Polar Equations
By: Lacy Gainey
Investigate r = a + b cos(kθ) for various values of a, b, and k.
Lets start by setting a and b equal to 1 and observing how different values of k affect the graph of r = a + b cos(kθ) .
Our parameter k seems to represent the number of pedals. ± a/b appears to represent the y-intercepts of the graphs with an odd number of petals.
Lets see if this holds for other values of a and b when k =5.
r = 2 + 2 cos(5θ)
r = 5 + 5 cos(5θ)
r = 10 + 10 cos(5θ)
As a/b increase the shape of the graph stays the same, but this size of rose increases.
We can conclude:
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Parameter k represents the number of petals in the rose.
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When k is odd and a=b, ± a/b represent the y-intercepts.
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When k is even, ± a/b does not represent the y-intercepts.
Next, lets looks at r = b cos(kθ) for various values of b and k.
Let b=1 and k=3
r = 1 cos(3θ)
Parameter k still seems to represent the number of petals, but +b appears to represent the x-intercept instead of the y-intercept.
Lets look at what happens when we change b, but hold k constant.
The video below illustrates -5 ≤ b ≤ 5.
Lets look at what happens when we change k and hold b constant.
Notice that when parameter k is even, there are 2k number of petals. Additionally, x-intercepts appear at –b and +b when k is even.
We can conclude:
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When parameter k is odd, k represents the number of petals in the rose.
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+b represents the x-intercept.
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When parameter k is even, 2k represents the number of petals in the rose.
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– b and +b represent the x-intercepts.
What do you think will happen if cos(kθ) is replaced with sin(kθ) in both equations.
The overall shape and size are the same, but it appears that the graph has been rotated 90° to the right.
