Root finding: bisection, Newton, secant

What you are seeing: a function $f(x)$ and the iterates of three classical root finders. Bisection halves the bracket each step (always convergent if $f$ changes sign on the bracket). Newton uses tangent lines (quadratic convergence near a simple root, but can diverge with bad initial guesses). Secant uses chord slopes (superlinear, golden-ratio order).