Section 10.8Graphs of Polar

Equations

Types of Polar Graphs

1. Circle with center not at the pole.2. Limaçon with and without a loop.3. Cardioid

Ways to Graph

1. Making a table of values.2. Using symmetry.3. Finding the maximum value of r.4. Finding the zeroes.

Using a Table

When the polar equation is of the form:

r = a cos or r = a sin The graph is a circle whose center is on the x-axis for cosine graphs and on the y-axis for sine graphs.The diameter of the circle is determined by a.

Let’s start from the beginning with a table to graph a curve.Graph r = 6cos θ using a table.

Let’s graph these points on a polar graph.

θ

r

6

3

20 2

356

43

32

53 2

6 3 3 3 0 3 3 3 6 3 0 3 6

The center is (3, 0).

Graphing using Symmetry

There are three types of symmetry that are used to graph on the polar coordinate system.

(r, θ)

1. The line 2

2

(r, − )

2. The polar axis(r, θ)

0

(r, -)

3. The pole(r, θ)

(r, + )

Quick Tests for Symmetry in Polar Coordinates

1. The graph of sin r fis symmetric with respect to the line .2

2. The graph of cos r gis symmetric with respect to the polar axis.

When the polar equation is of the form:

r = a ± b cos or r = a ± b sin (a > 0, b > 0)

The graph is a limaçon.If a < b, the limaçon has an inner loop.If a = b, the limaçon is a cardioid (heart-shaped)If a > b, the limaçon looks like a lima bean.

Example 1A limaçon with a loop

Use symmetry to sketch the graph r = 2 + 4sin

Since this is a sinegraph, we will onlylook at points in the 1st and 4th quad.and then use symmetry.

r

6

2

116

32

0 24

6

0

2

r

6

2

116

32

0 24

6

0

2

Now we will use symmetryto find pointsin the 2nd and3rd quadrants.Then draw thegraph.

Example 2A Cardioid

Use symmetry to sketch the graph r = 2 + 2cos

Since this is a sinegraph, we will onlylook at points in the 1st and 2nd quad.and then use symmetry.

r

3

2

23

0 43

2

1

0

r

3

2

23

0 43

2

1

0

Now we will use symmetryto find pointsin the 3rd and 4th quadrants.Then draw thegraph.

Example 2A limaçon without

a loop

Use symmetry to sketch the graph r = 4 − 2cos

Since this is a sinegraph, we will onlylook at points in the 1st and 2nd quad.and then use symmetry.

r

3

2

23

0 23

4

5

6

r

3

2

23

0 23

4

5

6

Now we will use symmetryto find pointsin the 3rd and 4th quadrants.Then draw thegraph.

End of 1st Day

2nd Day

When the polar equation is of the form:

r = a cos (n) or r = a sin (n) The graph is a rose curve.If n is an odd number, then1. there are n petals on the rose and 2. the petals are 2 divided by n apart.

If n is an even number, then1. there are 2n petals on the rose and2. the petals are 2 divided by 2n

apart.

The length of the petal is a units.

In previous math courses as well as Pre-Calculus you have learned how to graph on the rectangular coordinate system. You first learned how to graph using a table. Then you learned how to use intercepts, symmetry, asymptotes, periods, and shifts to help you graph.Graphing on the polar coordinate system will be done similarly.

Example 1

Graph r = 4cos 2This graph is symmetric to the polar axis.To graph this rose curve we are going to use another aid for graphing. This aid is finding the maximum value |r|and the zeroes of the graph.

To find the maximum value of |r| we must find the |r| when our trig function is equal to 1.In this example

cos 2 = 12 = 0, 2 = 0,

So our maximum value of |r| isr = 4cos 2(0)

r = 4(4, 0)

r = 4cos 2()r = 4(4, )

Where is cosine equal to 0?/2 or 3/2

So 2θ = /2, 3/2, 5/2, 7/2and

θ = /4, 3/4, 5/4, 7/4The zero points are (0, /4), (0, 3/4),(0, 5/4), and (0, 7/4).

How many petals are in this rose curve?

4 petalsHow far apart are each petal?

2/4 = /2 apartWhat are the coordinates of the tips of the petals?

(4, 0), (4, /2), (4, ), and (4, 3/2)

b. Graph r = 5 sin 3θFind the maximum value of |r|. This is where sin 3θ = 1.

5 93 , ,2 2 2

5 3, ,6 6 2

So our maximum value of |r| is

5sin3 6r

5 1r

5r

5 3So graph 5, , 5, , and 5, .6 6 2

55sin3 6r

5 1r

5r

35sin3 2r

5 1r

5r

Where is sine equal to 0?0 or

So 3θ = 0, , 2, 3, 4, 5and

θ = 0, /3, 2/3, , 4/3, 5/3The zero points are (0, 0), (0, /3), (0, 2/3), (0, ), (0, 4/3), and (0, 5/3).