EXM.1.AHL.TZ0.5

pestleMathematicsAAHLPaper 1EXM· ahl-2-13-rational-functionssource ↗

Let $f\left( x \right) = \frac{{2{x^2} - 5x - 12}}{{x + 2}}{\text{,}}\,\,x \in \mathbb{R}{\text{,}}\,\,x \ne - 2$ .

Find all the intercepts of the graph of $f\left( x \right)$ with both the $x$ and $y$ axes.

[4]

Write down the equation of the vertical asymptote.

[1]

As $x \to \pm \infty$ the graph of $f\left( x \right)$ approaches an oblique straight line asymptote.

Divide $2{x^2} - 5x - 12$ by $x + 2$ to find the equation of this asymptote.

[4]

Markscheme / solution

$x = 0 \Rightarrow y = - 6$ intercept on the $y$ axes is (0, −6) A1

$2{x^2} - 5x - 12 = 0 \Rightarrow \left( {2x + 3} \right)\left( {x - 4} \right) = 0 \Rightarrow x = \frac{{ - 3}}{2}\,\,{\text{or}}\,\,{\text{4}}$ M1

intercepts on the $x$ axes are $\left( {\frac{{ - 3}}{2}{\text{,}}\,\,0} \right)\,\,{\text{and}}\,\,\left( {4{\text{,}}\,\,0} \right)$ A1A1

[4 marks]

$x = - 2$ A1

[1 mark]

$f(x) = 2x - 9 + \frac{6}{{x + 2}}$ M1A1

So equation of asymptote is $y = 2x - 9$ M1A1

[4 marks]

Examiners’ report

[N/A]