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This animation is from the problem in your book r = 1-3cos(theta). I replaced the 1 with a t and allow t to varry from -5 to 5.

>	r:=proc(x) t+3*cos(1*x) end:

>	animate({ [r(th),th,th=0..2*Pi] },t=-5..5,coords=polar,numpoints=100,frames=100,thickness=3);

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