Exploring Polar Curves

Use the mouse to adjust the sliders so that, a = 2, i = 2, b = 0, and c = 0 and then change the value of t from 0° to 360°. Which directions do A and C move as t changes from 0° to 360°? Now change value of a from 1 to 5. What do you see?

Set a = 2, b = 0, and c = 0. Then change the value of i from 1 to 2, to 3, … , to 10. Observe how the number of petals change as you change the value of i. a. What is the corresponding value of i in the function r(x)? b. What is the relationship between i and number of petals? c. Can we create a shape with 6 or 10 petals?

Now set i = 1, b = 0, c = 1. a. Then slowly change the value of a from 1 to 2. Describe the shape of the graph as the value of a change. b. Then slowly change the value of a from 1 to 0. Describe the shape of the graph as the value of a change.

Now set a = 1, b = 0 and c = 1. As the value of i change from 1 to 5, observe how the number of petals change. a. What is the relationship between i and number of petals?  b. Make your guess about the function that generates 6 or 10 petals. Check your conjecture by changing the value of i. Is the relationship between i and the number of petals the same as in Task 2.

Set a = c = 0 and b = 2. Do Task 2 and 3 replace a by b and i by k. What is the relationship between b and k in the function f(x) = b sin(kx) and the shape of the graph?