The Quadratic Family

The quadratic family of equations includes all functions of the form f(x) = x² + p, where p is any real constant parameter. So all of the following are examples of quadratic functions:

Formula	Parameter
f(x) = x2 + 2	2
f(x) = x2 - 4	-4
f(x) = .5 + x2	.5

The Cartesian graphs of these functions basically look like the graph of f(x) = x², only shifted up or down. As we will see soon, this shifting causes great changes in the dynamics (types of orbits) of a quadratic function.

By using graphical analysis on various quadratic functions, you can see that as the parameter changes, the number of fixed points (where the function f(x) intersects the diagonal line y = x) changes as well. This changing in the number of fixed points is called a bifurcation. We will examine a particular type of bifurcation called a saddle-node.

As we decrease the value of the parameter, the saddle-node is the first bifurcation that we encounter. While p > .25, the function has no fixed points, as shown in the graphical analysis picture below.

Using .1 as the seed and .5 as the parameter, we can see that there are no fixed points (before the bifurcation).

When p = .25, the function has one fixed point, a saddle, as depicted below.

Using .1 and .75 as seeds and .25 as the parameter, we can see that there is one neutral fixed point (during the bifurcation).

While p < .25, the function has two fixed points, one attracting and one repelling, as shown below.

Using 1.25 and 1.5 as seeds and -.5 as the parameter, we can see that there are two fixed points (after the bifurcation).