Question

identify the graph of $r = 2 + \cos(\theta)$.

Explanation:

Step1: Recall Polar Curve Type

The equation \( r = a + b\cos(\theta) \) is a limaçon. For \( a > b > 0 \), it is a limaçon without an inner loop (since \( a = 2 \), \( b = 1 \), and \( 2 > 1 \)).

Step2: Analyze Maximum/Minimum \( r \)

• When \( \cos(\theta) = 1 \) (i.e., \( \theta = 0 \)), \( r = 2 + 1 = 3 \).
• When \( \cos(\theta) = -1 \) (i.e., \( \theta = \pi \)), \( r = 2 - 1 = 1 \).

Step3: Compare with Graphs

• The rightmost point (at \( \theta = 0 \)) should be at \( r = 3 \), and the leftmost (at \( \theta = \pi \)) at \( r = 1 \). The top-left graph (first row, first column) has a horizontal axis, rightmost near 3, leftmost near 1, matching the limaçon shape (oval - like, no inner loop). The other graphs: second row fourth has a loop (invalid, since \( a > b \)), others have different radii.

Answer:

The graph in the top - left (first row, first column)