$\"\"$

The polar curve is $\"\"$ , where $\"\"$ is a real number .

Graph:

Graph the family of the curves for differrent values of $\"\"$ .

$\"\"$

Graph the curves for $\"\"$ :

$\"\"$

Observe the graphs:

For $\"\"$ , the curve is a circle.

As $\"\"$ increases the left side of the curve gets flatter then has a dimple for $\"\"$ .

The curve has a cusp for $\"\"$ .

For $\"\"$ , the curve has a loop.

$\"\"$

The curve is a circle for $\"\"$ .

As $\"\"$ increases the left side of the curve gets flatten then has a dimple for $\"\"$ .

The curve has a cusp for $\"\"$ .

The curve has a loop for $\"\"$ .