Define the Logistic Map.  For the Final Exam, you will  modify the map.

Clear[a,x,f];
f[x_]:= a x (1-x)

The fixed points of f  satisfy fpf = 0

fpf = Factor[f[x]-x]

fp[x_] = a-2a x

We can plot the solution to the equation in terms of a and x using ContourPlot:

ContourPlot[fpf==0, {a,0,4}, {x, -.1, 1}]

That looks bad at the crossing of the two branches of fixed points.  define the two factors of fpf, to get fp1f = x and fp2x.

fp1f = x
fp2f = fpf/x

ContourPlot[{fp1f==0, fp2f ==0}, {a,0,4}, {x, -.01, 1}, FrameLabel ->{a,x}]

By hand, we found that the trivial fixed point changes stability at a = 1, x = 0, and the nontrivial period 1 point is superstable at a = 2, x = 1/2, and it changes stability at a = 3, x = 2/3.  So we can plot these points on tops of the curves.

ContourPlot[{fp1f==0, fp2f ==0}, {a,0,4}, {x, -.001, 1}, FrameLabel ->{a,x}]

Now we add the period 2 points, and we will want to use the computer algebra.
The fixed points of f^2 satisfy fpf2 = 0

fpf2 = Factor[f[f[x]] -x]

The period 2 points satisfy per2 = 0

per2 = fpf2/fpf

We see that the nontrivial fixed point (the orange branch) has a period doubling bifurcation at a = 3.

ContourPlot[{fp1f==0, fp2f ==0, per2 == 0}, {a,0,4}, {x, -.001, 1}, FrameLabel ->{a,x}]