Question

A family of curves is given by the equations \(r = 1 + c\sin n\theta \) , where \(c\) is a real number and \(n\) is a positive integer. How does the graph change as \(n\) increases? How does it change as \(c\) changes? Illustrate by graphing enough members of the family to support your conclusions.

Short answer

When \(n\) increases, the number of lobes increases and when \(c\) increase, the radius of circular shape increases.

Step-by-step solution

Step 1:

Consider a family of curves is represented by the equation

\(r = 1 + c\sin n\theta \)

Where \(c\) is a real number and \(n\) is a positive integer.

As \(n\) increases we have the following

As we can see from the graph the number of lobes increases receptively as \(n\) increases.

Step 2:

As \(c\) increases we have the following

The radius of the circular shape increases.