Graph each polar equation. Also, identify the type of polar graph.
r² = 4 cos 2θ

Verified step by step guidance

1. Recognize that the given equation is in polar form: \(r^{2} = 4 \cos 2\theta\). This suggests the graph might be a rose curve or a lemniscate because of the \(\cos 2\theta\) term and the squared radius.
2. Recall the general form for rose curves: \(r = a \cos n\theta\) or \(r = a \sin n\theta\). However, since we have \(r^{2}\) instead of \(r\), this indicates the graph is a lemniscate, which often has equations like \(r^{2} = a^{2} \cos 2\theta\) or \(r^{2} = a^{2} \sin 2\theta\).
3. To graph the equation, consider values of \(\theta\) where \(\cos 2\theta\) is positive or zero, because \(r^{2}\) must be non-negative. For angles where \(\cos 2\theta\) is negative, \(r^{2}\) would be negative, which is not possible for real \(r\).
4. Calculate \(r\) for several key angles by substituting values of \(\theta\) into \(r^{2} = 4 \cos 2\theta\), then take the positive and negative square roots to find \(r\). Plot these points in polar coordinates to sketch the graph.
5. Identify the graph type: since the equation matches the form of a lemniscate of Bernoulli (a figure-eight shape), conclude that the graph is a lemniscate centered at the pole with petals aligned along the \(\theta = 0\) and \(\theta = \pi/2\) directions.

Key Concepts

Polar Coordinates and Equations

Polar coordinates represent points using a radius (r) and an angle (θ) from the positive x-axis. Polar equations express relationships between r and θ, allowing the graphing of curves in the polar plane. Understanding how to interpret and plot these equations is essential for visualizing the graph.

Types of Polar Graphs

Common polar graphs include circles, limaçons, roses, and lemniscates. Each type has characteristic equations and shapes. Recognizing the form of the equation, such as r² = a cos 2θ, helps identify the graph type, which in this case is a lemniscate, a figure-eight shaped curve.

Trigonometric Identities and Symmetry in Polar Graphs

Trigonometric functions like cosine and sine influence the symmetry and shape of polar graphs. The multiple angle (2θ) affects the number of petals or loops. Using identities and understanding symmetry properties aids in sketching the graph accurately and predicting its features.