What type of limacon is graphed by the polar equation r=4+2 sin theta?

Question

Please identify the axis of symmetry and horizontal and vertical intercepts?

Answer 1

We look at the value of the constant versus the value of the term with the sign to determine the type of shape.

As 4 > 2, this is a dimpled limacon

We have y intercepts where θ = π/2 and 3π/2

Then, r = 4 + 2 sin θ = 6 and 2, respectively.

y intercepts are (0, -2) and (0, 6)

We have x intercepts where θ = 0 and π

Then, r = 4 + 2 sin θ = 4 for each

x intercepts are (4, 0) and (-4, 0)

This is symmetric about the y-axis.

We prove symmetry about the y-axis by replacing θ by π - θ, as sin θ = sin (π - θ) and cos θ = cos (π - θ)

r = 4 + 2 sin (π - θ) = 4 + 2 sin θ