# Activities

• • • ##### Subject Area

• Math: Precalculus: Polar Equations

• ##### Author 9-12

60 Minutes

• ##### Software

TI InterActive!™

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

## Exploring Polar Equations

#### 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.

• 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