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

Verified step by step guidance

1. Step 1: Recognize the form of the polar equation. The given equation is \( r = 2 + 2 \cos \theta \). This is a type of polar equation known as a limaçon, specifically a cardioid, because the coefficients of \( r \) and \( \cos \theta \) are equal.
2. Step 2: Understand the general form of a limaçon. A limaçon can be expressed as \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). In this case, \( a = 2 \) and \( b = 2 \), which indicates a cardioid since \( a = b \).
3. Step 3: Determine the symmetry of the graph. Since the equation involves \( \cos \theta \), the graph is symmetric about the polar axis (the horizontal axis in polar coordinates).
4. Step 4: Plot key points. Calculate \( r \) for key angles \( \theta \) such as \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \). For example, when \( \theta = 0 \), \( r = 2 + 2 \cos(0) = 4 \). When \( \theta = \pi \), \( r = 2 + 2 \cos(\pi) = 0 \).
5. Step 5: Sketch the graph. Use the calculated points and symmetry to sketch the cardioid. The graph will have a 'heart-shaped' appearance with a cusp at the pole (origin) when \( \theta = \pi \).

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance from a reference point (the pole) and an angle from a reference direction. In polar equations, 'r' denotes the radius (distance from the origin), and 'θ' represents the angle. Understanding how to convert between polar and Cartesian coordinates is essential for graphing polar equations.

Polar Equations

Polar equations express relationships between the radius 'r' and the angle 'θ'. The given equation, r = 2 + 2 cos θ, is a type of polar equation that can describe various shapes, such as circles or limaçons. Analyzing the form of the equation helps in identifying the type of graph it represents.

Graphing Polar Equations

Graphing polar equations involves plotting points based on the values of 'r' for different angles 'θ'. This process often requires evaluating the equation at various angles to see how 'r' changes, which helps in visualizing the graph. Identifying key features, such as symmetry and intercepts, is crucial for accurately representing the polar graph.