10.2 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

Graphing a polar Equation Using a Graphing Utility

• Solve the equation for r in terms of .

• Select a viewing window in POLar mode. In addition to setting Xmax, Xmin, Xscl, and so forth the polar mode requires setting max and min and step values for . Use radian mode and a square window.

• Enter the expression from Step1.

• Graph.

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.

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.

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.

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.

Theorem Tests for Symmetry

Symmetry with Respect to the Polar Axis (x-axis):

Theorem Tests for Symmetry

Theorem Tests for Symmetry

Symmetry with Respect to the Pole (Origin):

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.

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.

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.

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.

Lemniscates

are given by equations of the form

and have graphs that are propeller shaped.