Rational Functions

We have seen that a linear function has the form $y=mx+b$.
When a linear function is devided by another function, the result is a rational function.
Rational functions are characterised by asymptotes, which are lines the function gets close and close to but never reaches.
The rational functions we consider can be written in the form $y=\dfrac{ax+b}{cx+d}$. These functions have asymptotes which are horizontal and vertical.

Consider drawing the rational function $\dfrac{3-2x}{x-1}$.
\( \begin{align}
\dfrac{3-2x}{x-1} &= \dfrac{1-2x+2}{x-1} \\
&= \dfrac{1-2(x-1)}{x-1} \\
&= \dfrac{1}{x-1} – \dfrac{2(x-1)}{x-2} \\
&= \dfrac{1}{x-1} – 2 \\
\end{align} \)
The vertical asymptote is $x=1$, and the horizontal asymptote is $y=2$.

Example 1

Sketch the graphs of $y=\dfrac{1}{x}$, $\dfrac{3}{x}$ and $\dfrac{5}{x}$.

Example 2

Sketch the graphs of $y=\dfrac{1}{x}$ and $-\dfrac{1}{x}$.

Example 3

Sketch the graphs of $y=\dfrac{1}{x}$ and $\dfrac{1}{x-1}$.

Example 4

Sketch the graphs of $y=\dfrac{1}{x}$ and $\dfrac{1}{x+1}$.

Example 5

Sketch the graphs of $y=\dfrac{1}{x}$ and $\dfrac{1}{x}+1$.

Example 6

Sketch the graphs of $y=\dfrac{1}{x}$ and $\dfrac{1}{x}-1$.

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