In Exercises 35–44, test for symmetry and then graph each polar equation. r = 1 / 1−cos θ

Verified step by step guidance

1. Step 1: Test for Symmetry with Respect to the Polar Axis (x-axis).
2. Step 2: Test for Symmetry with Respect to the Line θ = π/2 (y-axis).
3. Step 3: Test for Symmetry with Respect to the Pole (origin).
4. Step 4: Analyze the behavior of the equation as θ approaches specific values.
5. Step 5: Sketch the graph based on the symmetry and behavior analysis.

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 pole), and 'θ' represents the angle. Understanding how to convert between polar and Cartesian coordinates is essential for graphing polar equations.

Symmetry in Polar Graphs

Symmetry in polar graphs can be tested by substituting specific values into the polar equation. A graph is symmetric about the polar axis if replacing θ with -θ yields the same r, and symmetric about the line θ = π/2 if replacing θ with π - θ gives the same r. Recognizing these symmetries helps in sketching the graph accurately.

Graphing Polar Equations

Graphing polar equations involves plotting points defined by the radius 'r' and angle 'θ'. The shape of the graph can vary significantly based on the equation's form. For the given equation, r = 1 / (1 - cos θ), understanding how to manipulate and evaluate the equation at various angles is crucial for accurately depicting the graph.