The thick red curve is y=f(x), initially f(x) = cosh(x).

The thinner red curve is the caustic formed by light parallel to the y-axis reflecting in y=f(X). (The caustic is the envelope of the reflected rays.  The envelope of a family of lines is the curve which is tangential to the entire family).

The blue curve is the parabola which is the 2nd order Taylor expansion of f(X) about the point J

I is the focus of this parabola.  What do you notice about Ias you drag J?