MTH 253 Calculus (Other Topics) Chapter 10 – Conic Sections and Polar Coordinates Section 10.6 – Graphing in Polar Coordinates Copyright © 2009 by Ron Wallace, all rights reserved.

Graphs of Polar Equations polar axis (r,  ) r  pole The angle  may be expressed in degrees or radians. If r = f(  ), then the graph of this equation consists of ALL of the points whose coordinates make this equation true.

Graphing Polar Equations  Reminder: How do you graph rectangular equations? Method 1:  Create a table of values.  Plot ordered pairs.  Connect the dots in order as x increases. Method 2:  Recognize and graph various common forms.  Examples: linear equations, quadratic equations, conics, … The same basic approach can be applied to polar equations.

Graphing Polar Equations Method 1: Plotting and Connecting Points 1.Create a table of values. 2.Plot ordered pairs. 3.Connect the dots in order as  increases. NOTE: Since most of these equations involve periodic functions (esp. sine and cosine), at some point the graph will start repeating itself (but not always).

Graphing Polar Equations wrt x-axis Replacing  with –  doesn’t change the function Replacing r with –r and  with  –  doesn’t change the function Symmetry Tests (r,) (r,-)=(-r, – )

Graphing Polar Equations wrt y-axis Replacing r with –r and  with – doesn’t change the function Replacing  with  –  doesn’t change the function Symmetry Tests (r,) (-r, -)=(r,-)

Graphing Polar Equations wrt the origin Replacing r with –r doesn’t change the function. Replacing  with    doesn’t change the function. Symmetry Tests (r,) (-r,) (r,  )

Slope of Polar Curves  To find the slope of … … remember … … therefore …

Slope of Polar Curves  Example: Find the equation of the tangent line to the following curve when  = /4

Graphing Polar Equations Recognizing Common Forms  Circles Centered at the origin: r = a  radius: a period = 360 Tangent to the x-axis at the origin: r = a sin   center: (a/2, 90) radius: a/2 period = 180  a > 0  above a < 0  below Tangent to the y-axis at the origin: r = a cos   center: (a/2, 90) radius: a/2 period = 180  a > 0  right a < 0  left r = 4 r = 4 sin  r = 4 cos  Note the Symmetries

Graphing Polar Equations Recognizing Common Forms  Flowers (centered at the origin) r = a cos n or r = a sin n  radius: |a|  n is even  2n petals petal every 180/n period = 360  n is odd  n petals petal every 360/n period = 180  cos  1 st 0  sin  1 st 90/n r = 4 sin 2  r = 4 cos 3  Note the Symmetries

Graphing Polar Equations Recognizing Common Forms  Spirals Spiral of Archimedes: r = k  |k| large  loose |k| small  tight r =  r = ¼ 

Graphing Polar Equations Recognizing Common Forms  Heart (actually: cardioid if a = b … otherwise: limaçon) r = a ± b cos  or r = a ± b sin  r = cos  r = cos  r = sin  r = sin  Note the Symmetries

Graphing Polar Equations Recognizing Common Forms  Leminscate a = 16 Note the Symmetries

Polar Graphs w/ Technology  TI-84  WinPlot

Intersections of Polar Curves  As with Cartesian equations, solve by the substitution method.  Warning: 2 polar curves may intersect, but at different values of . i.e. Setting the two equations equal to each other may not reveal ALL of the points of intersection. Solution: Always graph the equations.

Intersections of Polar Curves  Example: Find the points of intersection of … Note that 2 of the points are found by substitution, the third by the graph.