6.5: Graphing Polar Equations 1. y = cosx 2. y = sinx€¦ · Limacon Curves: r = a + b cos θ r =...

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March 14, 2012 6.5: Graphing Polar Equations 1. y = cosx 2. y = sinx

Transcript of 6.5: Graphing Polar Equations 1. y = cosx 2. y = sinx€¦ · Limacon Curves: r = a + b cos θ r =...

Page 1: 6.5: Graphing Polar Equations 1. y = cosx 2. y = sinx€¦ · Limacon Curves: r = a + b cos θ r = a + b sin θ--Lemniscates: r 2 = a 2 sin 2θ r 2 = a 2 cos 2θ Spiral of Archimedes:

March 14, 2012

6.5: Graphing Polar Equations

1. y = cosx 2. y = sinx

Page 2: 6.5: Graphing Polar Equations 1. y = cosx 2. y = sinx€¦ · Limacon Curves: r = a + b cos θ r = a + b sin θ--Lemniscates: r 2 = a 2 sin 2θ r 2 = a 2 cos 2θ Spiral of Archimedes:

March 14, 2012

r = cosθ

Page 3: 6.5: Graphing Polar Equations 1. y = cosx 2. y = sinx€¦ · Limacon Curves: r = a + b cos θ r = a + b sin θ--Lemniscates: r 2 = a 2 sin 2θ r 2 = a 2 cos 2θ Spiral of Archimedes:

March 14, 2012

r = sinθ

Page 4: 6.5: Graphing Polar Equations 1. y = cosx 2. y = sinx€¦ · Limacon Curves: r = a + b cos θ r = a + b sin θ--Lemniscates: r 2 = a 2 sin 2θ r 2 = a 2 cos 2θ Spiral of Archimedes:

March 14, 2012

y = 3cos θ

Page 5: 6.5: Graphing Polar Equations 1. y = cosx 2. y = sinx€¦ · Limacon Curves: r = a + b cos θ r = a + b sin θ--Lemniscates: r 2 = a 2 sin 2θ r 2 = a 2 cos 2θ Spiral of Archimedes:

March 14, 2012

Algebraic Relationship

r = cos O

Page 6: 6.5: Graphing Polar Equations 1. y = cosx 2. y = sinx€¦ · Limacon Curves: r = a + b cos θ r = a + b sin θ--Lemniscates: r 2 = a 2 sin 2θ r 2 = a 2 cos 2θ Spiral of Archimedes:

March 14, 2012

r = sin O

Page 7: 6.5: Graphing Polar Equations 1. y = cosx 2. y = sinx€¦ · Limacon Curves: r = a + b cos θ r = a + b sin θ--Lemniscates: r 2 = a 2 sin 2θ r 2 = a 2 cos 2θ Spiral of Archimedes:

March 14, 2012

Page 8: 6.5: Graphing Polar Equations 1. y = cosx 2. y = sinx€¦ · Limacon Curves: r = a + b cos θ r = a + b sin θ--Lemniscates: r 2 = a 2 sin 2θ r 2 = a 2 cos 2θ Spiral of Archimedes:

March 14, 2012

Symmetry:

1. line O =

2. polar axis

3. the pole

(r,θ)

(r,θ)

(r,θ)

Page 9: 6.5: Graphing Polar Equations 1. y = cosx 2. y = sinx€¦ · Limacon Curves: r = a + b cos θ r = a + b sin θ--Lemniscates: r 2 = a 2 sin 2θ r 2 = a 2 cos 2θ Spiral of Archimedes:

March 14, 2012

Use polar symmetry tests to determine if the graph is symmetric about the x-axis, the y-axis, or the origin.

r = 4 sin 3θ

Page 10: 6.5: Graphing Polar Equations 1. y = cosx 2. y = sinx€¦ · Limacon Curves: r = a + b cos θ r = a + b sin θ--Lemniscates: r 2 = a 2 sin 2θ r 2 = a 2 cos 2θ Spiral of Archimedes:

March 14, 2012

Find the maximum r-value of the following graphs.

1. r = 3 + 2cos θ 2. r = 4 - sin θ

Maximum r-value: value occurring at a point the maximum distance from the pole.

*To find: find values of |r| as opposed to the directed distance r.

Page 11: 6.5: Graphing Polar Equations 1. y = cosx 2. y = sinx€¦ · Limacon Curves: r = a + b cos θ r = a + b sin θ--Lemniscates: r 2 = a 2 sin 2θ r 2 = a 2 cos 2θ Spiral of Archimedes:

March 14, 2012

Analyze the graph of the polar curve.

r = -3 cos 4θ

Domain:___________________Range:____________________Continuity:_________________Boundedness: ______________Symmetry: ________________Maximum r-value: ___________Asymptotes:_______________

Page 12: 6.5: Graphing Polar Equations 1. y = cosx 2. y = sinx€¦ · Limacon Curves: r = a + b cos θ r = a + b sin θ--Lemniscates: r 2 = a 2 sin 2θ r 2 = a 2 cos 2θ Spiral of Archimedes:

March 14, 2012

Rose Curves: r = a cos nθr = a sin nθ

Limacon Curves: r = a + b cos θr = a + b sin θ

--

Lemniscates: r2 = a2sin 2θ

r2 = a2cos 2θ

Spiral of Archimedes: r = θ

where n is an integer > 1

n is even: 2n petalsn is odd: n petals

r-value max = a + b

where a > 0, b > 0