Section 10.8Graphs of Polar

Equations by Hand

Types of Polar Graphs

1. Circle with center not at the pole.2. Limaçon with and without a loop.3. Cardioid4. Rose Curve

Ways to Graph

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

and the zeros.

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.

Sketch the graph r = 6cos θ using all ways. Describe the graph in detail.A circle with the center at (3, 0) and a radius of 3.1. Find the maximum r value.

r = 6 cos 0 = 6(6, 0)

2. The way to use a table is to find an exact ordered pair on the graph.When is cosine ½?

Find r when

3

.3

6cos3r 3

3, 3

3. What symmetry will the graph have?

Symmetry with the polar axis.Now reflect all the points you have across the polar axis.

5. Finally find the zeros of the graph.r = 6 cos θ0 = 6 cos θ

2

0, 2

Now graph the circle.

Sketch the graph r = 4sin θ.

Sketch the graph r = 2 + 4sin Describe the graph in detail.This is a limaçon with a loop.Find the maximum r, find the zeros, graph all points possible with exact integer values for r, and finally use symmetry to find other points.

r = 2 + 4sin 1. Find the maximum r.

r = 6

2. Find the tip of the loop.

r = -2

2 4sin2r 6, 2

32 4sin 2r 32, 2

3. Find the zeros.0 = 2 + 4sin

4. Find the point with exact integer values for r.

r = 4

116 110, 6

2 4sin6r

4, 6

4. r = 2 + 4sin 0r = 2 (2, 0

5. Now use symmetry to find other points on the graph.

Sketch the graph r = 4 – 4cos θ.

Sketch the graph r = 6 – 4sin θ.

Sketch the graph r = 4sin 2.Describe the graph in detail.A rose curve Since n is even, there will be 2n petals.A rose curve with 4 petals.

Find the first petal by setting the equation equal to the maximum r then use the angle measure between petals to find the 4 petals.

360 2Each petal will be or apart.4 4

4 = 4sin 2θ1 = sin 2θ2θ = 90°θ = 45°So the tip of a petal is at (4, 45°).Use this information and the angle measure between petals to find the other 3 petals.

Sketch the graph r = 8cos 3.1. Since n is odd, there are n petals.2. So there are 3 petals.3. These 3 petals are 120° apart.4. Find the 1st petal the same way as

before.The 1st petal is at (8, 0).