Symmetry is an important concept in mathematics and it has a built-in aesthetic appeal. The following shows how different types of symmetry relate to one another. Start with the equation $f(x) = -x^2 + bx + c$. Without having to complete the square, we can find the axis of symmetry. Rewrite the equation as $f(x) = x(b - x) + c$. Now find $f(b - x) = (b-x)x + c = f(x)$.
Since $f(b-x) = f(x)$, we know that for every point $(x,y)$ on the curve, there is a point $(b - x,y)$ on the curve. The midpoint of these two points on the curve is $(b/2, y)$. The line $x=b/2$ is an axis of symmetry. We can generalize this by noticing that what made this work is that the equation for $f(x) = x(b-x) + c$ is symmetric in $x$ and $b -x$. We can get similar results for $f(x) = x^2 + bx + c$. $f(x) = -x(-b - x) + c$. Reasoning as before, it follows that $x = -b/2$ is an axis of symmetry.
Consider the function $f(x) = \arctan(x) + \arctan(6-x)$, which is symmetric in $x$ and $(6 - x)$. We can say immediately that the line $x=3$ is an axis of symmetry. One of the reasons that this works is that the function $g(x) = b - x$ is self-inverse, $g(g(x))=x$. If we take any function $f(x)$ that is symmetric in $x$ and a self-inverse function $g(x)$, we can say that $f(x) = f(g(x))$, although this will not usually result in an axis of symmetry.
Finally, there is a way of talking about self-inverse functions in terms of symmetry. The equation $y = b - x$ does not at first appear to have any symmetry, but if we write it as $x + y = b$, it is clear that it is symmetric in $x$ and $y$. The function for getting $y$ in terms of $x$ is the same as the function for getting $x$ in terms of $y$, causing $g(g(x)) = x$. For every point $(x, g(x))$ on a self-inverse curve there is a point $(g(x), x)$ on the curve. The midpoint is $((x+g(x))/2, (x+g(x))/2)$, which is on the line $y=x$. The line $y=x$ is an axis of symmetry for all self-inverse curves.