Section 6-5• symmetry for polar graphs

• analyzing a polar graph

• finding maximum r-values

• rose curves

• limaçon curves

• other polar graphs

Symmetry For Polar Graphs

• x-axis:

6cosr

Symmetry For Polar Graphs

• y-axis:

4 4sinr

Symmetry For Polar Graphs

• origin:

4cos(2 )r

Test For x-axis Symmetry

insert the following values into the equation and then simplify, if either case reduces to the same as the original then it has x-axis symmetry

),(or ),( rr

5) Chap. fromidentity triga (using )cos(6

)cos(6

r

rExample: test our earlier example 6cosr

Test For y-axis Symmetry

insert the following values into the equation and then simplify, if either case reduces to the same as the original then it has y-axis symmetry

),(or ),( rr

Example for y-axis: r = 4 + 4sinθ

t work)(didn' sin44

)sin(44

)sin(44

),(in plug

r

r

r

r

it worked) (yeah, sin44

)sin)1()(cos0(44

)sincoscos(sin44

)sin(44

),(in plug

r

r

r

r

r

Test For origin Symmetry

insert the following into the equation and then simplify, if either case reduces to the same as the original then it has origin symmetry

),(or ),( rr

Analyzing Polar Graphs

• analysis of a polar graph is not as extensive as with functions

• domain = possible θ’s (usually all reals)

• range = r values

• boundedness (varies)

• continuity (usually continuous)

• symmetry (just did this)

Maximum r-values

Two ways to find the range (including the maximum r-values)

• since sinθ and cosθ must be between –1 and 1, plug in these values to see what happens to r

• change the equation to y= format and graph the function, find the max and min of the graph

Example:

cos23r

when cosθ = -1 this becomes 1

when cosθ = 1 this becomes 5

thus, the range is [1 , 5]

and the max r-value is 5

Rose Curves

• format

• if n is even 2n petals

if n is odd n petals

• a is the length of the petals

• with cosθ then x-axis symmetry

with sinθ then y-axis symmetry

sometimes origin symmetry

)cos(or )sin( narnar

More examples of rose curves

5sin(3 )r

4cos(4 )r

Limaçon Curves

• format

• range is [a – b, a + b]

• with cosθ then x-axis symmetry

with sinθ then y-axis symmetry

• shape depends on a and b

cosor sin barbar

More examples of limaçon curves:

2 4sinr

5 3cosr

Other Polar Graphs

• Spiral of Archimedes: r = θ

• Lemniscates:

)2cos(or )2sin( 2222 arar

enter as cos(2 )r a