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. We could square both sides Now use our conversion: 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

Let's try another: Take the tangent of both sides Now use our conversion: We recognize this as a line with slope square root of 3. Multiply both sides by x To graph on a polar plot we'd go to where and make a line.

Let's try another: Now use our conversion: We recognize this as a horizontal line 5 units below the origin (or on a polar plot below the pole)

Sometimes converting to rectangular equations doesn't help us figure out what the graph would look like or it is not necessary. The only way we know how to convert is if there is an r in front of the sin  term so we'll multiply both sides by r. Now use our conversions:  I still don't know what the graph looks like! In these cases we'll plot points, choosing a  from the polar form and finding a corresponding r value.

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 (x axis) Replace  by -  and if you get original equation back Symmetry with Respect to the Line  =  /2 (y axis) Replace  by  -  and if you get original equation back (r,  ) (r, -  ) Symmetry with Respect to the Pole (Origin) Replace r by - r and if you get original equation back (r,  ) (r,  -  ) (r,  ) (- r,  ) These tests are sufficient but not necessary so if test fails you don't know anything.

Let's test for symmetry Polar Axis: FAILS Line  =  /2: Use the difference formula 0 This IS the original equation! Pole: Not the original equation. So this graph is symmetric with respect to the line  =  /2 (y axis). We will only need to choose  's on the right side of the graph then and we can use symmetry to get the other half.

Let's let each unit be 1/4. Let's plot the symmetric points This type of graph is called a cardioid.

Equations of carioids would look like one of the following: r = a(1 + cos  ) r = a(1 + sin  ) r = a(1 - cos  ) r = a(1 - sin  ) where a > 0 All graphs of cardioids pass through the pole.

Let's test for symmetry Polar Axis: YES! Line  =  /2: Use the difference formula 0 Not the original equation Pole: Not the original equation. So this graph is symmetric with respect to the polar axis (x axis). We will only need to choose  's on the top half of the graph then and we can use symmetry to get the other half. Let's try another:

Let's let each unit be 1. Let's plot the symmetric points This type of graph is called a limacon without an inner loop.

Equations of limacons without inner loops would look like one of the following: r = a +b cos  r = a +b sin  r = a - b cos  r = a - b sin  where a > 0, b > 0, and a > b These graphs DO NOT pass through the pole.

Let's test for symmetry Polar Axis: YES! Line  =  /2: Use the difference formula 0 Not the original equation Pole: Not the original equation. So this graph is symmetric with respect to the polar axis (x axis). We will only need to choose  's on the top half of the graph then and we can use symmetry to get the other half. Let's try another:

Let's let each unit be 1/2. Let's plot the symmetric points This type of graph is called a limacon with an inner loop.

Equations of limacons with inner loops would look like one of the following: r = a +b cos  r = a +b sin  r = a - b cos  r = a - b sin  where a > 0, b > 0, and a < b These graphs will pass through the pole twice.

Let's test for symmetry Polar Axis: YES! Line  =  /2: YES! Pole: Since graph is symmetric to both the polar axis and the line  =  /2 it will also be with respect to the pole. So this graph is symmetric with respect to the pole, the polar axis and the line  =  /2. We will only need to choose  's between 0 and  /2. Let's try another: cos is periodic so can drop the 2 

Let's let each unit be 1/2. Let's plot the symmetric points This type of graph is called a rose with 4 petals.

Equations of rose curves would look like one of the following: r = a cos(n  ) r = a sin(n  ) Where n even has 2n petals and n odd has n petals (n  0 or  1)

Let's test for symmetry Polar Axis: FAILS Line  =  /2: FAILS Pole: So this graph is symmetric with respect to the pole. Let's try another: sin is periodic so can drop the 2 

Let's let each unit be 1/4. This type of graph is called a lemniscate

Equations of lemniscates would look like one of the following: r 2 = a 2 cos(2  ) r 2 = a 2 sin(2  ) These graphs will pass through the pole and are propeller shaped.

Table 7 in your book on page 732 summarizes all of the polar graphs. You can graph these on your calculator. You'll need to change 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 r 1 instead of y 1. Your variable button should now put in  on TI-83's and it should be a menu choice in 85's & 86's.

Have fun plotting pretty pictures! Limacon With Inner Loop made with TI Calculator Rose with 7 petals made with graphing program on computer

Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. Shawna has kindly given permission for this resource to be downloaded from and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar