Section 10.8 Notes

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.

Graphing by Plotting Points

Let’s start from the beginning with a table to graph a curve. When graphing using a table use θ’s that will give you an “r” that is an integer or a terminating decimal.

Example 1

Graph r = 6cos θ using a table.

What θ’s would you use?

Which points are the exact same points?

Let’s graph these points on a polar graph.

θ

r

3

2

0

2

3

4

3

3

2

5

3

2

6 3 0 3 6 3 0 3 6

If we look at how the points are arranged,

it appears that this is a circle whose

center is at (3, 0) with a radius of 3.

The graph from the previous example is a circle whose center is not at the pole.

The equations for this graph are

r = a cos θ with center at (½a, 0)

r = a sin θ with center at

a = the diameter of the circle

12,

2

a

Graphing using Symmetry

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

1. The line 2

2

Replace (r, θ) by(r, − ) or (-r, -)

2. The polar axis

Replace (r, θ) by(r, -) or (-r, − )

0

3. The pole

Replace (r, θ) by(r, + ) or (-r, )

Quick Tests for Symmetry in Polar Coordinates

1. The graph of sin r f

is symmetric with respect to the line .2

2. The graph of cos r g

is symmetric with respect to the polar axis.

Example 2

Use symmetry to sketch the graph

r = 2 + 4sin This graph has

symmetry with

We will only look at

points in the 1st and

4th quadrants and

then use symmetry.

r

6

2

11

6

3

2

0 2

4

6

0

2

the line 2

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

The graph in the previous example is called a limaçon with a loop.

The equations for a graph of a limaçon with and without a loop are

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

If a > b, then there is no loop.If a < b, then there is a loop.

End of 1st Day

The two polar graphs we did today are

1. a circle whose center is not at the pole.

2. A limaçon with and without a loop.

2nd Day

Today we will look at two more polar

graphs.

1. Cardioid

r = a ± b sin r = a ± b cos where a = b

Example 3

Graph r = 4 + 4cos This graph is symmetric to the polar axis.

To graph this cardioid 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 = 1

= 0

So our maximum value of |r| is

r = 4 + 4cos(0)

r = 8

Now we need to find the zeroes of r.

This will be where the polar equation is

equal to 0.

In the example,

4 + 4cos = 0

Now graph the maximum point and the zero point.

r

2

3

3

6

r = 4 + 4cos θ

2

2

4

Using symmetry we can graph the points in the 3rd and 4th quadrants. Now we can draw the graph.

Now the 2nd graph:

2. Rose Curves

r = a sin nr = a cos n

If the trig ratio is cosine and n is odd the graph is symmetric with the polar axis. If the trig ratio is

sine and n is odd then the graph is symmetric with

When n is even, then both graphs are symmetric with the pole, polar axis, and

the line .2

the line .2

Furthermore, the number of petals on the curve depends on whether n is even or odd.

If n is odd, then there are n petals on the rose curve.

If n is even, then there are 2n petals on the rose curve.

Example 4

Graph r = 4sin 2.The graph will have 4 petals.

It will be symmetric with the pole, polar

axis, and the line .2

Maximum value of |r| = 4 when sin 2 = ±1.

The zeroes are found when sin 2 = 0.

3 5 72 , , ,

2 2 2 2

3 5 7so , , , .

4 4 4 4

2 0, ,2 ,3 3so 0, , , .

2 2

r = 4sin 2

We can graph the rose curve using this

information and finding one point where1

sin 22

26

12

112

6

11

12

1and sin 2 .

2

r = 4sin 2

Using symmetry we can now find other points to help us graph the petals.The rose curves also have symmetry on the line containing the maximum point.

r = 4sin 2

The polar graphs that you need to know are:

1. A circle whose center is the pole

2. A circle whose center is not the pole

3. A limaçon with and without a loop

4. A cardioid

5. A rose curve