1. Graphs of Polar Equations

2. Graphs of Polar Equations
Polar curves are actually just special cases of parametric curves. Keep in mind that polar curves are graphed in the (x, y) plane, despite the fact that they are given in terms of r and θ. That is why the polar graph of r = 4 cos θ is a circle rather than a cosine curve.
In function mode, points are determined by a vertical coordinate that changes as the horizontal coordinate moves left to right. In polar mode, points are determined by a directed distance from the pole that changes as the angle sweeps around the pole.

3. Since modern graphing calculators produce these graphs so easily in polar mode, we are frankly going to assume that you do not have to sketch them by hand. Instead we will concentrate on analyzing the properties of the curves.

4. Symmetry
Three types of symmetry figures to be considered will have are:
The x-axis (polar axis) as a line of symmetry.
The y-axis (the line θ = π/2) as a line of symmetry.
The origin (the pole) as a point of symmetry.
All three algebraic tests for symmetry in polar forms require replacing the pair (r, θ), which satisfies the polar equation, with another coordinate pair and determining whether it also satisfies the polar equation.

5. Symmetry Tests for Polar Graphs
The graph of a polar equation has the indicated symmetry if either replacement produces an equivalent polar equation.
To Test for Symmetry Replace By
About the x-axis, (r, θ) (r, -θ) or (-r, π – θ)
About the y-axis, (r, θ) (-r, -θ) or (r, π – θ)
About the x-axis, (r, θ) (-r, θ) or (-r, θ + π)
EXAMPLE 1 Testing for Symmetry
Use the symmetry tests to prove that the graph of r = 4 sin 3θ is symmetric about the y-axis.
EXAMPLE 2 Testing for Symmetry
Use the symmetry test to find out if the graph of r = 5 cos 4θ is symmetric.

6. Analyzing Polar Graphs
We analyze graphs of polar equations in much the same way that we analyze the graphs of rectangular equations. For example, the function r of Example 1 is a continuous function of θ. Also r = 0 when θ = 0 and when θ is any integer multiple of π/3. The domain of this function is the set of all real numbers.
Trace can be used to help determine the range of this polar function. It can be shown that -4 ≤ r ≤ 4.
Usually, we are more interested in the maximum value of |r| rather than the range of r in polar equations. In this case |r| ≤ 4 so we can conclude that the graph is bounded.

7. EXAMPLE 3 Finding Maximum r-Values
A maximum value for |r| is a maximum r-value for a polar equation. A maximum r-value occurs at a point on the cuve that is the maximum distance from the pole. In r = 4 sin 3θ, a maximum r-value occurs at (4, π/6) and (-4, π/2). In fact, we get a maximum r-value at every (r, θ) which represents the tip of one of the three petals.
To find the maximum r-values we must find maximum values |r| as opposed to the directed distance r. Example 3 shows one way to find maximum r-values graphically.
EXAMPLE 3 Finding Maximum r-Values
Find the maximum r-value of r = 2 +2 cos θ.
EXAMPLE 4 Finding Maximum r-Values
Identify the points on the graph of r = 3 cos 2θ for 0 ≤ θ ≤ 2π that give maximum r-values. F 7

8. Rose Curves
The curve in Example 1 is a 3-petal rose curve and the curve in Example 4 is a 4-petal rose curve. The graphs of the polar equations r = a cos nθ and r = a sin nθ, where n is an integer greater than 1, are rose curves. If n is odd, then there are n petals, and if n is even there are 2n petals.
EXAMPLE 5 Analyzing a Rose Curve
Analyze the graph of the rose curve r = 3 sin 4θ.

9. Symmetry: n even, symmetric about x-, y-axis, origin
Graphs of Rose Curves
The graphs of r = a cos nθ and r = a sin nθ, where n > 1 is an integer, have the following characteristics:
Domain: All reals
Range: [-|a|, |a|}
Continuous
Symmetry: n even, symmetric about x-, y-axis, origin
n odd, r = a cos nθ symmetric about x-axis
n odd, r = a sin nθ symmetric about y-axis
Bounded
Maximum r-value: |a|
No asymptotes
Number of petals: n, if n is odd
2n, if n is even F 9

10. r = a ± b sin θ and r = a ± b cos θ
Limaçon Curves
The limaçon curves are graphs are graphs of polar equation of the form
r = a ± b sin θ and r = a ± b cos θ
where a > 0 and b > 0. Limaçon is Old French for “snail”. There are four different shapes of limaçons.
Inner Loop: When a/b < 1 as in r = sin θ
Cardioid: When a/b = 1 as in r = sin θ
Dimpled: When 1 < a/b < 2 as in r = sin θ
Convex: When a/b ≥ 2 as in r = 2 + sin θ

11. EXAMPLE 6 Analyzing a Limaçon Curve
Analyze the graph of r = 3 – 3 sin θ
EXAMPLE 7 Analyzing a Limaçon Curve
Analyze the graph of r = cos θ F 11

12. Graphs of Limaçon Curves
The graphs r = a ± b sin θ and r = a ± b cos θ , where a > 0 and b > 0, have the following characteristics:
Domain: All reals
Range: [a – b , a + b]
Continuous
Symmetry: r = a ± b sin θ, symmetric about the y-axis
r = a ± b cos θ, symmetric about the x-axis
Bounded
Maximum r-value: a + b
No asymptotes F 12

13. Other Polar Curves
All the polar curves we have graphed so far have been bounded. The spiral in Example 8 is unbounded.
EXAMPLE 8 Analyzing the Spiral Archimedes
Analyze the graph of r = θ.
The lemniscate curves are graphs of polar equation of the form
r2 = a2 sin 2θ for [0, 2π]

14. EXAMPLE 9 Analyzing a Lemniscate Curve
Analyze the graph of r2 = 4 cos 2 θ F 14

15. Homework: Pg 540: 3-43 (o), 49, 51 F 15