Curve sketching with a parameter

In school-leaving exam exercises the function often depends on a real parameter $m$. The study requires distinguishing the possible cases for $m$, because the domain, the sign, the limits and the derivative can change qualitatively as $m$ varies.

The strategy is to reason by cases: one identifies the parameter values that modify the structure of the function (for example those that make a denominator vanish or change the sign of an expression), splits the set of values of $m$ into intervals and, for each one, carries out a complete study with its own graph. The interesting point is the qualitative transition: the value of $m$ at which the behaviour changes radically, for example from “with a vertical asymptote” to “without”.

Example — $f(x) = \dfrac{2x+1}{x^2+m+|m|}$

Case $m<0$: $|m|=-m$, so $m+|m|=0$ and $f(x) = \dfrac{2x+1}{x^2}$. Existence condition $x\ne 0$: the function has a vertical asymptote at $x=0$ and a horizontal asymptote $y=0$, with $f'(x) = -\dfrac{2}{x^2}-\dfrac{2}{x^3}$. The complete study (symmetries, sign with $f>0$ for $x>-\tfrac{1}{2}$ and $x\ne 0$, derivative, inflection point at $x=-\tfrac{3}{2}$) leads to the graph.

Case $m>0$: $m+|m|=2m>0$ and $f(x) = \dfrac{2x+1}{x^2+2m}$. The denominator is $>0$ for every $x$: domain $\mathbb{R}$, no vertical asymptote. The qualitative behaviour is completely different.

Case $m=0$: $f(x) = \dfrac{2x+1}{x^2}$, the same as the case $m<0$.

For each case one carries out the complete study and draws the graph. The qualitative transition occurs at $m=0$: for $m<0$ there is a vertical asymptote, for $m>0$ there is not.

In the simulation below the parameter $k$ controls the appearance and position of the stationary points of the family of cubics $f(x)=x^3-kx$: for $k\le 0$ the function is always increasing (no extrema), for $k>0$ a maximum and a minimum appear at $x=\pm\sqrt{k/3}$, while the inflection point stays at $x=0$.

Drag the parameter $k$ and watch how the stationary points of $f(x)=x^3-kx$ move.

Links

Topics: Curve sketching
Concepts: Asymptote · Domain · Parameter · Curve sketching
Methods: Curve sketching
Skills: Reasoning by cases · Sketching a function