $n$ is odd, the rose has $n$ petals.
• 🌀 Lemniscates: $r^2 = a^2\cos(2\theta)$ or $r^2 = a^2\sin(2\theta)$. The shape resembles a figure eight.

🧪 Practice Quiz

1. What shape does the polar equation $r = 5$ represent?
   1. Line
   2. Circle
   3. Rose
   4. Cardioid
2. Which of the following polar equations represents a line passing through the pole?
   1. $r = 5\cos(\theta)$
   2. $\theta = \frac{\pi}{4}$
   3. $r = 5$
   4. $r = 5 + 5\sin(\theta)$
3. What type of curve is represented by the polar equation $r = 3 + 3\cos(\theta)$?
   1. Lemniscate
   2. Rose
   3. Cardioid
   4. Spiral
4. How many petals does the rose curve $r = 4\sin(2\theta)$ have?
   1. 2
   2. 4
   3. 6
   4. 8
5. Which polar equation represents a lemniscate?
   1. $r = 2\theta$
   2. $r^2 = 4\cos(2\theta)$
   3. $r = 4\sin(3\theta)$
   4. $r = 4$
6. What is the shape of the graph of $\theta = \frac{\pi}{2}$?
   1. A horizontal line
   2. A vertical line
   3. A circle
   4. A spiral
7. What shape does the polar equation $r = 2 + 4\cos(\theta)$ represent?
   1. Rose Curve
   2. Lemniscate
   3. Limaçon
   4. Cardioid