Section 6.4 Graphs of Polar Equations MA.PC.6.3 2000 Graph equations in the polar coordinate plane.

Graphing a Polar Equation by Point Plotting A polar equation is an equation whose variables are 𝑟 and 𝜃. The graph of a polar equation is the set of all points whose polar coordinates satisfy the equation. We use polar grids to graph polar equations.

This type of graph is called a ROSE WITH 4 PETALS. 𝜽 𝒓=𝟐 cos (𝟐𝜽) 2 1 =2 𝜋 6 2 1 2 =1 𝜋 4 2 0 =0 𝜋 3 2 − 1 2 =−1 𝜋 2 2 −1 =−2 Let's let each unit be 1/2.

Classical Curves Circle Rose Lemniscate Limacon Cardioid

CIRCLES Equations of circles would look like one of the following: 𝒓=𝒂 cos 𝜽 𝒓=𝒂 sin 𝜽

ROSE Equations of rose curves would look like one of the following: 𝒓=𝒂 cos 𝑛𝜃 𝒓=𝒂 sin 𝑛𝜃 Where 𝑛 even has 2𝑛 petals and 𝑛 odd has 𝑛 petals.

LEMNISCATE Equations of lemniscates would like one of the following: 𝒓 𝟐 = 𝒂 𝟐 cos 2𝜃 𝒓 𝟐 = 𝒂 𝟐 sin 𝟐𝜽 These graphs will pass through the pole and are propeller shaped. Unit is 1/4

LIMAÇON Equations of limaçons would like one of the following: 𝒓=𝒂+𝒃 cos 𝜽 𝒓=𝒂+𝒃 sin 𝜽 If 𝑎 < 𝑏 there is an inner loop. If 𝑎 > 𝑏 there is an indentation. Unit is 1/2

CARDIOD Equations of cardiods would like one of the following: 𝒓=𝒂+𝒂 cos 𝜽 𝒓=𝒂+𝒂 sin 𝜽 All graphs of cardiods pass through the pole. Unit is 1/4

Example Graph the equation 𝑟=3+ cos 𝜃

Example Graph the equation 𝑟= sin 2𝜃

Example Graph the equation 𝑟=4(1− cos 𝜃 )

Example Graph the equation 𝑟 2 = cos 2𝜃

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