Jadonna Brewton

Fall 2000

Assignment #11

Problem # 2


r = 2a cos (kq) + b

(k is odd)


(Click on the picture to see a moving graph that shows values of k from -6 to 6.)

Make the following observations about the graphs of the equation when k is odd and the go on to investigate graphs of this equation when "k is even."


Since k is odd, recall that there are k petals with a petal lying on the positive x-axis. We can classify the graphs into three groups:

1. Single roses with closed petals that converge at the origin. This occurs when b = 0 and when b = 2a. The radius of the rose is given by r = 2a + b.

(graphs in black and green).

2. Double roses composed of outer and inner petals. This occurs when b < 2a. There are two radii to consider: the radius of the (1) outer rose is given by r = 2a + b. (2) the inner rose is given by r = 2a - b.

(graphs in purple, red, and blue)

3. Open-petaled roses with an opening or core in the center. This occurs when b > 2a. There are two radii to consider: the radius of the (1) rose is given by r = 2a + b. (2) the core is given by r = 2a - b.

(graphs in cyan and yellow)

 

NOTE: The graphs of the equations with negative values of b look the same. Perhaps they are simply reflections about the y-axis. In cases where the radius was r = 2a + b in a positive b graph, the radius for a negative b graph has a radius of r = 2a - b. And in cases where the radius was r = 2a - b in a positive b graph, the radius for a negative b graph has a rdius of r = 2a + b.

Also, recall that if a is negative, the graph is reflected about the y-axis. And remember that a negative value for k has no effect on the graph since cos Q = cos (- Q).


Single Roses

(with circles to display radii.)

 

 

return


Double Roses

(with circles to display radii of inner and outer roses)

 

return


Open-Petaled Roses

(with circles to display radii of rose and opening, or core.)

 

return


 

 

 

Other equations in write-up # 11

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