what type of limacon is graphed by the polar equation r=4+2sin (theta)?identify the axis of symmetry and horizontal and vertical intercepts.

Answer 1

The type of limacon graphed by the polar equation r=4+2sin(theta) is an eccentric limacon with an inner loop.

The axis of symmetry is the polar axis (the horizontal line r=0).

To find the horizontal intercept, set r=0 in the equation:
0 = 4 + 2sin(theta)
-2 = sin(theta)
theta = -pi/6 or 11pi/6

To find the vertical intercept, set theta = pi/2 in the equation:
r = 4 + 2sin(pi/2)
r = 4+2(1)
r = 6

So, the vertical intercept is at (pi/2,6).