By Gloria L. Jones
Case 1: r=a+b cos(kθ) where a=b and k is and odd integer.
Looking at the graphs of r=4+4cos θ, r=4+4 cos(3θ) and r=4+4 cos(5θ)
One can deduce rose-shaped graphs with k leaves of length 2a=2b. As k becomes larger, the leaves appear to grow thinner. Each leaf appears to be at a 360/kْ rotation from each other starting from the positive x-axis and all leaves seem to meet at the origin. The r=4+4 cos θ graph confirms a cardioid shape as opposed to the multi-leaved rose and all graph appear symmetric with respect to the x-axis. Looking at the graphs of r=4+4cos(θ), r=4+4 cos (-3θ) and r=4+4 cos (-5θ), one detects no change from the previous graphs thus confirming that cos (-θ) = cosθ.
Looking at the graphs of r=2+2 cos 7θ and r=-2-2 cos 7θ one notices a reflection
Case 2: r= a+b cos (kθ) where a=b and k is an even integer.
Looking at the graphs of r=4+4 cos (2θ), r=4+4 cos(4θ) and r=4+4 cos (6θ)
Case 3: r=a+b cos(kθ) where a=b and k is a non integer
One notices from the graphs of r=4+4 cos(1/2(θ)), r=4+4 cos (4/3(θ) and r=4+4 cos(11/4(θ)) that for non integer values of k, partial leaves occur though further analysis of specific patterns is left for a future investigation.
Case 4: r=a+b cos (kθ) where a>b and k is an odd integer
Looking at the graphs of r=3+8 cos (5θ) and r=2+4 cos (5θ) and r=-3+5 cos (5θ)
Case 5: r=a+b cos(kθ) where a<b and k is an even integer
When k is even, one notices the smaller leaves mentioned above with odd k are now no longer inside the larger leaves but are now between them: r=3+8 cos (6θ)
Case 6: r=a+b cos(kθ) where a>b and k is an integer
Looking at the graphs of r=4+2 cos(5θ), r=10+2 cos(5θ) and r=-16+2 cos (5θ)
we notice that when a>b, the graphs lose their rose shape though still appear to have k │sections.▓ The sections no longer meet at the origin (and, in fact, the larger the distance between a and b, the further the graph departs from the origin), and the x intercepts appear to be b+/-a.
Case 7: r=b cos(kθ) where k is odd
Looking at the graphs of r=3cosθ, r=3cos(3θ) and r=3cos(5θ)
Case 8: r=b cos(kθ) where k is even
Looking at the graphs of r=4cos(2θ), r=4cos(4θ) and r=4cos(6θ)
we now notice 2k leaves at a rotation of (360/2k)ْ from each other starting at the positive x-axis and at a length of |b| and symmetric with respect to both the x and the y-axes.
Finally, looking at replacing cos with sin in the previous investigations we notice from the graph of r=4+4 cos(45θ) and r=4+4 sin(45θ) and through many other investigations with Graphing Calc.
it appears that similar characteristics to the above cases also hold true though the sin graphs appear to have been rotated counterclockwise (90/kْ).