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]}