IB Questionbank

User interface language: English | Español

Date	May 2019	Marks available	6	Reference code	19M.1.AHL.TZ2.H_11
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 2
Command term	Sketch and State	Question number	H_11	Adapted from	N/A

Question

Consider the functions $f$ and $g$ defined by $f\left( x \right) = {\text{ln}}\left| x \right|$ , $x \in \mathbb{R}$ \ $\left\{ 0 \right\}$ , and $g\left( x \right) = {\text{ln}}\left| {x + k} \right|$ , $x \in \mathbb{R}$ \ $\left\{ { - k} \right\}$ , where $k \in \mathbb{R}$ , $k > 2$ .

The graphs of $f$ and $g$ intersect at the point P .

Describe the transformation by which $f\left( x \right)$ is transformed to $g\left( x \right)$ .

[1]

State the range of $g$ .

[1]

Sketch the graphs of $y = f\left( x \right)$ and $y = g\left( x \right)$ on the same axes, clearly stating the points of intersection with any axes.

[6]

Find the coordinates of P.

[2]

The tangent to $y = f\left( x \right)$ at P passes through the origin (0, 0).

Determine the value of $k$ .

[7]

Markscheme

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

translation $k$ units to the left (or equivalent) A1

[1 mark]

range is $\left( {g\left( x \right) \in } \right)\mathbb{R}$ A1

[1 mark]

correct shape of $y = f\left( x \right)$ A1

their $f\left( x \right)$ translated $k$ units to left (possibly shown by $x = - k$ marked on $x$ -axis) A1

asymptote included and marked as $x = - k$ A1

$f\left( x \right)$ intersects $x$ -axis at $x = - 1$ , $x = 1$ A1

$g\left( x \right)$ intersects $x$ -axis at $x = - k - 1$ , $x = - k + 1$ A1

$g\left( x \right)$ intersects $y$ -axis at $y = {\text{ln}}\,k$ A1

Note: Do not penalise candidates if their graphs “cross” as $x \to \pm \infty$ .

Note: Do not award FT marks from the candidate’s part (a) to part (c).

[6 marks]

at P ${\text{ln}}\left( {x + k} \right) = {\text{ln}}\left( { - x} \right)$

attempt to solve $x + k = - x$ (or equivalent) (M1)

$x = - \frac{k}{2} \Rightarrow y = {\text{ln}}\left( {\frac{k}{2}} \right)\,\,$ (or $y = {\text{ln}}\left| {\frac{k}{2}} \right|$ ) A1

P $\left( { - \frac{k}{2},\,\,{\text{ln}}\frac{k}{2}} \right)$ (or P $\left( { - \frac{k}{2},\,\,{\text{ln}}\left| {\frac{k}{2}} \right|} \right)$ )

[2 marks]

attempt to differentiate ${\text{ln}}\left( { - x} \right)$ or ${\text{ln}}\left| x \right|$ (M1)

$\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{1}{x}$ A1

at P, $\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{ - 2}}{k}$ A1

recognition that tangent passes through origin $\Rightarrow \frac{y}{x} = \frac{{{\text{d}}y}}{{{\text{d}}x}}$ (M1)

$\frac{{{\text{ln}}\left| {\frac{k}{2}} \right|}}{{ - \frac{k}{2}}} = \frac{{ - 2}}{k}$ A1

${\text{ln}}\left( {\frac{k}{2}} \right)\, = 1$ (A1)

$\Rightarrow k = 2{\text{e}}$ A1

Note: For candidates who explicitly differentiate ${\text{ln}}\left( {x} \right)$ (rather than ${\text{ln}}\left( { - x} \right)$ or ${\text{ln}}\left| x \right|$ , award M0A0A1M1A1A1A1.

[7 marks]

Examiners report

[N/A]

Syllabus sections

Topic 2—Functions » AHL 2.8—Transformations of graphs, composite transformations

Show 50 related questions

22M.2.AHL.TZ2.6f:
Determine the time when the arrow should be released to hit the ball before the ball reaches its maximum height.
19M.1.AHL.TZ2.H_11b:
State the range of $g$ .
SPM.1.AHL.TZ0.14a.ii:
Find the value of $b$ .
19M.1.SL.TZ2.S_4a.i:
Find the $y$ -intercept of the graph of $f\left( x \right) + 3$ .
19M.1.AHL.TZ2.H_11e:
The tangent to $y = f\left( x \right)$ at P passes through the origin (0, 0).

Determine the value of $k$ .
19M.1.SL.TZ2.S_4a.ii:
Find the $y$ -intercept of the graph of $f\left( {4x} \right)$ .
22M.2.AHL.TZ2.6e:
Determine the two positions where the path of the arrow intersects the path of the ball.
18N.1.SL.TZ0.S_8b:
Find the equation of the axis of symmetry of the graph of $f$ .
19M.1.SL.TZ2.S_4b:
Find the $x$ -intercept of the graph of $f\left( {2x} \right)$ .
22M.2.AHL.TZ1.6b:
Find the value of $p$ and the value of $q$ .
21N.1.AHL.TZ0.10a:
Find $g (0)$ .
18N.1.AHL.TZ0.H_3b:
Write down the least value of $a$ such that $g$ has an inverse.
18N.1.AHL.TZ0.H_3a:
For $a = - \frac{\pi }{2}$ , sketch the graph of $y = g\left( x \right)$ . Indicate clearly the maximum and minimum values of the function.
19M.2.AHL.TZ1.H_10a:
Write down the maximum and minimum value of $v$ .
18N.1.AHL.TZ0.H_3c.i:
For the value of $a$ found in part (b), write down the domain of ${g^{ - 1}}$ .
19M.2.SL.TZ1.S_8b:
Find the range of $g$ .
19M.1.AHL.TZ1.H_10a.i:
Find the remainder when $p\left( x \right)$ is divided by $\left( {x - 2} \right)$ .
19M.2.SL.TZ1.S_8a:
The range of $f$ is $k$ ≤ $f\left( x \right)$ ≤ $m$ . Find $k$ and $m$ .
18M.1.SL.TZ2.S_5a:
On the same axes, sketch the graph of $f\left( { - x} \right)$ .
SPM.1.AHL.TZ0.14a.i:
Write down the value of $a$ .
19M.1.AHL.TZ2.H_11d:
Find the coordinates of P.
19M.2.SL.TZ1.S_8d:
The equation $g\left( x \right) = 12$ has two solutions where $\pi$ ≤ $x$ ≤ ${\frac{{4\pi }}{3}}$ . Find both solutions.
18N.1.SL.TZ0.S_8c.ii:
Find the value of $k$ .
18N.1.AHL.TZ0.H_3c.ii:
For the value of $a$ found in part (b), find an expression for ${g^{ - 1}}\left( x \right)$ .
19M.2.SL.TZ1.S_8c.ii:
Find the period of $g$ .
19M.2.AHL.TZ1.H_10g:
Given that $p\left( t \right)$ can be written as $p\left( t \right) = a\,{\text{sin}}\left( {b\left( {t - c} \right)} \right) + d$ where $a$ , $b$ , $c$ , $d$ > 0, use your graph to find the values of $a$ , $b$ , $c$ and $d$ .
19M.2.AHL.TZ1.H_10c:
Sketch the graph of $y = p\left( t \right)$ for 0 ≤ $t$ ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.
19M.2.SL.TZ1.S_8c.i:
Find the value of $b$ and of $c$ .
19M.1.AHL.TZ1.H_10a.ii:
Find the remainder when $p\left( x \right)$ is divided by $\left( {x - 3} \right)$ .
19M.2.AHL.TZ1.H_10e:
Find ${p_{av}}$ (0.007).
20N.1.SL.TZ0.S_10a.ii:
Show that the equation of $L_{1}$ is $k x + p^{2} y - 2 p k = 0$ .
19M.1.AHL.TZ1.H_10b:
Prove that $p\left( x \right)$ has only one real zero.
19M.1.AHL.TZ1.H_10c:
Write down the transformation that will transform the graph of $y = p\left( x \right)$ onto the graph of $y = 8{x^3} + 12{x^2} + 16x - 24$ .
21M.1.AHL.TZ1.17:
The graph of the function $f (x) = \ln x$ is translated by $(\begin{matrix} a \\ b \end{matrix})$ so that it then passes through the points $(0, 1)$ and $(e^{3}, 1 + \ln 2)$ .

Find the value of $a$ and the value of $b$ .
20N.1.SL.TZ0.S_10a.i:
Find $f' (p)$ in terms of $k$ and $p$ .
19M.1.SL.TZ2.S_4c:
Describe the transformation $f\left( {x + 1} \right)$ .
19M.2.AHL.TZ1.H_10b:
Write down two transformations that will transform the graph of $y = v\left( t \right)$ onto the graph of $y = i\left( t \right)$ .
19M.2.AHL.TZ1.H_10d:
Find the total time in the interval 0 ≤ $t$ ≤ 0.02 for which $p\left( t \right)$ ≥ 3.
19M.1.AHL.TZ1.H_10d:
The random variable $X$ follows a Poisson distribution with a mean of $\lambda$ and $6{\text{P}}\left( {X = 3} \right) = 3{\text{P}}\left( {X = 2} \right) - 2{\text{P}}\left( {X = 1} \right) + 3{\text{P}}\left( {X = 0} \right)$ .

Find the value of $\lambda$ .
19M.1.AHL.TZ2.H_3:
Consider the function $f\left( x \right) = {x^4} - 6{x^2} - 2x + 4$ , $x \in \mathbb{R}$ .

The graph of $f$ is translated two units to the left to form the function $g\left( x \right)$ .

Express $g\left( x \right)$ in the form $a{x^4} + b{x^3} + c{x^2} + dx + e$ where $a$ , $b$ , $c$ , $d$ , $e \in \mathbb{Z}$ .
18M.1.SL.TZ2.S_5b:
Another function, $g$ , can be written in the form $g\left( x \right) = a \times f\left( {x + b} \right)$ . The following diagram shows the graph of $g$ .

Write down the value of a and of b.
19M.1.AHL.TZ2.H_11a:
Describe the transformation by which $f\left( x \right)$ is transformed to $g\left( x \right)$ .
SPM.1.AHL.TZ0.14b:
The outer dome of a large cathedral has the shape of a hemisphere of diameter 32 m, supported by vertical walls of height 17 m. It is also supported by an inner dome which can be modelled by rotating the curve $y = 33 - 0.08{x^3}$ through 360° about the $y$ -axis between $y$ = 0 and $y$ = 33, as indicated in the diagram.

Find the volume of the space between the two domes.
18N.1.SL.TZ0.S_8c.i:
Write down the value of $h$ .
20N.1.SL.TZ0.S_10c:
The graph of $f$ is translated by $(\begin{matrix} 4 \\ 3 \end{matrix})$ to give the graph of $g$ .
In the following diagram:
- point $Q$ lies on the graph of $g$
- points $C$ , $D$ and $E$ lie on the vertical asymptote of $g$
- points $D$ and $F$ lie on the horizontal asymptote of $g$
- point $G$ lies on the $x$ -axis such that $FG$ is parallel to $DC$ .
Line $L_{2}$ is the tangent to the graph of $g$ at $Q$ , and passes through $E$ and $F$ .

Given that triangle $EDF$ and rectangle $CDFG$ have equal areas, find the gradient of $L_{2}$ in terms of $p$ .
20N.1.SL.TZ0.S_10b:
Find the area of triangle $AOB$ in terms of $k$ .
19M.2.AHL.TZ1.H_10f:
With reference to your graph of $y = p\left( t \right)$ explain why ${p_{av}}\left( T \right)$ > 0 for all $T$ > 0.
18N.1.SL.TZ0.S_8d:
The graph of a second function, $g$ , is obtained by a reflection of the graph of $f$ in the $y$ -axis, followed by a translation of $\left( {\begin{array}{*{20}{c}} { - 3} \\ 6 \end{array}} \right)$ .

Find the coordinates of the vertex of the graph of $g$ .
EXN.1.AHL.TZ0.8b:
Write down a sequence of transformations that will transform the graph of $y = \cos x$ onto the graph of $y = f (x)$ .
21N.1.AHL.TZ0.10b:
On the same set of axes draw the graph of $y = g (x)$ , showing any intercepts and asymptotes.

Hide related questions

Topic 2—Functions

Question

Markscheme

Examiners report

Syllabus sections

View options