Match each equation with its polar graph from choices A–D.
r = cos 2θ

Verified step by step guidance

1. Recall that the equation given is in polar form: \(r = \cos 2\theta\). This is a type of rose curve, which generally has the form \(r = \cos n\theta\) or \(r = \sin n\theta\).
2. Identify the parameter \(n\) in the equation. Here, \(n = 2\), which means the rose curve will have \$2n\( petals if \)n$ is even, so it will have 4 petals.
3. Understand the shape: For \(r = \cos 2\theta\), the petals are symmetrically placed around the origin, with petals aligned along the angles where \(\cos 2\theta\) reaches its maximum values.
4. To match the graph, look for a polar plot with 4 petals evenly spaced around the origin, each petal corresponding to the angles where \(\cos 2\theta = 1\) or \(-1\).
5. Compare the given graph options A–D to this description, selecting the one that shows a 4-petal rose curve centered at the origin with petals aligned along the axes.

Key Concepts

Polar Coordinates and Graphs

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian (x, y). Understanding how equations like r = cos 2θ plot points based on θ is essential to visualize and match the graph correctly.

Rose Curves and Their Properties

Equations of the form r = cos(nθ) or r = sin(nθ) produce rose curves with petals. When n is even, the curve has 2n petals; when n is odd, it has n petals. Recognizing this helps identify the shape and number of petals in the graph.

Symmetry in Polar Graphs

Polar graphs like r = cos 2θ exhibit symmetry about the polar axis or other lines. Understanding symmetry properties aids in matching the equation to its graph by predicting the orientation and repetition of petals.