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Date	May 2019	Marks available	5	Reference code	19M.1.AHL.TZ2.H_3
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 2
Command term	Express	Question number	H_3	Adapted from	N/A

Question

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 $a$ , $b$ , $c$ , $d$ , $e \in \mathbb{Z}$ .

Markscheme

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

$g\left( x \right) = f\left( {x + 2} \right)\left( { = {{\left( {x + 2} \right)}^4} - 6{{\left( {x + 2} \right)}^2} - 2\left( {x + 2} \right) + 4} \right)$ M1

attempt to expand ${{{\left( {x + 2} \right)}^4}}$ M1

${\left( {x + 2} \right)^4} = {x^4} + 4\left( {2{x^3}} \right) + 6\left( {{2^2}{x^2}} \right) + 4\left( {{2^3}x} \right) + {2^4}$ (A1)

$= {x^4} + 8{x^3} + 24{x^2} + 32x + 16$ A1

$g\left( x \right) = {x^4} + 8{x^3} + 24{x^2} + 32x + 16 - 6\left( {{x^2} + 4x + 4} \right) - 2x - 4 + 4$

$= {x^4} + 8{x^3} + 18{x^2} + 6x - 8$ A1

Note: For correct expansion of $f\left( {x - 2} \right) = {x^4} - 8{x^3} + 18{x^2} - 10x$ award max M0M1(A1)A0A1.

[5 marks]

Examiners report

[N/A]

Syllabus sections

Topic 2—Functions » SL 2.11—Transformation of functions

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19M.1.AHL.TZ2.H_11b:
State the range of $g$ .
19M.1.AHL.TZ2.H_11c:
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.
19M.2.AHL.TZ1.H_10a:
Write down the maximum and minimum value of $v$ .
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.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.TZ2.H_11d:
Find the coordinates of P.
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 $d$ .
22M.1.SL.TZ2.7d:
Find the values of $d$ for which $g$ is an increasing function.
17M.2.SL.TZ2.S_3a:
Write down the range of $f$ .
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 $y = i\left( t \right)$ .
20N.1.SL.TZ0.S_4b:
The $x$ -intercept of the graph of $f$ is $(5, 0)$ .

On the following grid, sketch the graph of $f$ .
16N.2.SL.TZ0.S_4b:
Hence, find the area of the region enclosed by the graphs of $f$ and $g$ .
19M.1.AHL.TZ2.H_11a:
Describe the transformation by which $f\left( x \right)$ is transformed to $g\left( x \right)$ .
20N.1.SL.TZ0.S_10b:
Find the area of triangle $AOB$ in terms of $k$ .
21M.1.SL.TZ1.1b:
Let $g (x) = \frac{1}{2} f (x) + 1$ for $- 4 \leq x \leq 6$ . On the axes above, sketch the graph of $g$ .
21M.1.AHL.TZ2.12a:
Describe a sequence of transformations that transforms the graph of $y = arctan x$ to the graph of $y = arctan (2 x + 1) + \frac{π}{4}$ for $x \in ℝ$ .
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 $p$ .
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.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.
19N.1.SL.TZ0.S_3b:
The graph of $f\left( x \right) = {x^2}$ is transformed to obtain the graph of $g$ .

Describe this transformation.
17M.2.SL.TZ2.S_3c:
Write down the domain of $g$ .
20N.1.SL.TZ0.S_10a.i:
Find $f' (p)$ in terms of $k$ and $p$ .
20N.1.SL.TZ0.S_4a:
Find the value of $a$ .
17M.2.SL.TZ1.S_10a.i:
Write down the value of $q$ ;
19N.1.SL.TZ0.S_3a:
Find the value of $b$ .
16N.2.SL.TZ0.S_4a:
Find the value of $p$ and of $q$ .
EXN.1.SL.TZ0.8a:
Describe a sequence of transformations that transforms the graph of $y = \sqrt{x}$ for $x \geq 0$ to the graph of $y = - 1 - \sqrt{x + 3}$ for $x \geq - 3$ .

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Topic 2—Functions