There’s more? Yes, another way of visualizing recursion, this time as reflection off the y=x (45° angle) line.
Here is yet another way of watching what happens when we plug a function’s output back into its input. When you start at a y-value and make it the new x-value, this is the same as moving horizontally to where y=x, which is the 45° diagonal line. Then you move up (or down) to the new value of the function. Each time you click on the button, you draw a dotted line from the old value to the new value.
For many (but not all) values of a the attracting value is the intersection of the graph with the diagonal. Using this visualization helps explain why the orbits all tend to zero when the value of a is less than one: the graph of f(x) falls below the diagonal, and only intersects it at the origin. Similarly for the tent map: until its maximum is 1/2, the entire function is below the diagonal, so zero is the only attractor. But after that, the attractor jumps up in a discontinuous leap: the diagonal intersects the second half (the descending part) of the function. (See the Universality page.)
Sometimes the intersection becomes a repelling point, and the values move away from it. Figuring out how and when this change happens is part of understanding the bifurcation pattern.