10.8 polar equations and graphs. an equation whose variables are polar coordinates is called a polar...
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10.8 Polar Equations and Graphs
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An equation whose variables are polar coordinates is called a polar equation. The graph of a polar equation consists of all points whose polar coordinates satisfy the equation.
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Identify and graph the equation: r = 2
Circle with center at the pole and radius 2.
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Theorem
Let a be a nonzero real number, the graph of the equation
is a horizontal line a units above the pole if a > 0 and |a| units below the pole if a < 0.
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Theorem
Let a be a nonzero real number, the graph of the equation
is a vertical line a units to the right of the pole if a > 0 and |a| units to the left of the pole if a < 0.
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Theorem
Let a be a positive real number. Then,
Circle: radius a; center at (0, a) in rectangular coordinates.
Circle: radius a; center at (0, -a) in rectangular coordinates.
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Theorem
Let a be a positive real number. Then,
Circle: radius a; center at (a, 0) in rectangular coordinates.
Circle: radius a; center at (-a, 0) in rectangular coordinates.
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Theorem Tests for Symmetry
Symmetry with Respect to the Polar Axis (x-axis):
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Theorem Tests for Symmetry
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Theorem Tests for Symmetry
Symmetry with Respect to the Pole (Origin):
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The tests for symmetry just presented are sufficient conditions for symmetry, but not necessary.
In class, an instructor might say a student will pass provided he/she has perfect attendance. Thus, perfect attendance is sufficient for passing, but not necessary.
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Symmetry:
Polar axis:
Symmetric with respect to the polar axis.
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The test fails so the graph may or may not be symmetric with respect to the above line.
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The pole:
The test fails, so the graph may or may not be symmetric with respect to the pole.
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Identify points on the graph:
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Cardioids (a heart-shaped curves)
are given by an equation of the form
r a(1 cos) r a(1 sin )
r a(1 cos) r a(1 sin )
where a > 0. The graph of cardioid passes through the pole.
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Limacons without the inner loop
are given by equations of the form
where a > 0, b > 0, and a > b. The graph of limacon without an inner loop does not pass through the pole.
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Limacons with an inner loop
are given by equations of the form
where a > 0, b > 0, and a < b. The graph of limacon with an inner loop will pass through the pole twice.
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Rose curves
are given by equations of the form
and have graphs that are rose shaped. If n is even and not equal to zero, the rose has 2n petals; if n is odd not equal to +1, the rose has n petals.
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Lemniscates
are given by equations of the form
and have graphs that are propeller shaped.