1. 9.7 Graphs of Polar Equations

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Warm-Up
Find the maximum and minimum values of
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Warm-Up #2
Find the maximum and minimum values of
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4. Polar Curves
A Polar Curve consists of all the points (r,θ) satisfying a given equation F(r,θ) = 0.
Often one can solve r from the equation and represent the polar curve in the form r = f(θ), where r represents the distance from the pole as a function of theta.
Some curves are easier to describe with polar coordinates:
(Circle centered at the origin)
(Line through the origin)

5. Graphs of Polar Equations
Example:
Graph the polar equation r = 2cos . 1 2 3 2 –2 –1 1 r 
The graph is a circle of radius 1 whose center is at point (x, y) = (1, 0).
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Graphs of Polar Equations

6. Examples of Polar Curves
For a polar curve that has symmetry
with respect to x-axis:
If (r, q) is on the graph,
so is (r, -q).

7. Examples of Polar Curves
For a polar curve that has symmetry
with respect to y-axis:
If (r, q) is on the graph,
so is (r, p-q)
or (-r, -q).

8. Example: Zeros and Maximum r-values
Find the zeros and the maximum value of for the graph of r = 2cos . 1 2 3
These are the zeros of r.
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Example: Zeros and Maximum r-values

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You try: Analyze the graph of r = 4sin. State the symmetry, find the zeros and maximum value of .
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You try again: Analyze the graph of r = 1-2cos. State the symmetry, find the zeros and maximum value of
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You try again: Analyze the graph of State the symmetry, find the zeros and maximum value of
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12. Special Polar Graphs: Limaçon
Each polar graph below is called a Limaçon. –3 –5 5 3 –5 5 3 –3
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Special Polar Graphs: Limaçon

13. Special Polar Graphs: Lemniscate
Each polar graph below is called a Lemniscate. –5 5 3 –3 –5 5 3 –3
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Special Polar Graphs: Lemniscate

14. Special Polar Graphs: Rose Curve
Each polar graph below is called a Rose curve. –5 5 3 –3 –5 5 3 –3 a a
The graph of will have n petals if n is odd, and 2n petals if n is even.
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Special Polar Graphs: Rose Curve

15. 9.7 Classifying Polar Equations

16. What do you see?

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What do you see?
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