Question

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

Explanation:

Step1: Recall Polar Curve Type

The polar equation \( r = a\cos(n\theta) \) (or \( r = a\sin(n\theta) \)) represents a rose curve. For \( r = 2\cos(2\theta) \), \( n = 2 \) (even), so the number of petals is \( 2n = 4 \). The amplitude \( a = 2 \), so the maximum \( r \)-value (petal length) is 2.

Step2: Analyze Graphs

• Check petal count: 4 petals (since \( n = 2 \), even). Eliminate graphs with 3 petals (bottom right).
• Check petal length: Maximum \( r = 2 \), so petals should extend to \( r = 2 \). The top - right graph has 4 petals and petal length consistent with \( r = 2 \). The top - left has a petal near \( r = 2 \) but shape (petals' orientation/width) and the bottom - left has larger scale (numbers go to 6, inconsistent with \( r = 2 \)) are off.

Answer:

The top - right graph (the graph in the top - right quadrant of the four - graph grid)