The polar coordinate system offers a fresh way to plot points and graph equations. Instead of using x and y, we use distance from the origin (r) and angle (θ). This system unlocks new ways to describe and visualize curves, especially those with circular or spiral shapes.

Graphing polar equations involves plotting points, checking for symmetry, and understanding how changes to r and θ affect the shape. We can convert between polar and rectangular coordinates, opening up different perspectives on the same curve. This versatility makes polar coordinates a powerful tool for describing complex shapes.

Polar Coordinate System

Characteristics of polar graphs

• Symmetry: Polar graphs can exhibit reflectional symmetry about the polar axis (vertical line through pole) if $f(\theta) = f(-\theta)$ or rotational symmetry about the pole (origin) if $f(\theta) = f(\theta + \pi)$
• Boundedness: Polar graphs are bounded if the distance $r$ from the pole has a finite maximum value, otherwise they extend infinitely
• Loops and cusps: Self-intersections in the graph create loops ($r = 1 + \cos(\theta)$) while sharp corners or points form cusps ($r = 1 + \cos(2\theta)$)

Conversion of coordinate systems

• Polar to rectangular: Substitute $x = r \cos(\theta)$ and $y = r \sin(\theta)$ to convert polar coordinates $(r, \theta)$ to rectangular coordinates $(x, y)$
• Rectangular to polar: Use $r = \sqrt{x^2 + y^2}$ for distance and $\theta = \tan^{-1}(\frac{y}{x})$ for angle, adjusting $\theta$ based on the quadrant of the point $(x, y)$
• The angle $\theta$ in polar coordinates is typically expressed in radian measure

Transformation of polar equations

• Polar to rectangular: Replace $r$ with $\sqrt{x^2 + y^2}$ and $\theta$ with $\tan^{-1}(\frac{y}{x})$ in the polar equation and simplify using trigonometric identities and algebra
• Rectangular to polar: Substitute $x = r \cos(\theta)$ and $y = r \sin(\theta)$ into the rectangular equation and manipulate to isolate $r$ as a function of $\theta$ (polar form)

Graphing Polar Equations

Graphing techniques for polar equations

• Direct plotting: Generate a table of $\theta$ values (usually multiples of $\frac{\pi}{6}$ or $\frac{\pi}{4}$) and calculate corresponding $r$ values, then plot points $(r, \theta)$ in the polar plane and connect smoothly
• Rectangular conversion: Transform the polar equation to rectangular form, graph the resulting equation in the $xy$-plane, and identify key features like symmetry and boundedness

Symmetry and periodicity in polar curves

• Symmetry tests:
  1. Reflectional symmetry about polar axis if $f(\theta) = f(-\theta)$
  2. Rotational symmetry about pole if $f(\theta) = f(\theta + \pi)$
  3. Symmetry about origin if $f(\theta) = -f(\theta + \pi)$
• Periodicity: A polar curve repeats itself every $p$ units if $f(\theta) = f(\theta + p)$ for some constant $p$, with common periods being $2\pi$, $\pi$, and $\frac{2\pi}{n}$ for integer $n$
• Constant effects:
  1. Multiplying $f(\theta)$ by $a$ scales the graph by factor $|a|$, reflecting across pole if $a < 0$
  2. Adding $b$ to $\theta$ rotates the graph by angle $b$, counterclockwise if $b > 0$ and clockwise if $b < 0$

Polar Graphs and Curves

• A polar graph is the visual representation of a polar equation in the polar coordinate system
• The set of all points satisfying a polar equation forms a polar curve
• Polar graphs can reveal unique patterns and shapes not easily visible in rectangular coordinates