Graphing r = 2 + 2cos(θ)

Calculus 2 · Axiom Academy

Build a table of r-values, plot the points, and recognize the cardioid. Graph the polar curve . Find the maximum distance from the origin, locate where the curve passes through the pole, and identify the shape. Nice work — you graphed a cardioid from scratch. Here's the playbook: Build a table first: evaluate r at convenient angles (multiples of ) to learn how the curve breathes in and out. Find the pole crossings: where r = 0 the curve passes through the origin. For that's , producing a cusp. Recognize the family: and are cardioids — here a = 2 , so the max is r = 2a = 4 . Use symmetry: a cosine cardioid is symmetric about the polar axis ( x -axis); a sine cardioid is symmetric about . The same recipe sketches any polar curve: tabulate r , mark the pole crossings, then exploit symmetry. Meet or and you'll know them on sight.

This is the written version of the interactive lesson above. See the full Calculus 2 course.