I am sorry for the effort made, but I think we all agree that the depicted instance of a run of secant method on an instance candidate of f : R --> R and x0,x1 is not very helpful in motivating the method. The issue is simply that it cannot be seen straight-forwardly why this method is constructed like so. Furthermore, it appears unfathomable as to why this method would converge at all, as judged from the current presentation in the image.
Instead, applying x0=-4, x1=-3 to f(x)=x*x-1 (or some deviation thereof), it could be seen more directly that the iterates mimic a successive linearization approach and therefor under reasonable assumptions converge.
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