# 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 can be written in $y=\dfrac{ax+b}{cx+d}$. 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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