Graphs of Polar Equations

Pre-Calculus · Axiom Academy

LESSON Graphs of Polar Equations A polar equation gives the radius for each angle. Sweep the angle, plot the points, and the curve draws itself. 1. The Recipe: One Angle at a Time To plot a polar equation, pick an angle , compute , and place a point that distance out along that angle's ray. Watch it for the cardioid : as the angle rotates to each value, the radius arm stretches to and drops a point. the radius is whatever the angle feeds in 2. Sweep the Angle, Trace the Curve A handful of points only suggests the shape. Now let run continuously from 0 to . The radius arm glides, its tip never lifts, and the cardioid traces itself in one smooth pass — the cusp appearing exactly where the radius collapses to r = 0 . . The point sits at its maximum distance 2a on the positive x -axis. . The radius collapses to the origin, carving the heart's sharp dent. r shrinks smoothly then grows back , so the trace curves around continuously. At we return to r = 2 — one full sweep closes the curve. From the Greek kardia , "heart." Any equation of the form or traces this heart shape, reaching a maximum distance 2a from the origin. The and the / choice just rotate which way the cusp points. 3. The Same Sweep Predicts the Shape

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