Question

identify the graph of $r = 2\cos(\theta)$. images of four graphs are shown, with the first row having two circular graphs and the second row having a lemniscate and a circular graph

Explanation:

Step1: Recall Polar Circle Formula

The polar equation \( r = 2a\cos(\theta) \) represents a circle with diameter \( 2a \), centered at \( (a, 0) \) in polar coordinates (or \( (a, 0) \) in Cartesian, \( x = a, y = 0 \)). For \( r = 2\cos(\theta) \), \( a = 1 \), so diameter is \( 2 \), radius \( 1 \), centered at \( (1, 0) \) (Cartesian: \( x = 1, y = 0 \)).

Step2: Analyze Graphs

• Top - Left: Circle centered left, radius ~2? No.
• Top - Right: Circle with numbers 1 - 5, not centered at \( x = 1 \).
• Bottom - Left: Lemniscate (infinity symbol), not a circle.
• Bottom - Right: Circle centered at \( x = 1 \) (since it touches \( x = 2 \) and is around \( x = 1 \)), radius 1. Matches \( r = 2\cos(\theta) \) (center \( (1,0) \), radius 1).

Answer:

The graph at the Bottom - Right.