Solved examples: Graphing cardioids and circles in polar coordinates

📚 Quick Study Guide

• 🧭 Polar Coordinates: A system using a distance ($r$) from the origin and an angle ($\theta$) from the positive x-axis to define a point.
• ❤️ Cardioid Equation: $r = a(1 \pm \cos \theta)$ or $r = a(1 \pm \sin \theta)$, where $a$ determines the size. The $\pm$ and $\cos$ or $\sin$ determine the orientation.
• ⭕ Circle Equation: $r = a \cos \theta$ or $r = a \sin \theta$, where $a$ is the diameter of the circle. The $\cos$ gives a circle on the x-axis, and $\sin$ gives a circle on the y-axis.
• 📐 Graphing Strategy: Create a table of values for $\theta$ (e.g., $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$) and calculate corresponding $r$ values. Plot the points $(r, \theta)$ and connect them smoothly.
• 💡 Tips: Remember that negative $r$ values are plotted in the opposite direction of $\theta$. Consider symmetry to reduce calculations.

Practice Quiz

1. What type of curve is represented by the polar equation $r = 3(1 + \cos \theta)$?
   1. Circle
   2. Line
   3. Cardioid
   4. Spiral


2. Which of the following polar equations represents a circle centered on the y-axis?
   1. $r = 5 \cos \theta$
   2. $r = 5 \sin \theta$
   3. $r = 5$
   4. $\theta = 5$


3. What is the maximum value of $r$ for the cardioid $r = 2(1 - \sin \theta)$?
   1. 1
   2. 2
   3. 4
   4. 6


4. The cardioid $r = a(1 + \cos \theta)$ is symmetric with respect to which axis?
   1. x-axis
   2. y-axis
   3. Origin
   4. None of the above


5. What does the equation $r = 4 \cos \theta$ represent?
   1. A circle with radius 4 centered at the origin.
   2. A circle with radius 2 centered at (2, 0).
   3. A cardioid along the x-axis.
   4. A line parallel to the y-axis.


6. For the cardioid $r = 1 + \sin \theta$, at what angle $\theta$ does $r$ equal 0?
   1. 0
   2. $\frac{\pi}{2}$
   3. $\pi$
   4. $\frac{3\pi}{2}$


7. Which of the following equations represents a circle passing through the origin with a diameter of 6 along the x-axis?
   1. $r = 6 \sin \theta$
   2. $r = 3 \cos \theta$
   3. $r = 6 \cos \theta$
   4. $r = 3 \sin \theta$