9.2 Polar Equations and Graphs. Steps for Converting Equations from Rectangular to Polar form and vice versa Four critical equivalents to keep in mind.

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Presentation transcript:

9.2 Polar Equations and Graphs

Steps for Converting Equations from Rectangular to Polar form and vice versa Four critical equivalents to keep in mind are:

Convert the equation: r = 2 to rectangular form Since we know that, square both sides of the equation.

We still need r 2, but is there a better choice than squaring both sides?

Convert the following equation from rectangular to polar form. and Since

Convert the following equation from rectangular to polar form.

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 Circle with center at the pole and radius 2.

The graph is a straight line at extending through the pole.

The graph is a horizontal line at y = -2

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.

Graph:

Theorem Let a be a positive real number. Then, Circle: radius ; center at (, 0) in rectangular coordinates. Circle: radius ; center at (-, 0) in rectangular coordinates.

Theorem Let a be a positive real number. Then, Circle: radius ; center at (0, ) in rectangular coordinates.

Cardioids (heart-shaped curves) where a > 0 and passes through the origin

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. a represents the length of the petals.

Lemniscates are given by equations of the form and have graphs that are propeller shaped.