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

Verified step by step guidance

1. Recognize that the given polar equation is of the form \(r = a + b \cos \theta\), which typically represents a limaçon curve. The values of \(a\) and \(b\) will determine the specific shape (cardioid, dimpled limaçon, or limaçon with an inner loop).
2. Identify the constants: here, \(a = 2\) and \(b = 2\). Since \(a = b\), this suggests the graph is a cardioid, a special type of limaçon.
3. To graph the equation, create a table of values by choosing several values of \(\theta\) (for example, \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{2}\), \(\pi\), etc.) and calculate the corresponding \(r\) values using \(r = 2 + 2 \cos \theta\).
4. Plot the points \((r, \theta)\) on polar coordinates, where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis. Connect these points smoothly to reveal the shape of the graph.
5. Confirm the shape by noting that when \(\theta = 0\), \(r\) is maximum (\(r = 4\)), and when \(\theta = \pi\), \(r\) is minimum (\(r = 0\)), which matches the behavior of a cardioid.

Key Concepts

Polar Coordinates and Equations

Polar coordinates represent points in a plane using a radius and an angle (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Polar equations express relationships between r and θ, allowing the graphing of curves based on these parameters.

Types of Polar Graphs

Common polar graphs include circles, limaçons, cardioids, roses, and spirals. The equation r = 2 + 2 cos θ is a limaçon, characterized by its shape depending on the coefficients; it can have an inner loop, dimple, or be convex.

Graphing Polar Equations

To graph a polar equation, calculate r for various values of θ, plot the points in polar coordinates, and connect them smoothly. Understanding symmetry and key features like maximum and minimum r values helps in sketching the graph accurately.