Today in Precalculus
• Go over homework
• Notes: Graphs of Polar Equations
• Homework
Rose Curves
1. r = 4sin3θ
2. r = 4cos3θ
y
y
x



x

3. r = 4sin2θ
4. r = 4cos2θ



y
y
x
x







Conclusions
General form of the equations:
r = acosnθ
r = asinnθ
Number of petals: if n is odd, n petals
if n is even, 2n petals
Position of petals:
cos: if n is odd, one petal on positive x-axis
if n is even, petals on each axis
sin: if n is odd, one petal on half of y-axis
if n is even, no petals on axis
Length of petal is a
Limaçon Curves
5. r = 2 – 3sinθ
6. r = 2 + 2sinθ
y
y
x




x





8. r = 2 – 1cosθ
7. r = 3 + 2cosθ
y
y
x





x





Conclusions
General form of the equations:
r = a ± bcosθ
r = a ± bsinθ
a
when  1, there is an inner loop
b
a
when  1, touches the origin "cardioid
b
a
when 1   2, called a "dimpled limacon"
b
a
when  2, it is a "convex limacon"
b
Analyzing polar graphs
• The domain is the set of possible inputs for 
• The range is the set of outputs for r. The
domain and range can be read from the “trace”
or “table” features on your calculator.
• The maximum r-value is the maximum distance
from the pole. This can be found using trace, or
by knowing the range of the function.
• Symmetry can be about the x-axis, y-axis, or
origin, just as it was in rectangular equations.
• Continuity, boundedness, and asymptotes are
analyzed the same way they were for
rectangular equations.
r = 4sin3θ
Domain: All reals
Range: [-4, 4]
Maximum r-value: 4
Symmetry: y-axis
Continuity: continuous
Boundedness: bounded
Asymptotes: none
y
x



r = 4cos3θ
Domain: All reals
Range: [-4, 4]
Maximum r-value: 4
Symmetry: x-axis
Continuity: continuous
Boundedness: bounded
Asymptotes: none
y
x




r = 2 – 3sinθ
Domain: All reals
Range: [-1, 5]
Maximum r-value: 5
Symmetry: y-axis
Continuity: continuous
Boundedness: bounded
Asymptotes: none
y
x





r = 2 +2sinθ
Domain: All reals
Range: [0, 4]
Maximum r-value: 4
Symmetry: y-axis
Continuity: continuous
Boundedness: bounded
Asymptotes: none
y
x




r = 3 +2cosθ
Domain: All reals
Range: [1,5]
Maximum r-value: 5
Symmetry: x-axis
Continuity: continuous
Boundedness: bounded
Asymptotes: none
y
x





r = 2 – 1cosθ
Domain: All reals
Range: [1,3]
Maximum r-value: 3
Symmetry: x-axis
Continuity: continuous
Boundedness: bounded
Asymptotes: none
y
x





Homework
Worksheet
Quiz Monday, April 13