A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<ol class="abc-list"><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0"><div class="list-item-paragraph">The graphs of $y=2|5x-p|$ and $y=qx$, where&nbsp;$p,q&gt;0$, intersect at only one point.</div><ol class="i-list"><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0-sub-0" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0/h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0-sub-0"><div class="list-item-paragraph"><p>On a single diagram, sketch the two graphs.</p><p>State the range of values of $q$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0-sub-1" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0/h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0-sub-1"><div class="list-item-paragraph"><p>Solve $2|5x-p|&lt;10x$. Deduce the solution set of the inequality $\left|\frac{10}{x}-2p\right|&lt;\frac{10}{x}$.</p><p>Leave both answers in terms of $p$.</p><span class="inline-marks">[3]</span></div></li></ol></li></ol>
<ol class="abc-list" start="2"><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0"><div class="list-item-paragraph"><p>The diagram shows a sketch of the curve $y=2\mathrm{f}(-x-1)$.</p><p>The curve has asymptotes with equation $x=-1$, $x=1$ and $y=0$.</p><p>The curve has a stationary point of inflexion at $(0,0)$, a maximum point at $(-2,3)$ and cuts the $x$-axis at $(-1.5,0)$.</p></div><div class="graph-block" data-graph-config="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" data-width="460" data-height="360" style="width:460px;height:360px;max-width:100%;display:block;margin:0 auto;"></div><div class="list-item-paragraph">On separate diagrams, sketch the graphs of</div><ol class="i-list"><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0-after-1-item-0" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0/h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0-after-1-item-0"><div class="list-item-paragraph">$y=\frac{1}{2\mathrm{f}(-x-1)}$,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0-after-1-item-1" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0/h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0-after-1-item-1"><div class="list-item-paragraph"><p>$y=\mathrm{f}(x)$,</p><span class="inline-marks">[3]</span></div></li></ol><div class="list-item-after-content">stating clearly the equation of any asymptotes, and the coordinates of any turning points and axial intercepts.</div></li></ol>

The graphs of $y=2|5x-p|$ and $y=qx$, where $p,q>0$, intersect at only one point.

Parameter Conditions for Graphs and Equations

Inequalities involving Substitution

Graphical Method

Reciprocal Graph, $y=\frac{1}{f(x)}$

Recovering Original Graphs from Transformations

Graphs of Modulus Function

Inequalities involving Modulus Functions

2025 DHS Promo Q11

Graphing Techniques

Inequalities

Transformations of Graphs

<p>A curve, $y=f(x)$, undergoes the transformations $A$, $B$, $C$ and $D$ in succession:</p>
<ol class="ABC-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">A translation by $5$ units in the negative $x$-direction.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">A scaling by a scale factor of $3$ parallel to the $x$-axis.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">A reflection about the $y$-axis.</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">A translation by $2$ units in the negative $y$-direction.</div></li></ol>
<p>The equation of the resulting curve is $y=9-x+2\ln\left(5-\dfrac{x}{3}\right)$, where&nbsp;$x&lt;15$.</p>
<p>Find the equation of the original curve $y=f(x)$, and state its domain.<span class="inline-marks">[6]</span></p>

A curve, $y=f(x)$, undergoes the transformations $A$, $B$, $C$ and $D$ in succession:

The equation of the resulting curve is $y=9-x+2\ln\left(5-\dfrac{x}{3}\right)$, where $x<15$.

Find the equation of the original curve $y=f(x)$, and state its domain.

Domain of a Function

Recovering a logarithmic function after graph transformations

Functions

<p>A curve, $y=f(x)$, undergoes the transformations $A$, $B$, $C$ and $D$ in succession:</p>
<ol class="ABC-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">A translation by $2$ units in the negative $x$-direction.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">A scaling by a scale factor of $3$ parallel to the $x$-axis.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">A reflection about the $y$-axis.</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">A translation by $5$ units in the positive $y$-direction.</div></li></ol>
<p>The equation of the resulting curve is $y=12-\dfrac{4}{3}x+e^{\frac{2}{3}x-4}$.</p>
<p>Find the equation of the original curve $y=f(x)$.<span class="inline-marks">[5]</span></p>

The equation of the resulting curve is $y=12-\dfrac{4}{3}x+e^{\frac{2}{3}x-4}$.

Find the equation of the original curve $y=f(x)$.

Recovering a function after successive graph transformations

<ol class="abc-list"><li data-list-item-id="item-1769597605016-2gv61rxm1"><div class="list-item-paragraph">A curve with equation $y=\mathrm{f}(x)$ undergoes the following sequence of transformation.</div><div class="list-item-paragraph">$\qquad A$:	A translation of $3$ units in the positive $y$-direction.<br>$\qquad B$:	A translation of $2$ units in the positive $x$-direction.<br>$\qquad C$: 	A stretch with scale factor $2$ parallel to the $x$-axis.</div><div class="list-item-paragraph">The resulting curve has equation $y=\mathrm{g}(x)$ where&nbsp;$\mathrm{g}(x)=2-\frac{4}{3x-7}$. Find $\mathrm{f}(x)$ in terms of $x$.  		<span class="inline-marks">[3]</span></div><div class="list-item-paragraph">Another curve with equation $y=\frac{1}{\mathrm{f}(x)}$ undergoes the same sequence of transformations $A$, $B$ and $C$. <br>Will the equation of the resulting curve be$y=\frac{1}{\mathrm{g}(x)}$? Justify your answer<span class="inline-marks">[1]</span></div></li></ol>

Composite Transformations

2015 IJC P1 Q8

<p>A graph with equation $y=\mathrm{f}\left( x \right)$ undergoes $3$ successive transformations as follows:<br><br>I:	A scaling parallel to $x$-axis with scale factor $\frac{1}{2}$,<br>II:	A translation of $3$ units in the direction of the positive $x$-axis, <br>III:	A scaling parallel to $y$-axis with scale factor $2$.<br><br>The resulting equation is $y=2\ln \left( 2x-6 \right)$. Determine the equation of the graph $y=\mathrm{f}\left( x \right)$, showing your workings clearly.<br></p>

A graph with equation $y=\mathrm{f}\left( x \right)$ undergoes $3$ successive transformations as follows:

I:	A scaling parallel to $x$-axis with scale factor $\frac{1}{2}$,
II:	A translation of $3$ units in the direction of the positive $x$-axis, 
III:	A scaling parallel to $y$-axis with scale factor $2$.

The resulting equation is $y=2\ln \left( 2x-6 \right)$. Determine the equation of the graph $y=\mathrm{f}\left( x \right)$, showing your workings clearly.


2006 CJC Promo Q3 [Modified]

<p>A curve, $y=\mathrm{f}(x)$ undergoes the transformations $A$, $B$, $C$ and $D$ in succession:</p>
<ol class="ABC-list"><li data-list-item-id="item-A" data-list-item-path="item-A"><div class="list-item-content"><span class="list-item-text">Translation by $4$ units in the negative $x$ direction.</span></div></li><li data-list-item-id="item-B" data-list-item-path="item-B"><div class="list-item-content"><span class="list-item-text">Scaling by a scale factor of $2$ parallel to the $x$-axis.</span></div></li><li data-list-item-id="item-C" data-list-item-path="item-C"><div class="list-item-content"><span class="list-item-text">Reflection about the $y$-axis.</span></div></li><li data-list-item-id="item-D" data-list-item-path="item-D"><div class="list-item-content"><span class="list-item-text">A translation of $1$ unit in the positive direction of the $y$-axis.</span></div></li></ol>
<p>The equation of the resulting curve is $y=10-x+e^{\frac{x}{2}-4}$.</p>
<p>Find the equation of the curve $y=\mathrm{f}(x)$.</p>

A curve, $y=\mathrm{f}(x)$ undergoes the transformations $A$, $B$, $C$ and $D$ in succession:

Translation by $4$ units in the negative $x$ direction.

Scaling by a scale factor of $2$ parallel to the $x$-axis.

A translation of $1$ unit in the positive direction of the $y$-axis.

The equation of the resulting curve is $y=10-x+e^{\frac{x}{2}-4}$.

Find the equation of the curve $y=\mathrm{f}(x)$.

<p>A curve, $y = \mathrm{f}(x)$ undergoes the transformations A, B, C and D in succession:<br></br></p>
<p>A: Translation by 5 units in the positive x-direction.<br></p>
<p>B: Scaling by a scale factor of $\frac{1}{3}$ parallel to the x-axis.<br></p>
<p>C: Translation of 5 units in the negative y-direction.<br></p>
<p>D: Scaling by a scale factor of $\frac{1}{3}$ parallel to the y-axis.<br></br></p>
<p>The equation of the resulting curve is <br></p>
<p style="text-align: center">$y = 3^{3x - 6} + \frac{1}{3}\cos(3x - 5) - \frac{5}{3}$.</p>
<p>Find the equation of the curve $y = \mathrm{f}(x)$.</p>

A curve, $y = \mathrm{f}(x)$ undergoes the transformations A, B, C and D in succession:<br></br>

A: Translation by 5 units in the positive x-direction.


B: Scaling by a scale factor of $\frac{1}{3}$ parallel to the x-axis.


C: Translation of 5 units in the negative y-direction.


D: Scaling by a scale factor of $\frac{1}{3}$ parallel to the y-axis.<br></br>

$y = 3^{3x - 6} + \frac{1}{3}\cos(3x - 5) - \frac{5}{3}$.

Find the equation of the curve $y = \mathrm{f}(x)$.

<p>A curve undergoes the transformations $A$, $B$ and $C$ in succession:<br></br></p>
<p>$A$: a translation of $2$ units in the negative direction of the $x$-axis.<br></p>
<p>$B$: a scaling parallel to the $x$-axis by a factor $2$.<br></p>
<p>$C$: a translation of $1$ unit in the positive direction of the $y$-axis.</p>
<p>The equation of the resulting curve is $y = 2x + 9 - \frac{2}{x + 4}$.</p>
<p>Find the equation of the curve before the three transformations were affected.</p>

A curve undergoes the transformations $A$, $B$ and $C$ in succession:<br></br>

$A$: a translation of $2$ units in the negative direction of the $x$-axis.


$B$: a scaling parallel to the $x$-axis by a factor $2$.


$C$: a translation of $1$ unit in the positive direction of the $y$-axis.

The equation of the resulting curve is $y = 2x + 9 - \frac{2}{x + 4}$.

Find the equation of the curve before the three transformations were affected.

<p>A curve, $y = \mathrm{f}(x)$ undergoes the transformations A, B and C in succession:<br></br></p>
<p>A: a translation of 4 units in the positive direction of the x-axis.</p>
<p>B: a scaling parallel to the x-axis by a factor $\frac{1}{3}$.</p>
<p>C: a translation of 2 units in the negative direction of the y-axis.<br></br></p>
<p>The equation of the resulting curve is $y = e^{9x - 12} + 3x - 6$.</p>
<p>Find the equation of the curve $y = \mathrm{f}(x)$.</p>

A curve, $y = \mathrm{f}(x)$ undergoes the transformations A, B and C in succession:<br></br>

A: a translation of 4 units in the positive direction of the x-axis.

B: a scaling parallel to the x-axis by a factor $\frac{1}{3}$.

C: a translation of 2 units in the negative direction of the y-axis.<br></br>

The equation of the resulting curve is $y = e^{9x - 12} + 3x - 6$.

<ol class="abc-list"><li data-list-item-id="item-1760863599796-fblx92j0j" data-list-item-path="item-1760863599796-fblx92j0j"><div class="list-item-paragraph">The diagram shows the curve $y=\mathrm{f}\left( x \right)$ that passes through the origin and $\left( 1,0 \right)$. It has a minimum point $\left( \frac{1}{2},-\frac{1}{2} \right)$ and asymptotes $x=-1$ and $y=3$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770971230272-a2rzphld6gk.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA5NzEyMzAyNzItYTJyenBobGQ2Z2sucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTI2NywiZXhwIjoxNzkyODM5NTY3fQ.jj1sEP0tIYNMZReEVV3iBycH_s-nrGVGWEgKuJ5e80Q" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div><div class="list-item-paragraph">Sketch the curve $y=\frac{1}{\mathrm{f}\left( x \right)}$, showing clearly the coordinates of any stationary points, axial intercepts and equations of all asymptotes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760863600545-qm7zas2fz" data-list-item-path="item-1760863600545-qm7zas2fz"><div class="list-item-paragraph">A curve with equation $y=\mathrm{h}\left( x \right)$ undergoes the following sequence of transformations.</div><div class="list-item-paragraph" style="margin-left: 2rem">Step $1$: A translation of $2$ units in the positive direction of the $x$-axis.</div><div class="list-item-paragraph" style="margin-left: 2rem">Step $2$: A scaling parallel to the $x$-axis by factor of $3$.</div><div class="list-item-paragraph" style="margin-left: 2rem">Step $3$: A reflection in the $y$-axis.</div><div class="list-item-paragraph">The resultant curve has equation $y=-\frac{{{x}^{2}}}{6x+45}$. Find $\mathrm{h}\left( x \right)$.<span class="inline-marks">[3]</span></div></li></ol>

2024 NJC Promo Q4

<ol class="abc-list"><li><div class="list-item-paragraph">The transformations $A$, $B$ and $C$ are given as follows:</div><div class="list-item-paragraph">$A$: A reflection in the $x$-axis.</div><div class="list-item-paragraph">$B$: A translation of $1$ unit in the positive $y$-direction.</div><div class="list-item-paragraph">$C$: A translation of $2$ units in the negative $x$-direction.<br></br></div><div class="list-item-paragraph">A curve undergoes in succession, the transformations $A$, $B$ and $C$, and the equation of the resulting curve is $y=\frac{2x+1}{2x+2}$. Determine the equation of the curve before the transformations, expressing your answer as a single fraction.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">The diagram shows the curve $y=\mathrm{f}\left( x \right)$. The lines $x=-2$, $x=2$ and $y=\frac{3}{2}x$ are asymptotes to the curve. The curve has turning points at $\left( -3,-5 \right)$ and $\left( 3,5 \right)$. It also has a stationary point of inflexion at the origin $O$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1771832461999-rgkqf92t5u.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzE4MzI0NjE5OTktcmdrcWY5MnQ1dS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODM5MjY3LCJleHAiOjE3OTI4Mzk1Njd9.sXT4OPqBM0jRGpLGR9IMLh4Qby5Ax4Pgjy6NljLWj6s" alt="List item image" class="aligncenter custom-width" style="max-width: min(390px, 100%); height: auto;" /></div><ol class="i-list"><li><div class="list-item-paragraph">State the range of values of $x$ for which the graph is concave downwards.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Sketch the graph of $y=\frac{1}{\mathrm{f}\left( x \right)}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Sketch the graph of $y={{\mathrm{f}}'}\left( x \right)$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

Interpreting Gradients and Concavity

Sketching Derivative Graphs

2024 DHS Promo Q6

Applications of Differentiation

<ol class="abc-list" start="2"><li><div class="list-item-paragraph">The curve ${{C}_{1}}$ undergoes the transformations in the order given below:</div><div class="list-item-paragraph">1. A translation of $2$ units in the negative $x$ direction.</div><div class="list-item-paragraph">2. A stretch parallel to the $x$ axis, factor 2, $y$ axis invariant.</div><div class="list-item-paragraph">3. A translation of 1 unit in the positive $y$ direction.</div><div class="list-item-paragraph">The resulting curve ${{C}_{2}}$ has equation $y=\frac{{{x}^{2}}+9x+22}{x+4},$ $x\in R$, $x\ne -4$</div><div class="list-item-paragraph">Find, in the simplest form, the equation for ${{C}_{1}}$.</div><p style="text-align: right;">[4]</p></li></ol>

2024 VJC P1 Q8

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760435700775-vhzlzf8ue7s.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA0MzU3MDA3NzUtdmh6bHpmOHVlN3MucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTI2NywiZXhwIjoxNzkyODM5NTY3fQ.kPzS-qDzqxZxiaW-BYxvYmveEdLdHItPlSWqRkx2atY" style="max-width: min(600px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">The diagram above shows the graph of $y=\mathrm{f}\left( x \right)$. The curve crosses the $x$-axis at $\left( -3.5,\,\,0 \right)$ and has a minimum point at $\left( -2,\,\,0 \right)$. The lines $x=-3$, $x=0$ and $y=3$ are asymptotes of the curve.</div><div class="list-item-paragraph">On separate diagrams, sketch the graphs in <strong>(i)</strong> and <strong>(ii)</strong>, stating the equations of any asymptotes and the coordinates of the points where the curve crosses the axes and of any turning points in exact form, if possible.</div><ol class="i-list"><li><div class="list-item-paragraph">$y=\mathrm{f}\left( \left| x \right| \right)$, and<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">$y=\mathrm{f}\,\,\mathrm{ }\!\!'\!\!\mathrm{ }\left( x \right)$.<span class="inline-marks">[4]</span></div></li></ol></li><li><div class="list-item-paragraph">The transformations A, B and C are given as follows:</div><div class="list-item-paragraph">A: $ \qquad $ Reflection about the $x$-axis;</div><div class="list-item-paragraph">B: $ \qquad $ Translation of $4$ units in the positive $x$-direction;</div><div class="list-item-paragraph">C: $ \qquad $ Scaling parallel to the $x$-axis by a factor of $2$.</div><div class="list-item-paragraph">A curve undergoes in succession, the transformations A, B and C, and the equation of the resulting curve is $y=\frac{1}{3}{{\left( x+1 \right)}^{2}}$.</div><div class="list-item-paragraph">Determine the original equation of the curve.<span class="inline-marks">[3]</span></div></li></ol>

Graphs of $\lvert f(x)\rvert$ and $f(\lvert x\rvert)$

Sketching Graphs of $f(\lvert x\rvert)$

2023 SAJC Promo Q6

<ol class="abc-list"><li><div class="list-item-paragraph">A curve $C$ undergoes, in succession, the following transformations.</div><div class="list-item-paragraph" style="margin-left: 3rem">$A$: Translation of $4$ units in the negative $y$- direction.</div><div class="list-item-paragraph" style="margin-left: 3rem">$B$: Stretching parallel to the $x$-axis by a factor of $2$.</div><div class="list-item-paragraph" style="margin-left: 3rem">$C$: Reflection in the $x$-axis.</div><div class="list-item-paragraph">The equation of the resulting curve is $y=\frac{{{x}^{2}}}{4}+4$.</div><div class="list-item-paragraph">Determine the equation of the original curve $C$ before the transformations were carried out.</div><p style="text-align: right;">[3]</p></li></ol>

2022 TMJC Promo Q6

<p>The graph of $y=\mathrm{f}\left( 3-\frac{x}{2} \right)$ has $2$ stationary points at $\left( 8,-2 \right)$ and $\left( 14,2 \right)$ and intersects the $x$-axis at $x=6$, $x=10$ and $x=18$ as shown in the diagram below.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760247593346-p6yc7tkbcjq.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAyNDc1OTMzNDYtcDZ5Yzd0a2JjanEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTI2NywiZXhwIjoxNzkyODM5NTY3fQ.XlzoI1_UhcEA77pDjJFc30qd82R669NBCXB2kbUCRsM" width="400" class="aligncenter" /></figure>
<p>Sketch, on separate clearly labelled diagrams, the graphs of</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$y=\frac{1}{\mathrm{f}\left( 3-\frac{x}{2} \right)}$</div></li><li><div class="list-item-paragraph">$y=\mathrm{f}\left( x \right)$</div></li></ol>
<p>Show clearly the asymptotes, stationary points and points of intersection with the axes in your diagrams.</p>

The graph of $y=\mathrm{f}\left( 3-\frac{x}{2} \right)$ has $2$ stationary points at $\left( 8,-2 \right)$ and $\left( 14,2 \right)$ and intersects the $x$-axis at $x=6$, $x=10$ and $x=18$ as shown in the diagram below.

Sketch, on separate clearly labelled diagrams, the graphs of

Show clearly the asymptotes, stationary points and points of intersection with the axes in your diagrams.

Graph Transformations of $f(ax+b)$

2009 AJC P1 Q11

<ol class="abc-list" start="2"><li data-list-item-id="item-1760096914383-ydc2robbe" data-list-item-path="item-1760096914383-ydc2robbe"><div class="list-item-paragraph">A graph with equation $y=\,\mathrm{f}(x)$ undergoes in succession, the following transformations:<span class="inline-marks">[3]</span></div><div class="list-item-paragraph">A: Scaling parallel to the $y$-axis by a factor of $\frac{1}{\mathrm{e}}$</div><div class="list-item-paragraph">B: Translation of 1 unit in the direction of the negative $x$-axis</div><div class="list-item-paragraph">C: Scaling parallel to the $x$-axis by a factor of $\frac{1}{2}$</div><div class="list-item-paragraph">The equation of the resulting curve is given by $y={{\mathrm{e}}^{-2(x+1)}}$. Find the equation $y=\,\mathrm{f}(x)$.</div></li></ol>

2012 SRJC P1 Q10

<p>A graph with the equation $y=\mathrm{f}(x)$ undergoes, in succession, the following transformations:</p>
<p style="margin-left: 3rem">$A$: A translation of $1$ unit in the direction of the positive $x$- axis. </p>
<p style="margin-left: 3rem">$B$: A stretch parallel to the $x$- axis by factor $\frac{1}{2}$.</p>
<p style="margin-left: 3rem">$C$: A reflection in the $y$- axis.</p>
<p>The equation of the resulting curve is $y=\frac{2}{2{{x}^{2}}+2x+1}$. Determine the equation $y=\mathrm{f}(x)$.</p>

A graph with the equation $y=\mathrm{f}(x)$ undergoes, in succession, the following transformations:

$A$: A translation of $1$ unit in the direction of the positive $x$- axis. 

$B$: A stretch parallel to the $x$- axis by factor $\frac{1}{2}$.

The equation of the resulting curve is $y=\frac{2}{2{{x}^{2}}+2x+1}$. Determine the equation $y=\mathrm{f}(x)$.

ACJC Tutorial 3 Question 9

<ol class="abc-list"><li><div class="list-item-paragraph">The curve with equation $y=\mathrm{f}(x)$ is transformed by a translation of 2 units in the negative $x$-direction, followed by a scaling with scale factor $3$ parallel to the $x$-axis, and followed by a translation of $4$ units in the positive $y$-direction. The equation of the resulting curve is $y=2{{\mathrm{e}}^{{{x}^{2}}-36}}$.</div><div class="list-item-paragraph">Find the equation of the curve $y=\mathrm{f}(x)$.</div><p style="text-align: right;">[3]</p></li></ol>

2023 YIJC Promo Q8

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The curve ${{C}_{1}}$ has equation $y=\mathrm{f}\left( x \right)$, where&nbsp;$\mathrm{f}\left( x \right)={{x}^{3}}-4x+1$.</span></div><div class="list-item-paragraph">A second curve ${{C}_{2}}$ undergoes, in succession, the following transformations to get the resulting curve ${{C}_{1}}$:</div><div class="list-item-paragraph" style="margin-left: 3rem">$A$: A translation of $1$ unit in the negative direction of the $y$-axis</div><div class="list-item-paragraph" style="margin-left: 3rem">$B$: A reflection in the $y$-axis</div><div class="list-item-paragraph" style="margin-left: 3rem">$C$: A scaling of factor $\frac{1}{4}$ parallel to the $x$-axis</div><div class="list-item-paragraph">Determine the equation of ${{C}_{2}}$, showing your workings clearly.</div></li><li><div class="list-item-paragraph">The diagram shows the curve $y=\mathrm{f}\left( x \right)$. The curve passes through $\left( 4,0 \right)$ and $\left( 0,-2 \right)$, and has a maximum point $\left( 6,a \right)$ where&nbsp;$a&gt;0$. It has asymptotes $x=2$, $y=\frac{1}{2}a$ and $y=0$.</div><div class="list-item-paragraph">Sketch the graph of</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1762971341160-kgk7n9us39.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjI5NzEzNDExNjAta2drN245dXMzOS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODM5MjY3LCJleHAiOjE3OTI4Mzk1Njd9.e4-NApOf2ORi4FE8K5149vpd-_hg6j7_NnztRAHb6F0" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div><ol class="i-list"><li><div class="list-item-paragraph">$y=\frac{1}{\mathrm{f}\left( x \right)}$,<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">$y=\mathrm{f}\left( \left| x \right| \right)$,<span class="inline-marks">[3]</span></div><div class="list-item-paragraph">showing clearly the stationary points, axial intercepts and equations of all asymptotes.</div></li></ol></li></ol>

The curve ${{C}_{1}}$ has equation $y=\mathrm{f}\left( x \right)$, where $\mathrm{f}\left( x \right)={{x}^{3}}-4x+1$.

2018 AJC Promo Q6

<ol class="abc-list" start="2"><li data-list-item-id="item-1783336573852-2c9vo3bt7" data-list-item-path="item-1783336573852-2c9vo3bt7"><div class="list-item-paragraph">The graph of $y=\mathrm{g}\left( x \right)$ undergoes the following sequence of transformations,<br>$A$: Translation of $1$ unit in the positive $x$-direction,<br>$B$: Reflection in the $y$-axis,<br>$C$: Scaling by factor $2$ parallel to the $x$-axis,<br>$D$: Translation of $1$ unit in the negative $y$-direction.<br>Given that the equation of the resulting graph is $y=\frac{x}{2}+\frac{2}{x}$, find $y=\mathrm{g}\left( x \right)$, showing your working clearly.<span class="inline-marks">[4]</span></div></li></ol>

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