10.7 Graphs of Polar Functions

Just like any other function (quadratic, logarithmic, etc.), polar functions have certain rules and characteristics that govern their symmetry, shifting, asymptotes, periods, shifts, and general shape.

A simple way to quickly determine the shape of a polar graph without the aid of a graphing calculator is to plot its points:

	0	π/6	π/3	π/2	2π/3	5π/6	π	7π/6	3π/2	11π/6	2π
r	0	2	2√3	4	2√3	2	0	-2	-4	-2	0

Quickly plotting points as show above allows a quick sketch of the graph to be drawn:

In general, however, all simple sine and cosine polar equations of the form r = acos(θ) or r = asin(θ) will be circles with a diameter of a that are either tangent to the polar axis and symmetric with respect to the line θ = π/2 (sine functions) or tangent to the line θ = π/2 and symmetric with respect to the polar axis (cosine functions).

There are three important types of symmetry to consider that can make graphing or predicting the shape of a polar function much easier.

Figure A: This is a polar function that is symmetric with respect to the line θ = π/2 (the cartesian y-axis). This can be tested for algebraically by replacing (r, θ) with either (r, π - θ) or (-r, -θ). If this substitution results in an equivalent equation, the equation is symmetric with respect to θ = π/2.

Figure B: This is a polar function that is symmetric with respect to the polar axis (the cartesian x-axis). This can be tested for algebraically by replacing (r, θ) with either (r, -θ) or (-r, π - θ). If this substitution results in an equivalent equation, the equation is symmetric with respect to the polar axis. 

Figure C: This is a polar function that is symmetric with respect to the pole (the origin). This can be tested for algebraically by replacing (r, θ) with either (r, π + θ) or (-r, θ). If this substitution results in an equivalent equation, the equation is symmetric with respect to the pole.

Special Polar Graphs:

Limaçons and cardioids with the form r = a ± bcos(θ) or r = a ± bsin(θ) where a > 0 and b > 0:

(A)(B)(C)    (D)

Figure A: This limaçon with the equation r = 1 + 2cos(θ) is  looped because a/b < 1.

Figure B: This cardioid with the equation r = 1 + cos(θ) is heart-shaped because a/b = 1.

Figure C: This limaçon with the equation r = 3 + 2cos(θ) is dimpled because 1 < a/b < 2.

Figure D: This limaçon with the equation r = 4 + 2cos(θ) is convex because a/b ≥ 2.

Rose curves with the form r =  acos(nθ) or r = asin(nθ) where n is an integer and a is the length of each petal:

(A)(B)

(C)(D)

Figure A: This rose curve with the equation r = cos(3θ) has 3 petals because n = 3. When rose curves have n as an odd number, the number of petals will equal n.

Figure B: This rose curve with the equation r = cos(4θ) has 8 petals because n = 4. When rose curves have n as an even number, the number of petals will equal 2n.

Figure C: This rose curve with the equation r = sin(5θ) has 5 petals because n = 5.

Figure D: This rose curve with the equation r = sin(2θ) has 4 petals because n = 2.

As a rule, rose curves are symmetric to the polar axis, the line θ = π/2, and the pole and have zeroes when r = 0 or θ = π/4, 3π/4.


Lemniscates with the form r² = a²sin(2θ) or r² = a²cos(2θ) where a is the length of each protrusion:

(A)(B)

Figure A: This lemniscate with the equation r² = sin(2θ) is mostly contained in the first and third quadrants due to the sine in its equation.

Figure B: This lemniscate with the equation r² = cos(2θ) lies equally in all four quadrants due to the cosine in its equation.

As a rule, lemniscates are symmetric with respect to the pole and have zeroes when θ = 0, π/2.