Given the polar equation \( r = 1 - 2\cos(\theta) \), identify the type of conic section and sketch the graph.
Answer
Step 3: The graph of a cardioid opening in the negative x-direction starts at the pole (0,0) and extends left along the x-axis. The point farthest from the pole is 2 units to the left of the pole.
Step 1 ：Step 1: Observe the given equation. In polar coordinates, the equation of a cardioid can be written as \( r = a(1 \pm \cos(\theta)) \) or \( r = a(1 \pm \sin(\theta)) \).
Step 2 ：Step 2: Comparing the given equation with the standard form, it is clear that the given equation represents a cardioid. The value of 'a' is -2, which means the cardioid opens in the negative x-direction.
Step 3 ：Step 3: The graph of a cardioid opening in the negative x-direction starts at the pole (0,0) and extends left along the x-axis. The point farthest from the pole is 2 units to the left of the pole.
