Section 10.8
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Transcript of Section 10.8
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).