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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