Polar Spiral creates Polar Flower!

by

Priscilla Alexander

This write-up is for students learning about the constants of polar coordinates.

Investigate

Case 1: When a and b are equal, and n varies between real numbers:

The domain and range of the flower is equal to

2b, if b is an integer.

The amount of pedals formed by the flower is never more than the value of n.

Now see what happens when n is an integer.

Case 2: When a and b are not equal and n varies between real numbers:

Two flowers are created. There is a smaller one and a larger one.

The length of the larger pedal is a+b

The center of the two flowers are at the origin.

The amount of pedals formed by the flower is never more than the value of n.

Now see what happens when n is an integer:

Something Interesting!

When the following expression is given, the graph shows a curve that resembles a reflection of the spiral created in the golden rectangle.

Investigate

What makes this graph different from the cosine graph is the way the sine graph approaches the figure that appears to look like an flower.

The graph to the left is the start of the sine change . The graph to the right is the start of the cosine change.

Instead of the have a starting interception point a zero and two on the x-axis, the interception points are at negative one and one. Also, instead of intercepting the y-axis at negative one and one, the graph intercepts at zero and two.

However, everything else remains the same as the cosine graph.

See movie when a and b are the same and n varies between real numbers.

See movie when a and b are the same and n varies between integers.

See movie when a and b are different and n varies between real numbers.

See what happens when n varies between integers.

Conclusion, both of the graphs produce the same picture, except the beginnings of the sine graph starts changing on the y-axis. Where as, the beginning of the cosine graph changes start on the x-axis.

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