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Date	November 2018	Marks available	4	Reference code	18N.2.AHL.TZ0.H_8
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Determine	Question number	H_8	Adapted from	N/A

Question

Consider the function $f\left( x \right) = \frac{{ax + 1}}{{bx + c}}$ , $x \ne - \frac{c}{b}$ , where $a$ , $b$ , $c \in \mathbb{Z}$ .

The following graph shows the curve $y = {\left( {f\left( x \right)} \right)^2}$ . It has asymptotes at $x = p$ and $y = q$ and meets the $x$ -axis at A.

On the following axes, sketch the two possible graphs of $y = f\left( x \right)$ giving the equations of any asymptotes in terms of $p$ and $q$ .

[4]

a.

Given that $p = \frac{4}{3}$ , $q = \frac{4}{9}$ and A has coordinates $\left( { - \frac{1}{2},\,\,0} \right)$ , determine the possible sets of values for $a$ , $b$ and $c$ .

[4]

b.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

either graph passing through (or touching) A A1

correct shape and vertical asymptote with correct equation for either graph A1

correct horizontal asymptote with correct equation for either graph A1

two completely correct sketches A1

[4 marks]

a.

$a\left( { - \frac{1}{2}} \right) + 1 = 0 \Rightarrow a = 2$ A1

from horizontal asymptote, ${\left( {\frac{a}{b}} \right)^2} = \frac{4}{9}$ (M1)

$\frac{a}{b} = \pm \frac{2}{3} \Rightarrow b = \pm 3$ A1

from vertical asymptote, $b\left( {\frac{4}{3}} \right) + c = 0$

$b$ = 3, $c$ = −4 or $b$ = −3, $c$ = 4 A1

[4 marks]

b.

Examiners report

[N/A]

a.

[N/A]

b.

Syllabus sections

Topic 2—Functions » SL 2.8—Reciprocal and simple rational functions, equations of asymptotes

Show 26 related questions

18N.2.AHL.TZ0.H_8a:
On the following axes, sketch the two possible graphs of $y = f\left( x \right)$ giving the equations of any asymptotes in terms of $p$ and $q$ .
22M.1.SL.TZ2.4a.ii:
Write down the equation of the horizontal asymptote.
22M.1.SL.TZ2.4a.i:
Write down the equation of the vertical asymptote.
22M.1.SL.TZ2.4b:
On the set of axes below, sketch the graph of $y = f (x)$ .

On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.
22M.1.AHL.TZ2.3a.i:
Write down the equation of the vertical asymptote.
21N.1.SL.TZ0.3c:
Sketch the graph of $f$ on the axes below.
19M.1.AHL.TZ2.H_5a:
Sketch the graph of $y = \frac{{x - 4}}{{2x - 5}}$ , stating the equations of any asymptotes and the coordinates of any points of intersection with the axes.
21N.1.AHL.TZ0.2a.i:
the vertical asymptote of the graph of $f$ .
21N.1.AHL.TZ0.2a.ii:
the horizontal asymptote of the graph of $f$ .
21N.1.AHL.TZ0.2c:
Sketch the graph of $f$ on the axes below.
21N.1.SL.TZ0.3a.i:
the vertical asymptote of the graph of $f$ .
21N.1.SL.TZ0.3a.ii:
the horizontal asymptote of the graph of $f$ .
21N.1.SL.TZ0.3b.i:
the $x$ -axis.
21N.1.AHL.TZ0.2b.ii:
the $y$ -axis.
18N.2.SL.TZ0.S_3a.i:
For the graph of $f$ , find the $y$ -intercept.
22M.1.SL.TZ2.4c:
Hence, solve the inequality $0 < \frac{2 x - 1}{x + 1} < 2$ .
22M.1.AHL.TZ2.3a.ii:
Write down the equation of the horizontal asymptote.
22M.1.AHL.TZ2.3c:
Hence, solve the inequality $0 < \frac{2 x - 1}{x + 1} < 2$ .
22M.2.SL.TZ1.8a.ii:
find the equation of the horizontal asymptote.
18N.2.SL.TZ0.S_3b:
Hence or otherwise, write down $\mathop {{\text{lim}}}\limits_{x \to \infty } \left( {\frac{{6x - 1}}{{2x + 3}}} \right)$ .
18M.2.SL.TZ2.S_7b:
Write down the equation of the horizontal asymptote to the graph of f.
18M.1.AHL.TZ2.H_10b.i:
Express $g\left( x \right)$ in the form $A + \frac{B}{{x - 2}}$ where A, B are constants.
21N.1.AHL.TZ0.2d:
The function $g$ is defined by $g (x) = \frac{a x + 4}{3 - x}$ , where $x \in ℝ, x \neq 3$ and $a \in ℝ$ .

Given that $g (x) = g^{- 1} (x)$ , determine the value of $a$ .
22M.2.SL.TZ1.8a.i:
write down the equation of the vertical asymptote.
21N.1.SL.TZ0.3b.ii:
the $y$ -axis.
21N.1.AHL.TZ0.2b.i:
the $x$ -axis.

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Topic 2—Functions