Let \(f(x)\) be a function that is continuous on a closed interval \([a,b]\), and where \(f(a)\) and \(f(b)\) take opposite signs (one value is positive and the other negative). By the Intermediate Value Theorem, this function must have a root on the interval \([a,b]\). The bisection method is an algorithm that allows us to approximate this root.
After choosing a starting interval \([a,b]\) on which the required conditions exist, we calculate the midpoint \(c = \frac{a+b}{2}\) of the interval. This midpoint becomes the current approximation of the root. If \(f(a)\) and \(f(c)\) take opposite signs, the root must be within the interval \([a,c]\). Otherwise, \(f(b)\) and \(f(c)\) will take opposite signs, and we must have a root in the interval \([c,b]\). We then repeat this process using whichever of these intervals contains the root, dividing the interval in half at each stage. After enough iterations, the intervals will grow narrow enough that repeating the process does not result in any significant change to the approximation.

Enter a function and a starting interval that contains a root. Repeatedly click on the "Keep Searching" button to find the next approximation of the root. The method is considered complete when the approximation ceases to change.

This applet was created using JavaScript and the Konva graphics library.