How to graph a cardioid in polar coordinates

A cardioid reaches the pole at a cusp and curves around into a heart-like shape. In its equation, the constant and the sine or cosine coefficient have the same magnitude. We can use symmetry and a few radius values to sketch it.

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Use the sign and trig function to find the orientation

For positive $a$, the curves $r=a+a\cos \theta$ and $r=a-a\cos \theta$ are symmetric about the horizontal axis. The plus form extends right and the minus form extends left. The sine versions are symmetric about the vertical axis, extending up for plus and down for minus.

Find key points for r equal to four plus four cosine theta

Key radius values

Angle	Radius
$0$	$8$
$\frac{\pi }{2}$	$4$
$\pi$	$0$
$\frac{3\pi }{2}$	$4$
$2\pi$	$8$

The farthest point is $(8,0)$. The values at $\theta =\frac{\pi }{2}$ and $\theta =\frac{3\pi }{2}$ give $(0,4)$ and $(0,-4)$. At $\theta =\pi$, the radius is zero and the curve reaches its cusp at the pole.

The cardioid has horizontal symmetry, a cusp at the pole, and its farthest point at (8, 0).

Reflect the curve by changing the sign

For $r=4-4\cos \theta$, the farthest point is $(-8,0)$ and the cusp is still at the pole. The graph is the reflection of the plus-cosine cardioid across the vertical axis.

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