Graphing Polar Equations

Types of Polar Graphs

• Circles
• Limacons (Limaçons)
• Rose Curves
• Lemniscates

Graphing Circles

Equation: ( r = a \cos \theta )

• Positive A: Circle directed towards the right.
• Negative A: Circle directed towards the left.
• Diameter: ( A )
• Radius: ( \frac{A}{2} )
• Example: ( r = 4 \cos \theta )
  • Circle on the right side.
  • Center at 2 (because ( \frac{4}{2} = 2 ))
  • Travel 4 units right, then 2 units up and down. Starts at origin.

Equation: ( r = a \sin \theta )

• Positive A: Circle above the x-axis centered on the y-axis.
• Negative A: Circle below the x-axis centered on the y-axis.
• Example: ( r = 2 \sin \theta )
  • Center at (0, 1), diameter is 2, radius is 1.

Limacons

Equation: ( r = a \pm b \sin \theta ) or ( r = a \pm b \cos \theta )

• Positive Sine: Opens towards positive y-axis (upward).
• Negative Sine: Opens downward.
• Positive Cosine: Opens towards the positive x-axis (right).
• Negative Cosine: Opens towards the left.

Types of Limacons

1. With Inner Loop: ( \frac{A}{B} < 1 )
2. Cardioid (Heart Shape): ( \frac{A}{B} = 1 )
3. Dimpled Limacon (No Inner Loop): ( 1 < \frac{A}{B} < 2 )
4. Convex Limacon: ( \frac{A}{B} \geq 2 )

Example: ( r = 3 + 5 \cos \theta )

• Type: Limacon with inner loop
• Direction: Positive cosine, opens right
• Intercepts:
  • Y-intercepts: ( A ), (-A)
  • X-intercepts: ( B-A ) and ( A+B )

Example: ( r = 2 - 5 \sin \theta )

• Type: Limacon with inner loop
• Direction: Negative sine, opens downward
• Intercepts:
  • X-intercepts: ( A ), (-A)
  • Y-intercepts: ( B-A ) and ( A+B )

Example: ( r = 3 - 7 \cos \theta )

• Type: Limacon with inner loop
• Direction: Negative cosine, opens left
• Intercepts:
  • Y-intercepts: ( A ), (-A)
  • X-intercepts: ( B-A ) and ( A+B )

Example: ( r = 3 + 3 \cos \theta )

• Type: Cardioid (Heart Shape)
• Direction: Positive cosine, opens right
• Intercepts:
  • Y-intercepts: ( A ), (-A)
  • X-intercepts: Origin and ( A+B )

Important Concepts

• Cosine and Sine Association:
  • Cosine: Associated with x-values (horizontal symmetry).
  • Sine: Associated with y-values (vertical symmetry).
• Intercepts:
  • A: Determines intercepts based on equation's association.
  • ( A+B ) and ( B-A ): Important for determining the points of the limacon.

Conclusion

Understanding the characteristics of polar equations and how they graphically present themselves can make the process of plotting these equations intuitive. Each equation type has specific rules about the direction and shape of the graph based on the parameters involved.