1. Warm Up—begin after the Quiz
Find the absolute max & min value in
and where they occur: 2.
Determine if the graph of the function is symmetric about x, y or the origin: 3.
Use trig ID’s to simplify:
Hint: Use graph and table, see if you can answer in Radians
Hint: symmetry about…
x- axis (x,-y) when (x,y)
y-axis f(-x) = f(x)
Origin f(-x) = -f(x)
Hint: use the sum and difference formulas for sine and cosine…remember those?

2. Graphs of Polar Equations
Section 6.5
Graphs of Polar Equations

3. 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.
Function Mode Polar Mode
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.

4. Checking for Symmetry
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. Recall: we learned algebraic tests for rectangular equations in section 1.2 to check for symmetry:
x- axis show (x,-y) when (x,y)
y-axis show f(-x) = f(x)
Origin show f(-x) = -f(x) OR
Even show f(-x) = f(x)
Odd show f(-x) = -f(x)

5. 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, (90°).
The origin (the pole) as a point of symmetry, (looks the same upside-down and right-side up).
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.

6. Symmetry Test 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 With
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 pole, (origin) (r, θ) (-r, θ) or (r, θ + π)

7. Symmetry Test for Polar Graphs
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. First look at the graph to verify what you are trying to prove: If your graph doesn’t look like this, try resetting your calculator. As you can see the graph looks to be symmetric about y.

8. Analyzing Polar Graphs
We analyze graphs of polar equations in much the same way that we analyze the graphs of rectangular equations. Looking at example 1, r = 4 sin 3θ, Trace can be used to help determine the range (max/min y-value) 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.

9. Maximum r-value Alright, so who wondered where π/6 and π/2 come from??
A maximum value for |r| is a maximum r-value for a polar equation. A maximum r-value occurs at a point on the curve 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.
Alright, so who wondered where π/6 and π/2 come from??
Using the FUNC graph, you can see
that y=4, when x=.532 (aprox.) 
You can use: CALC MAX/MIN

10. Maximum r-value
To find the maximum r-values we must find maximum values |r| as opposed to the directed distance r. Example 2 shows one way to find maximum r-values graphically. EXAMPLE 2 Finding Maximum r-Values Find the maximum r-value of r = 2 +2 cos θ. Because we are interested in values of r, use the graph of the rec. eqn in function mode. You will see that the max value of r or y, is 4, which occurs when x or π, is a multiple of 2 π

11. Maximum r-value
EXAMPLE 3 Finding Maximum r-Values Identify the points on the graph of r = 3 cos 2θ for 0 ≤ θ ≤ 2π that give maximum r-values. Polar Function Abs val. Function Using trace we can see that there are 4 points that have a max distance of 3 from the pole

12. Rose Curves
The curve in Example 1 is a 3-petal rose curve and the curve in Example 3 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. r = 4 sin 3θ r = 3 cos 2θ

13. Rose Curves 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

14. Rose Curves EXAMPLE 4 Analyzing a Rose Curve
Analyze the graph of the rose curve r = 3 sin 4θ.
Domain:
Range:
Continuous:
Symmetry:
Bounded:
Max r-value:
Asymptotes:
Number of Petals:

15. Limaçon Curves (“LEE-ma-sohn”)
The limaçon curves 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 = cos θ
Convex: When a/b ≥ 2 as in r = 2 + cos θ

16. Limaçon Curves 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

17. Limaçon Curves
EXAMPLE 5 Analyzing a Limaçon Curve Analyze the graph of r = 3 – 3 sin θ
cardiod
Domain:
Range:
Continuous:
Symmetry:
Bounded:
Max r-value:
Asymptotes:

18. Other Polar Curves-Spirals
All the polar curves we have graphed so far have been bounded. The spiral in Example 6 is unbounded. EXAMPLE 6 Analyzing the Spiral Archimedes Analyze the graph of r = θ for [0, 10π] Note: these are not bounded
We choose 10π, so that we can see more spirals, Notice that the spiral crosses the axis 10 times!
Domain:
Range:
Continuous:
Symmetry:
Bounded:
Max r-value:
Asymptotes:

19. Other Polar Curves-Lemniscate
From Latin meaning, “decorated with ribbons” The lemniscate curves are graphs of polar equation of the form r2 = a2 sin 2θ for [0, 2π] Example Graph: or

20. Other Polar Curves-Lemniscate
Example 7 Analyzing a Lemniscante Curve Analyze the graph of r2 = 4 cos 2θ for [0, 2π]
Domain:
Range:
Continuous:
Symmetry:
Bounded:
Max r-value:
Asymptotes:

21. Project Due 3/30 or 4/2 (Friday/Monday)
Homework: Pg 540: 3-43 (o), 49, 51
Project Due 3/30 or 4/2 (Friday/Monday)
It is still due, even if you are out for a school related absence!!
Test 4/2 or 4/3 (Monday/Tuesday)
Review due test day, go to Help on-line for selected solutions
ReQuiz next Tuesday or Thursday Morning!!