Rational and reciprocal functionsIB MYP Maths Extended: Revision notes

The graph of $y=\frac kx$ has two separate branches. If $k>0$ the branches lie in the first and third quadrants (both coordinates positive, or both negative). If $k<0$ they lie in the second and fourth. It is undefined at $x=0$, so the graph never crosses the $y$-axis. It never crosses the $x$-axis either, because $\frac kx=0$ has no solution. Instead the graph gets closer and closer to the axes: these lines are asymptotes, $x=0$ and $y=0$. Example: $y=\frac6x$ passes through $(2,3)$, $(3,2)$, $(-3,-2)$ and $(6,1)$. For every point, $xy=6$. This is inverse proportion.

Translating $y=\frac kx$ by $a$ to the right and $b$ up gives $y=\frac{k}{x-a}+b$.

• Vertical asymptote: $x=a$ (where the denominator is zero).
• Horizontal asymptote: $y=b$ (because $\frac{k}{x-a}\to0$ for large $|x|$). Example: $g(x)=\frac{2}{x-3}+1$ has asymptotes $x=3$ and $y=1$.

For the $y$-intercept, put $x=0$. For $g(x)=\frac{2}{x-3}+1$: $g(0)=\frac{2}{-3}+1=\frac13$, so the intercept is $\left(0,\frac13\right)$. For the $x$-intercept, solve $g(x)=0$: $\frac{2}{x-3}=-1$, so $x-3=-2$ and $x=1$, giving $(1,0)$. A graph may have both, one or neither intercept. $y=\frac kx$ has none.

A rational function is one polynomial divided by another, such as $h(x)=\frac{x+1}{x-2}$. Rewrite it to see its shape: $\frac{x+1}{x-2}=\frac{(x-2)+3}{x-2}=1+\frac{3}{x-2}$. So the asymptotes are $x=2$ and $y=1$. Find the intercepts: $x$-intercept where the numerator is zero ($x=-1$), $y$-intercept at $h(0)=-\frac12$. To sketch: draw the asymptotes as dashed lines, plot the intercepts and a couple of points, and draw one branch on each side of the vertical asymptote.

Many real situations are reciprocal: the time $T=\frac{240}{v}$ for a journey at speed $v$, or a cost per person $c=\frac{1200}{n}+15$. In context:

• The domain is limited by the situation (for example $10\leq n\leq50$ seats).
• The horizontal asymptote often has a meaning, such as a fixed cost per person that can never be avoided.
• Do not use the model outside its domain: the equation may no longer describe reality.