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This is Activity 11 from the EXPLORATIONS Book: TI InterActive! Math for High School

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