# 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 $\dfrac{1}{x}-1$.

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Chain Rule Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume

## Responses