Polar Equations and Graphs

8/6/2019 Polar Equations and Graphs

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When trying to figure out the graphs of polar equations we

can convert them to rectangular equations particularly if

we recognize the graph in rectangular coordinates.

7!r We could square both sides

492 !r

Now use our conversion:

222yxr !

4922 ! yx

We recognize this as a circle

with center at (0, 0) and a

radius of 7.

On polar graph paper it will centered at the origin and out 7


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Let's try another:

3

TU ! Take the tangent of both sides

!

3tantan TU


3!x

y

We recognize this as a line with slope square root of 3.

3!

x

y!Utan

Multiply both

sides byx

xy 3!

To graph on a polar plot

we'd go to whereand make a line. 3

TU !


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Let's try another: 5sin !r


We recognize this as a

horizontal line 5 units belowthe origin (or on a polar plot

below the pole)

Usinry !

5!y


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Sometimes converting to rectangular equations doesn't

help us figure out what the graph would look like or it is

not necessary.

Usin1!rThe only way we know how toconvert is if there is an rin front of

the sin U term so we'll multiply both

sides by r.Usin2 rrr !

Now use our conversions:

222yxr ! 22 yxr ! Usinry !

yyxyx ! 2222 I still don't

know what the

graph looks

like!In these cases we'll plot points, choosing

a U from the polar form and finding a

corresponding rvalue.


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Before we do this, if we knew something about the

symmetry of the graph we may not have to find as many

points. TESTS FOR SYMMETRY

Symmetry with

Respect to the Polar

Axis (xaxis)

Replace U by - U

and if you get

original equation

back

Symmetry with

Respect to the Line

U = T/2 (yaxis)

Replace U by T - U

and if you get

original equation

back

(r,U)

(r, -U)

Symmetry with

Respect to the Pole

(Origin)

Replace rby - r

and if you get

original equation

back

(r,U)(r,T-U)

(r,U)

(- r,U)

These tests are sufficient but not necessary so if test fails you don't know anything.


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Usin1!rLet's test for symmetry

Polar Axis: U! sin1r Usin1!rFAILS

Line U = T/2: UT! sin1r

UTUT sincoscossin1 !r

Use the difference formula

0 -1

Usin1!r

This IS the original equation!

Pole: Usin1! r

Not the original equation.

So this graph is symmetric

with respect to the line U = T/2

(yaxis). We will only need tochoose U's on the right side of

the graph then and we can use

symmetry to get the other half.


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U Usin1!r

2

T 211 !

3

T 87.1

2

31 }

Let's let each unit be 1/4.

6

T

5.1211 !

0 101 !

6

T

2

1

2

11 !

3

T

13.02

31 }

2

T

011 !Let's plot the symmetric points

This type of graph is called a

cardioid.


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Equations of carioids would look like one of the following:

r= a(1 + cosU) r= a(1 + sinU)

r= a(1 - cosU) r= a(1 - sinU)

wherea > 0

All graphs of cardioids pass through the pole.


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Let's test for symmetry

Polar Axis: U! cos23r

YES!

Line U = T/2: UT! cos23r

UTUT sinsincoscos23 !r


-1 0

Ucos23!r

Not the original equation

Pole: Ucos23! r



with respect to the polar axis

(xaxis). We will only need to

choose U's on the top half of



Let's try another: Ucos23!r

Ucos23!r


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Equations of limacons without inner loops would look like

one of the following:

r= a +b cosU r= a +b sinU

r= a - b cosU r= a - b sinU

wherea > 0, b > 0, and a >b

These graphs DO NOT pass through the pole.


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Polar Axis: U! cos21r

YES!

Line U = T/2: UT ! cos21r

UTUT sinsincoscos21 !r


-1 0

Ucos21!r

Not the original equation

Pole: Ucos21! r



with respect to the polar axis

(xaxis). We will only need to

choose U's on the top half of



Let's try another: Ucos21!r

Ucos21!r


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U

0 3121 !

6

T

73.22

321 }


3

T

2

2

121 !

2

T 1021 !

3

2T0

2

121 !

6

5T

73.02

321 }

T 1121 !

Let's plot the symmetric points


limacon with an innerloop.Ucos21!r


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Equations of limacons with inner loops would look like

one of the following:

r= a +b cosU r= a +b sinU

r= a - b cosU r= a - b sinU

wherea > 0, b > 0, and a < b

These graphs will pass through the pole twice.


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Polar Axis: U! 2cos2rYES!

Line U = T/2: UT! 2cos2r

UU 2cos22cos2 !!rYES!

Pole:

Since graph is symmetric

to both the polar axis and

the line U =T/2 it will also

be with respect to the pole.


with respect to the pole, the

polar axis and the line U =T/2.

We will only need to choose

U's between 0 and T/2.

Let's try another: U2cos2!r

U2cos2!r

UT 22cos2 !

cos is periodic so can drop the 2T


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U

0 212 !

6

T1

2

12 !


4

T

002 !

3

T1

2

12 !

2

T

212 !

Let's plot the symmetric points

This type of graph is called a rose

with 4 petals. U2cos2!r


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Equations of rose curves would look like one of the

following:

r= a cos(nU) r= a sin(nU)

Wheren even has 2n petals and

n odd has n petals

(n {

0 ors

1)


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Polar Axis: U! 2sin42rFAILS

Line U = T/2: UT! 2sin42r

UU 2sin22sin42 !!r FAILS

Pole:


with respect to the pole.

Let's try another: U2sin42 !r

U2sin42 !r

UT 22sin4 !

sin is periodic so can drop the 2T

U2sin42

! r

U2sin42 !r


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U

0 004 !

6

T

322

34 !


4

T

4

14 !

3

T32

2

34 !

2

T

004 !


lemniscate U2sin42 !r r

0

9.1s

2s

9.1s

0


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Equations of lemniscates would look like one of the

following:

r2 = a2 cos(2U) r2 = a2 sin(2U)

These graphs will pass through the pole and are

propeller shaped.


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Table 7 in your book on page 732

summarizes all of the polar graphs.

You can graph these on your calculator. You'll need tochange to polar mode and also you must be in radians.

If you are in polar function mode when you hit your

button to enter a graph you should see r1 instead ofy1.

Your variable button should now put in U on TI-83's and

it should be a menu choice in 85's & 86's.


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Have fun plotting pretty pictures!

Limacon With InnerLoop

made with TI Calculator

Rose with 7 petals made

with graphing program on

computer


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Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah

USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloadedfrom www.mathxtc.com and for it to be modified to suit the Western

Australian Mathematics Curriculum.

Stephen Corcoran

Head of Mathematics

St Stephens School Carramar

www.ststephens.wa.edu.au

Polar Equations and Graphs

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