Math Algebra • Functions
Drag to pan. Mouse wheel zooms (cursor-centered). Double-click resets view. Fixed points are labeled at intersections \(f(x)=x\). Cobweb shows iteration \(x_{n+1}=f(x_n)\) if enabled.
A fixed point is a value x* such that f(x*)=x*. On a graph, fixed points occur where the curve y=f(x) intersects the line y=x.
It rewrites the problem as g(x)=f(x)-x=0, scans the chosen window for sign changes in g(x), and then refines each candidate root using bisection until the tolerance is met.
If |f'(x*)|<1, the fixed point is locally attractive, meaning iterations x(n+1)=f(x(n)) tend to move toward x* when starting nearby. If |f'(x*)|>1, it is repelling, and values move away.
The cobweb alternates vertical moves to y=f(x) and horizontal moves to y=x, visualizing the iteration x0, x1=f(x0), x2=f(x1), and so on. Tightening steps toward a single intersection indicates attraction, while expanding steps indicates repulsion.
A period-2 point satisfies f(f(x))=x but is not itself a fixed point, so the iteration alternates between two distinct values. Fixed points satisfy f(x)=x and remain unchanged under one application of f.