Polar Coordinates and Graphing

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Polar Coordinates and Graphing
r directed distance
directed angle
Polar Axis
O
Counterclockwise from polar axis to .
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Plotting Points in the Polar Coordinate System
The point lies two units
from the pole on the terminal side of the angle
.
The point lies three units
from the pole on the terminal side of the angle
.
The point coincides with the
point .
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Multiple Representations of Points
4
Try plotting the following points in polar
coordinates and find three additional polar
representations of the point
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Graphing a Polar Equation
circle
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Using Symmetry to Sketch aPolar Graph
Symmetry with respect to the line
Symmetry with respect to the Polar Axis.
Symmetry with respect to the Pole
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Tests for Symmetry
The graph of a polar equation is symmetric with
respect to the following if the given
substitution yields an equivalent equation.
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Using Symmetry to Sketch
Thus, the graph is symmetric with respect to the
polar axis, and you need only plot points from 0
to ?.
Lima?on
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Symmetry Test Fails
Unfortunately the tests for symmetry can
guarantee symmetry, but there are polar curves
that fail the test, yet still display symmetry.
Lets look at the graph of
Try the symmetry tests. What happens?
Original Replacement
New Equation Equation

• Not symmetric
• about the polar axis.

• Not symmetric
• about the pole.

All of the tests indicate that no symmetry
exists. Now, lets look at the graph.
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You can see that the graph is symmetric with
respect to the line
Spiral of Archimedes
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Quick Test for Symmetry
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Determine the effect of a on the graph of
.
Click on the graph below to open an interactive
page. Please note the interactive page may open
with the graph to the far left. You may need to
drag the graph slightly to the right.
13
Determine the effect of a on the graph of
.
Click on the graph below to open an interactive
page. Please note the interactive page may open
with the graph to the far left. You may need to
drag the graph slightly to the right.
14
Determine the effect of a and b on the
graph of
Click on the graph below to open an interactive
page. Please note the interactive page may open
with the graph to the far left. You may need to
drag the graph slightly to the right.
15
Determine the effect of a on the graph of
Click on the graph below to open an interactive
page. Please note the interactive page may open
with the graph to the far left. You may need to
drag the graph slightly to the right.
16
Determine the effect of a and b on the
graph of
Click on the graph below to open an interactive
page. Please note the interactive page may open
with the graph to the far left. You may need to
drag the graph slightly to the right.
17
Summary of Special Polar Graphs
Limacons
Convex Limacon
Limacon with inner loop
Cardiod (heart-shaped)
Dimpled Limacon
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Rose Curves
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Circles and Lemniscates
Circles
Lemniscates
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Analyzing Polar Graphs
Analyze the basic features of
Type of Curve Rose Curve with 2b petals 4
petals Symmetry Polar axis, pole, and Maximum
Value of r r 3 when Zeros of r r
0 when
You can use this same process to Analyze any
polar graph.
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Analyze and Sketch the graph
Type of curve Symmetry Maximum r Zeros of
r
Cardioid Limacon
Convex Limacon
sym. _at_ polar axis
none