Analyze and graph R=3/ 1+3 cos theta?

The first source link can help with the analysis. It tells us the form of the polar equation for a hyperbola is
.. r = a(e^2-1)/(1-e*cos(θ)) ... where e is the eccentricity
Our equation is
.. r = 3/(1+3cos(θ)
from which we deduce
.. e = |-3| = 3 ... greater than 1 indicates a hyperbola
.. a = 3/(e^2-1) = 3/8
.. c = a·e = 9/8
and
.. b^2 = c^2 - a^2 = 81/64 - 9/64 = 72/64 = 9/8

The graph of
.. r = 3/(1 + 3*cos(θ))
is that of a hyperbola with one focus at the origin and the center at (9/8, 0).
The asymptotes will have slope ±b/a = ±2√2.
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A graph is shown at the second source link. Clicking on the empty circle to the left of the second equation will cause the graph of that equation in rectangular coordinates to be displayed. You will see that it overlays the polar equation graph.

The rectangular equation is equivalent to
.. (8x-9)^2 - 8y^2 = 9