Activity Overview
In this activity, students will explore the polar equations
r(θ) = a + bcos(nθ) and r(θ) = a + bsin(nθ). They learn that by changing values a, b, and n, different polar curves can be generated.
Before the Activity
See the attached PDF file for detailed instructions for this activity
Print pages 89 - 95 from the attached PDF file for your class
During the Activity
Distribute the pages to the class.
Follow the Activity procedures:
Define r(θ) = bcos(θ) for b = 2 and graph it
Change value of b to 3, -4, and -5 and observe the different graphs
Observe that change in the absolute value of b results in an increase in the radius of the circle; for negative value of b the circle is reflected about the y-axis
Rose Polar Curve
Define r(θ) = 4cos(nθ) for n = 1 and graph it
Change the value of n to 2, 3, and 4 and observe the different graphs
Note that if n is odd, the number of rose leaves is equal to n; if n is even, the number of leaves is twice n
Observe that as the value of b increases the length of each leaf increases; for negative value of b the rose is reflected about the y-axis
Limacon Curve
Define r(θ) = a + bcos (θ) for a = 1, b = 2, n = 1 and graph it
Change the values of a and b and observe the graphs
Observe that the absolute value of b is always greater than the absolute value of a, the greater the sum of the absolute values of a and b the larger the outer loop
Also, for negative value of b the limacon is reflected about the y-axis
Cardioid Curve
Define r(θ) = a + bcos(θ) for a = 2, b = 2, n = 1 and graph it
For a = b = 4, and a = b = 5 graph the function
Observe that the absolute values of a and b are equal and their sum results in a larger loop
Change the value of b to -3 and graph the function
Observe that for negative value of b the cardioid is reflected about the y-axis
Solve other polar coordinate problems
After the Activity
Students complete the Student Activity sheet.
Review student results:
As a class, discuss questions that appeared to be more challenging
Re-teach concepts as necessary