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Date	November Example questions	Marks available	3	Reference code	EXN.1.SL.TZ0.8
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Describe	Question number	8	Adapted from	N/A

Question

The following diagram shows the graph of $y = - 1 - \sqrt{x + 3}$ for $x \geq - 3$ .

A function $f$ is defined by $f (x) = - 1 - \sqrt{x + 3}$ for $x \geq - 3$ .

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 $x \geq - 3$ .

[3]

State the range of $f$ .

[1]

Find an expression for $f^{- 1} (x)$ , stating its domain.

[5]

Find the coordinates of the point(s) where the graphs of $y = f (x)$ and $y = f^{- 1} (x)$ intersect.

[5]

Markscheme

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

for example,

a reflection in the $x$ -axis (in the line $y = 0$ ) A1

a horizontal translation (shift) $3$ units to the left A1

a vertical translation (shift) down by $1$ unit A1

Note: Award A1 for each correct transformation applied in a correct position in the sequence. Do not accept use of the “move” for a translation.

Note: Award A1A1A1 for a correct alternative sequence of transformations. For example,

a vertical translation (shift) down by $1$ unit, followed by a horizontal translation (shift) $3$ units to the left and then a reflection in the line $y = - 1$ .

[3 marks]

range is $f (x) \leq - 1$ A1

Note: Correct alternative notations include $] - \infty, - 1]$ , $(- \infty, - 1]$ or $y \leq - 1$ .

[1 mark]

$- 1 - \sqrt{y + 3} = x$ M1

Note: Award M1 for interchanging $x$ and $y$ (can be done at a later stage).

$\sqrt{y + 3} = - x - 1 (= - (x + 1))$ A1

$y + 3 = {(x + 1)}^{2}$ A1

so $f^{- 1} (x) = {(x + 1)}^{2} - 3 (f^{- 1} (x) = x^{2} + 2 x - 2)$ A1

domain is $x \leq - 1$ A1

Note: Correct alternative notations include $] - \infty, - 1]$ or $(- \infty, - 1]$ .

[5 marks]

the point of intersection lies on the line $y = x$

EITHER

${(x + 1)}^{2} - 3 = x$ M1

attempts to solve their quadratic equation M1

for example, $(x + 2) (x - 1) = 0$ or $x = \frac{- 1 \pm \sqrt{1^{2} - 4 (1) (- 2)}}{2} (x = \frac{- 1 \pm 3}{2})$

$- 1 - \sqrt{x + 3} = x$ M1

${(- 1 - \sqrt{x + 3})}^{2} = x^{2} \Rightarrow 2 \sqrt{x + 3} + x + 4 = x^{2}$

substitutes $2 \sqrt{x + 3} = - 2 (x + 1)$ to obtain $- 2 (x + 1) + x + 4 = x^{2}$

attempts to solve their quadratic equation M1

for example, $(x + 2) (x - 1) = 0$ or $x = \frac{- 1 \pm \sqrt{1^{2} - 4 (1) (- 2)}}{2} (x = \frac{- 1 \pm 3}{2})$

THEN

$x = - 2, 1$ A1

as $x \leq - 1$ , the only solution is $x = - 2$ R1

so the coordinates of the point of intersection are $(- 2, - 2)$ A1

Note: Award R0A1 if $(- 2, - 2)$ is stated without a valid reason given for rejecting $(1, 1)$ .

[5 marks]

Examiners report

[N/A]

Syllabus sections

Topic 2—Functions » SL 2.11—Transformation of functions

Show 28 related questions

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

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Topic 2—Functions

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