graph the polar equation. 20) r = 2 + 2sin θ

Explanation:

Step1: Find values of (r) for key - angles

We know that in polar coordinates, we can find (r) for different values of (\theta). Let's consider (\theta = 0,\frac{\pi}{2},\pi,\frac{3\pi}{2}). When (\theta = 0), (r=2 + 2\sin(0)=2+0 = 2). When (\theta=\frac{\pi}{2}), (r=2 + 2\sin(\frac{\pi}{2})=2 + 2=4). When (\theta=\pi), (r=2+2\sin(\pi)=2 + 0=2). When (\theta=\frac{3\pi}{2}), (r=2+2\sin(\frac{3\pi}{2})=2-2 = 0).

Step2: Analyze the general form of the polar equation

The polar equation (r = a\pm b\sin\theta) with (a = b=2) represents a cardioid. The general shape of a cardioid (r=a + b\sin\theta) is symmetric about the (y) - axis in the polar - coordinate system.

Step3: Plot the points and draw the curve

Plot the points ((2,0)), ((4,\frac{\pi}{2})), ((2,\pi)), ((0,\frac{3\pi}{2})) on the polar - coordinate plane. Then, connect these points smoothly to form a heart - shaped curve (cardioid) symmetric about the (y) - axis.

Answer:

The graph is a cardioid symmetric about the (y) - axis with key points ((r,\theta)=(2,0)), ((4,\frac{\pi}{2})), ((2,\pi)), ((0,\frac{3\pi}{2})) and the general shape of (r = 2+2\sin\theta).