[Math] relation between curvature and second derivative for a quadratic function

Question

Sorry if this has already been answered in a different post. We often think of the second derivative of a function in terms of the curvature of its graph. The graph of a quadratic function like $f(x) = x^2$ seems more "curvy" near $x = 0$ than for large values of $x$ (where its graph looks more like a straight line). However, the second derivative of such function does not depend on $x$ ($f''(x) =2$). Why is it that the second derivative of a quadratic function does not seem to reflect the change of curvature of its graph? Thank you for your help!

Answer 1

Curvature $\kappa$ depends on $y'$ too: $$\kappa=\dfrac {\left|\,y''\,\right|}{\left[1+\left(\,y'\right)^{\,2}\right]^{\,3/2}}$$