In Exercises 55-58, use a graphing utility to find the...

In Exercises $55-58,$ use a graphing utility to find the smallest value of $\theta$ max, with $\theta \min =0,$ such that the entire curve is graphed exactly once without retracing.
$$r=1-2 \cos \theta$$

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Key Concepts

Tracing and Periodicity in Polar Graphs

Because polar graphs can be traced more than once as the angle parameter changes, it is important to identify the minimal interval of the angle variable over which the entire curve is traced exactly once. This eliminates redundant parts of the graph and ensures an accurate representation of the curve.

Understanding Negative Radius Values

In polar coordinates, a negative radius indicates that the point is plotted in the direction opposite to the angle given. This concept is crucial when analyzing polar graphs since negative values can lead to overlapping sections of the curve, affecting how the complete graph is traced without repetition.

Polar Coordinates

Polar coordinates offer an alternative to Cartesian coordinates by describing points in the plane using a radius and an angle. This system is particularly useful for representing curves where relationships between the distance from a central point and an angle are more natural than x and y coordinates.

Graphing Polar Equations

Graphing polar equations involves plotting points whose positions are determined by a radius and an angle. It requires understanding how varying the angle, often within a specific interval, affects the radius and the overall shape of the graph, and it may involve handling cases where the radius becomes negative.

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