After learning about inverse functions and quadratic equations, I thought students might enjoy seeing a connection between the two.

Suppose we are given the function f(x)= -x**2 + 6x + 10, and we want to find the axis of symmetry. Instead of completing the square we rewrite the function as f(x) = -x(6-x) + 10.

What do we get for f(6 – x)? We notice that 6-x is self-inverse and that f contains terms for both x and 6-x . For f(6-x), x gets mapped to 6 -x and 6-x gets mapped to x. f(6-x)= -(6-x)x + 10, which is the same as f(x), so f(6-x)=f(x). For every point (x,y) on the curve there will be a point (6-x,y) on the curve. The midpoint of (x,y) and (6-x,y) is (3,y), so the line x=3 is a line of symmetry.

This technique can be extended to all quadratic expressions and can be used for other types of functions, like f(x)=arctan(x)*arctan(6-x), which will also have a line of symmetry for x=3.