Graphs of Rational Functions (Edexcel IGCSE Further Pure Maths): Revision Note

• A horizontal line should only intersect this type of graph once at most

• The only horizontal line that should not intersect the graph is the horizontal asymptote

  • This can be used to check your sketch in an exam

The function $\mathrm{f}$ is defined by $\mathrm{f}\left(x\right)=\frac{10-5x}{x+2}$ for $x\ne -2$.

(a) Write down the equations of the horizontal and vertical asymptotes of the graph of $y=\mathrm{f}\left(x\right)$.

The horizontal asymptote is the limiting value as x becomes very large in the positive or negative direction

When that happens, the '10' in the numerator and '2' in the denominator no longer affect the value of the function much

For large values of $x$

$\mathrm{f}\left(x\right)\approx \frac{-5x}{x}=-5$

That means the horizontal asymptote will be at  $y=-5$

Horizontal asymptote:  $\mathit{y}\mathbf{=}\mathbf{-}\mathbf{5}$

The vertical asymptote occurs where the denominator becomes zero

$\begin{array}{rcl}x+2& =& 0\\ x& =& -2\end{array}$

Vertical asymptote:  $\mathit{x}\mathbf{=}\mathbf{-}\mathbf{2}$

(b) Find the coordinates of the intercepts of the graph of $y=\mathrm{f}\left(x\right)$ with the coordinate axes.

The y-intercept occurs when $x=0$

$\mathrm{f}\left(0\right)=\frac{10-5\left(0\right)}{\left(0\right)+2}=\frac{10}{2}=5$

y-intercept:  $\left(0,5\right)$

The x-intercept occurs when $y=\mathrm{f}\left(x\right)=0$

$\begin{array}{rcl}\frac{10-5x}{x+2}& =& 0\\ 10-5x& =& 0\\ 5x& =& 10\\ x& =& 2\end{array}$

x-intercept:  $\left(2,0\right)$

(c) Sketch the graph of  $y=\mathrm{f}\left(x\right)$.

Don't forget to include (and label) the asymptotes, and to label the axis intercepts