9.2 Polar Equations and Graphs

1
9.2 Polar Equations and Graphs
2
Steps for Converting Equations from Rectangular
to Polar form and vice versa
Four critical equivalents to keep in mind are
3
Convert the equation r 2 to rectangular form
Since we know that
, square both sides of the equation.
4
We still need r2, but is there a better choice
than squaring both sides?
5
Convert the following equation from rectangular
to polar form.
Since
and
6
Convert the following equation from rectangular
to polar form.
7
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.
8
(No Transcript)
9
Identify and graph the equation r 2
Circle with center at the pole and radius 2.
10
(No Transcript)
11
The graph is a straight line at extending
through the pole.
12
(No Transcript)
13
The graph is a horizontal line at y -2
14
(No Transcript)
15
Theorem
Let a be a nonzero real number, the graph of the
equation
is a horizontal line a units above the pole if a
gt 0 and a units below the pole if a lt 0.
16
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 gt 0 and a units to the left of the
pole if a lt 0.
17
Graph
18
(No Transcript)
19
(No Transcript)
20
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.
21
(No Transcript)
22
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.
23
Cardioids (heart-shaped curves) where a gt 0 and
passes through the origin
24
(No Transcript)
25
Limacons without the inner loop
are given by equations of the form
where a gt 0, b gt 0, and a gt b. The graph of
limacon without an inner loop does not pass
through the pole.
26
(No Transcript)
27
Limacons with an inner loop
are given by equations of the form
where a gt 0, b gt 0, and a lt b. The graph of
limacon with an inner loop will pass through the
pole twice.
28
(No Transcript)
29
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.
30
(No Transcript)
31
Lemniscates
are given by equations of the form
and have graphs that are propeller shaped.
32
(No Transcript)
33
(No Transcript)