Classify the limacon represented by the equation r = 3 + 2sin (theta)

Look to the attached figures

Step-by-step explanation:

* Lets study the limacon curve  

- Equations of the form:  

# r = a + b sin θ  

# r = a – b sin θ  

# r = a + b cos θ  

# r = a – b cos θ  

All will produce limacons.  

* Lets examine what happens for various values of a and b.  

- When the value of a is less than the value of b, the graph is

 a limacon with and inner loop.  ⇒ figure (1)  

- When the value of a is greater than the value of b, the graph is  

 a dimpled limacon.  ⇒ figure (2)

- When the value of a is greater than or equal to the value of 2b,  

 the graph is a convex limacon.  ⇒ figure (3)

- When the value of a equals the value of b, the graph is a special

 case of the limacon.  It is called a cardioid.  ⇒ figure (4)

* Notice that, in each of the graphs of the liamsons, changing

 from sine to cosine does not affect the shape of the graph just its  

 orientation.  

- Equations using sine will be symmetric to the vertical axis  

- Equations using cosine are symmetric to the horizontal axis.  

* The sign of b will also affect their orientation.

- The large of the graph under the x-axis or left to the y-axis b is -ve

- The large of the graph over the x-axis or right to the y-axis b is +ve

* Lets study our problem

∵ r = 3 + 2sinФ

∴ a = 3 and b = 2

- a > b

∴ the graph is   a dimpled limacon

∴ It is symmetric around the y-axis (sinФ)

∵ b = 2 ⇒ positive value

∴ The large of the graph over the x-axis

* Look at the attached graph ⇒ figure (5)