Parametric study of x over (x²+k)

Statement

Parametric study: $f(x) = \dfrac{x}{x^2+k}$ with $k>0$. How does the graph change as $k$ varies?

Solution

Domain: $\mathbb{R}$ (since $k>0$, the denominator is always positive). Symmetry: $f(-x)=-f(x)$, an odd function. Horizontal asymptote: $y=0$.

First derivative: $f'(x)=\dfrac{(x^2+k)-x\cdot 2x}{(x^2+k)^2}=\dfrac{k-x^2}{(x^2+k)^2}$. It vanishes at $x=\pm\sqrt{k}$:

• maximum at $x=\sqrt{k}$, with $f(\sqrt k)=\dfrac{\sqrt k}{2k}=\dfrac{1}{2\sqrt k}$;
• minimum at $x=-\sqrt{k}$, with $f=-\dfrac{1}{2\sqrt k}$.

Effect of $k$: as $k$ increases, the extremum points $\pm\sqrt k$ move away from the origin and their height $\dfrac{1}{2\sqrt k}$ decreases: the graph keeps the same shape (odd, with a positive and a negative “hump”) but flattens and widens. The inflection points are at $x=0$ and $x=\pm\sqrt{3k}$.

Links

Topics: Derivatives
Concepts: Asymptote · Maximum minimum · Parameter
Skills: Reasoning by cases · Studying a function
Exercise type: Function study