Section 5.2 – Polar Equations and Graphs. An equation whose variables are polar coordinates is...
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Transcript of Section 5.2 – Polar Equations and Graphs. An equation whose variables are polar coordinates is...
Section 5.2 – Polar Equations and Graphs
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.
Identify and graph the equation: r = 2
r 2
r2 4
x y2 2 4
Circle with center at the pole and radius 2.
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Identify and graph the equation: =3
tan tan
3
31
yx
31
y x 3
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3
Identify and graph the equation: r sin 2
sin sin yr
y r
y 2
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Theorem
Let a be a nonzero real number, the graph of the equation
r asin
is a horizontal line a units above the pole if a > 0 and units below the pole if a < 0.
a
Theorem
Let a be a nonzero real number, the graph of the equation
r acos
is a vertical line a units to the right of the pole if a > 0 and units to the left of the pole if a < 0.
a
Identify and graph the equation: r 4cos
r r2 4 cos
x y x2 2 4
x x y2 24 0
x x y2 24 4 4
x y 2 42 2
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Theorem
Let a be a positive real number. Then,
r a2 sin Circle: radius a; center at (0, a) in rectangular coordinates.
r a 2 sin Circle: radius a; center at (0, -a) in rectangular coordinates.
Theorem
Let a be a positive real number. Then,
r a2 cos Circle: radius a; center at (a, 0) in rectangular coordinates.
r a 2 cos Circle: radius a; center at (-a, 0) in rectangular coordinates.
Theorem Tests for Symmetry
Symmetry with Respect to the Polar Axis (x-axis):
In a polar equation, replace by If
an equivalent equation results, the graph
is symmetric with respect to the polar
axis.
.
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r,
r,
Theorem Tests for Symmetry
Symmetry with Respect to the Line (y-axis):
In a polar equation, replace by
If an equivalent equation results, the
graph is symmetric with respect to the
line = 2
.
.
2
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r, r,
Theorem Tests for Symmetry
Symmetry with Respect to the Pole (Origin):
In a polar equation, replace by If
an equivalent equation results, the graph
is symmetric with respect to the pole.
r r .
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r,
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Other Graphs
You will run into some other graphs
• Cardioids • Limacons – without inner loop• Limacons – with inner loop• Lemniscate• Rose
See table 7 on pages 341 & 342