Question

Identify, graph, and state the symmetries for the polar equation r=2+2sintheta.

Answer 1

Answer:

Graph = Cardioid

Axis of symmetry = y-axis

Critical points= $\frac{\pi}{2} , \frac{3\pi }{2}$

Step-by-step explanation:

General equation for this type of cardioid is:

a ± b sinθ

Condition for a cardioid = $\frac{a}{b} = 1$

Axis of symmerty according to the graph of 2 + 2 sinθ is along y-axis.

Critical points:

r = 2 + 2 sinθ  ⇒ r = 2(1 + sinθ) ⇒ r' = 2 cosθ  

 ∵derivative of 1 + sinθ = cosθ

For finding critical point the derivative is equal to zero,

2 cosθ = 0    ⇒    cosθ = 0

the value of cosθ is equal to zero at intervals: $\frac{\pi}{2} , \frac{3\pi }{2}$

So, critical points = $\frac{\pi}{2} , \frac{3\pi }{2}$