How do you graph ##r=8cos2theta## on a graphing utility?

Use the Cartesian form.

The period for the graph is ##(2pi)/2=pi##.

In one half period ##theta in [-pi/4, pi/4], r>=0; r<0##. for the other half

##theta in (pi/4, pi/2]##. In the double period theta in [0, 2pi], two

loops are created.

The cartesian form of the given equation is

##(x^2+y^2)sqrt(x^2+y^2)=9(x^2-y^2)## that befits the graphic utility

that is readily available here.

graph{(x^2+y^2)^1.5-8(x^2-y^2)=0 [-10, 10, -5, 5]}