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H1 Math

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N(A) A Math

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PSLE

PSLE Foundation Mathematics

<ol class="abc-list"><li data-list-item-id="item-1760863688588-bx4gr9sr2" data-list-item-path="item-1760863688588-bx4gr9sr2"><div class="list-item-paragraph">Express $4{{x}^{2}}-8x+4$ in the form of $A{{\left( x-B \right)}^{2}}$ where&nbsp;$A$ and $B$ are constants to be determined.<span class="inline-marks">[1]</span></div><div class="list-item-paragraph">Hence or otherwise, state a sequence of transformations that would transform the curve with equation $y=3{{e}^{{{x}^{2}}}}$ onto the curve with equation $y=3{{\mathrm{e}}^{4{{x}^{2}}-8x+4}}-10$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760863689345-6y9m4dd2h" data-list-item-path="item-1760863689345-6y9m4dd2h"><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770971282168-et04ven2rd8.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA5NzEyODIxNjgtZXQwNHZlbjJyZDgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTI3OCwiZXhwIjoxNzkyODM5NTc4fQ.oaoVqvBSOGiwVjrfEmpyn1Ppzb4moHtxPi3bh5OF-b0" alt="List item image" class="aligncenter custom-width" style="max-width: min(450px, 100%); height: auto;" /></div><div class="list-item-paragraph">The diagram shows the curve $y=\mathrm{f}\left( x \right)$. The curve has a turning point at $(6,-5)$ and crosses the $x$-axis at $(-4,0)$ and $(0,0)$. The lines $y=-2$ and $x=-3$ are the asymptotes to the curve.</div><div class="list-item-paragraph">On separate diagrams, sketch the graphs of</div><ol class="i-list"><li data-list-item-id="subitem-1760863721581-w7fma0673" data-list-item-path="item-1760863689345-6y9m4dd2h/subitem-1760863721581-w7fma0673"><div class="list-item-paragraph">$y=3\left| \mathrm{f}\left( x \right) \right|$ ,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760863722684-mwchgp7p8" data-list-item-path="item-1760863689345-6y9m4dd2h/subitem-1760863722684-mwchgp7p8"><div class="list-item-paragraph">$y=\frac{1}{\mathrm{f}\left( x \right)}$ ,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760863723223-delsuva6q" data-list-item-path="item-1760863689345-6y9m4dd2h/subitem-1760863723223-delsuva6q"><div class="list-item-paragraph">$y={{\mathrm{f}}'}\left( x \right)$ ,<span class="inline-marks">[3]</span></div></li></ol></li></ol>
<p style="margin-left: 3rem">Labelling clearly the equation(s) of any asymptote(s), coordinates of any axial intercept(s) and turning point(s) where applicable.</p>

Labelling clearly the equation(s) of any asymptote(s), coordinates of any axial intercept(s) and turning point(s) where applicable.

Composite Transformations

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

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

Graphs of Modulus Function

Sketching Derivative Graphs

2024 CJC Promo Q9

Transformations of Graphs

Applications of Differentiation

<p>The function $y=\mathrm{f}\left( x \right)$ is a one-to-one function.</p>
<p>The graphs of $y=\mathrm{f}\left( \left| x \right| \right)$ and $y=\left| \mathrm{f}\left( x \right) \right|$ are shown below in Figure 1 and 2 respectively.</p>
<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/1760435914280-9o1denki78a.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA0MzU5MTQyODAtOW8xZGVua2k3OGEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTI3OCwiZXhwIjoxNzkyODM5NTc4fQ.3exPmruBdMrXwCCl2c58rOTh2uQ5okrroVh9NpmODIU" style="max-width: min(700px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-1760435897187-klomepur2" data-list-item-path="item-1760435897187-klomepur2"><div class="list-item-paragraph">State the range of values of $x$ for which $\mathrm{f}\left( x \right)\le 0$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760435897713-f83zntrag" data-list-item-path="item-1760435897713-f83zntrag"><div class="list-item-paragraph">State the equation(s) of the asymptote(s) of the graph of $y=\frac{1}{\mathrm{f}\left( x \right)}$.<span class="inline-marks">[2]</span></div></li></ol>

The function $y=\mathrm{f}\left( x \right)$ is a one-to-one function.

The graphs of $y=\mathrm{f}\left( \left| x \right| \right)$ and $y=\left| \mathrm{f}\left( x \right) \right|$ are shown below in Figure 1 and 2 respectively.

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

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<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA0MzU3MDA3NzUtdmh6bHpmOHVlN3MucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTI3OCwiZXhwIjoxNzkyODM5NTc4fQ.L9adwom5tDWS0GLAajFD96SVBeHatt-C-vYBbQ79V7Y" 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>

Recovering Original Graphs from Transformations

2023 SAJC Promo Q6

<ol class="abc-list" start="2"><li><div class="list-item-paragraph">The diagram below shows a sketch of the curve with equation $y=\mathrm{g}(x)$. The curve has asymptotes with equations$x=0$ and $y=2$. The curve crosses the $x$-axis at $A(2,0)$ and $C(7,0)$ and has a turning point at $B(4,-2)$.</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/1760334470912-zsw0p6a0r2.jpg?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAzMzQ0NzA5MTItenN3MHA2YTByMi5qcGciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODM5Mjc4LCJleHAiOjE3OTI4Mzk1Nzh9.EOII41LUlyNdRBlUPPTylQQUWlSzSEsmdSR1kH_L36g" alt="List item image" class="aligncenter custom-width" style="max-width: min(450px, 100%); height: auto;" /></div><div class="list-item-paragraph">On separate diagrams, sketch the curves with equations</div><ol class="i-list"><li><div class="list-item-paragraph">$y=\left| \mathrm{g}(-x) \right|$,<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">$y=\frac{1}{\mathrm{g}(x)}$.<span class="inline-marks">[4]</span></div></li></ol></li></ol>
<p style="margin-left: 3rem">For each curve, state clearly the equations of any asymptotes, the coordinates of any turning points and the coordinates of any points where the curves crosses the $x$- and $y$- axes.</p>

For each curve, state clearly the equations of any asymptotes, the coordinates of any turning points and the coordinates of any points where the curves crosses the $x$- and $y$- axes.

2021 NJC Lecture Test

<ol class="abc-list"><li data-list-item-id="item-1759919560416-0x54j67eb"><div class="list-item-paragraph">The diagram shows the graph of $y=\mathrm{f}\left( x \right)$. The curve passes through the origin, has a minimum point at $\left( -3,0 \right)$ and its equations of asymptotes are $x=-2$ and $y=2$.</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/1759919573279-x5glibx4w3r.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTk5MTk1NzMyNzkteDVnbGlieDR3M3IucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTI3OCwiZXhwIjoxNzkyODM5NTc4fQ.udWT3cBIrStOjvwBYrc--YfzVc8XfODltbeEUDiD2IU" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div><div class="list-item-paragraph">Sketch the following graphs on separate axes.</div><ol class="i-list"><li data-list-item-id="subitem-1759919583922-pzh5mmhqt"><div class="list-item-paragraph">$y=\frac{1}{\mathrm{f}\left( x \right)}$<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1759919616181-qb3lwb6qk"><div class="list-item-paragraph">$y=\mathrm{f}\left( -\left| x \right| \right)$<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-1759919600654-hc3l0n9wa"><div class="list-item-paragraph">Describe a series of transformations that transforms the graph of ${{x}^{2}}+{{\left( y+1 \right)}^{2}}=1$ onto the graph of ${{\left( x-1 \right)}^{2}}+\frac{{{y}^{2}}}{9}=1$.<span class="inline-marks">[3]</span></div></li></ol>

2019 RVHS Promo Q5

<p>The graphs of $y=\mathrm{f}'\left( x \right)$ and $y=\left| \mathrm{f}\left( x \right) \right|$ are shown below.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="250" data-height="250" style="width:250px;height:250px;max-width:100%;display:block;margin:0 auto;"></div></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="250" data-height="250" style="width:250px;height:250px;max-width:100%;display:block;margin:0 auto;"></div></div></div></div>
<ol class="i-list"><li data-list-item-id="item-1759654477096-7tpu0jawd" data-list-item-path="item-1759654477096-7tpu0jawd"><div class="list-item-paragraph">State the nature of all stationary points of the graph of $y=\mathrm{f}\left( x \right)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759654479864-006rf6uoz" data-list-item-path="item-1759654479864-006rf6uoz"><div class="list-item-paragraph">State the range of values of $x$ where</div><ol class="abc-list"><li data-list-item-id="subitem-1759654504662-9fevmd1o4" data-list-item-path="item-1759654479864-006rf6uoz/subitem-1759654504662-9fevmd1o4"><div class="list-item-paragraph">$\operatorname{f}$ is decreasing,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1759654506069-07ssuq5um" data-list-item-path="item-1759654479864-006rf6uoz/subitem-1759654506069-07ssuq5um"><div class="list-item-paragraph">$\mathrm{f}$ is concave upwards.<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="item-1759654526311-tig8bvc3b" data-list-item-path="item-1759654526311-tig8bvc3b"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$ indicating clearly the asymptotes, coordinates of the stationary points and the intersections with the axes.<span class="inline-marks">[3]</span></div></li></ol>

The graphs of $y=\mathrm{f}'\left( x \right)$ and $y=\left| \mathrm{f}\left( x \right) \right|$ are shown below.

Interpreting Gradients and Concavity

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<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjI5NzEzNDExNjAta2drN245dXMzOS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODM5Mjc4LCJleHAiOjE3OTI4Mzk1Nzh9.gkWGLWTpdZEb6UVkyEAYEKF8HGaVFcWRjEdVGCbyPuY" 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 $\mathrm{f}\left( x \right)={{x}^{3}}-4x+1$.

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<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/1759226098256-lwfehnddjz.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTkyMjYwOTgyNTYtbHdmZWhuZGRqei5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODM5Mjc4LCJleHAiOjE3OTI4Mzk1Nzh9.Dg7Nsb0kAb0tC5o_3IrnrRkGB631Gfki5lGkd-TaBp8" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The diagram shows the curve $y=\mathrm{f}\left( x \right)$. There are two vertical asymptotes with equations $x=-2$ and $x=2$ respectively. The curve crosses the $x$-axis at the point $A$ and has a maximum turning point at $B$ where it crosses the $y$-axis.</p>
<p>The curve also has a minimum turning point at $C$. The coordinates of $A$, $B$ and $C$ are $\left( a,0 \right)$, $\left( 0,-10 \right)$ and $\left( p,q \right)$ respectively, where&nbsp;$a$, $p$ and $q$ are constants</p>
<p>Sketch the following curves and state the equations of the asymptotes, the coordinates of the turning points and of points where the curve crosses the axes, if any. Leave your answers in terms of $a$, $p$ or $q$ where necessary.</p>
<ol class="i-list"><li data-list-item-id="item-1759226109016-pkmxk2gk7" data-list-item-path="item-1759226109016-pkmxk2gk7"><div class="list-item-paragraph">$y=\frac{1}{\mathrm{f}\left( x \right)}$, and<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759226109622-kqp84utzn" data-list-item-path="item-1759226109622-kqp84utzn"><div class="list-item-paragraph">$y=\mathrm{f}\left( 2-\left| x \right| \right)$. .<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows the curve $y=\mathrm{f}\left( x \right)$. There are two vertical asymptotes with equations $x=-2$ and $x=2$ respectively. The curve crosses the $x$-axis at the point $A$ and has a maximum turning point at $B$ where it crosses the $y$-axis.

The curve also has a minimum turning point at $C$. The coordinates of $A$, $B$ and $C$ are $\left( a,0 \right)$, $\left( 0,-10 \right)$ and $\left( p,q \right)$ respectively, where $a$, $p$ and $q$ are constants

Sketch the following curves and state the equations of the asymptotes, the coordinates of the turning points and of points where the curve crosses the axes, if any. Leave your answers in terms of $a$, $p$ or $q$ where necessary.

$y=\frac{1}{\mathrm{f}\left( x \right)}$, and

$y=\mathrm{f}\left( 2-\left| x \right| \right)$. .

Graph Transformations of $f(ax+b)$

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<ol class="abc-list"><li><div class="list-item-paragraph">Describe a sequence of three transformations which transforms the graph of $y=x-\frac{1}{2x+4}$ to the graph of $y=-2\left( x+2 \right)+\frac{1}{x}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">The diagram shows the graph of $y=\mathrm{f}\left( x \right)$. The curve passes through the points $\left( -3,-2 \right)$, $\left( -1,0 \right)$, $\left( 0,1 \right)$, $\left( 1,2 \right)$ and $\left( 3,0 \right)$, and has a horizontal asymptote at $y=-\frac{1}{2}$.</div><div class="list-item-paragraph">On separate diagrams that are clearly labelled, sketch the graphs 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/1764583914468-zia0m90by9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjQ1ODM5MTQ0NjgtemlhMG05MGJ5OS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODM5Mjc4LCJleHAiOjE3OTI4Mzk1Nzh9.kK2lzYDuC7RgAjGXw_wSBd2xoAgv6YjZVLIaB6CogCU" alt="List item image" class="aligncenter custom-width" style="max-width: min(450px, 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">[4]</span></div></li><li><div class="list-item-paragraph">$y=\mathrm{f}\left( \left| x \right|-1 \right)$,<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">$y=\mathrm{f}\left( 2x+1 \right)$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

Describe a sequence of three transformations which transforms the graph of $y=x-\frac{1}{2x+4}$ to the graph of $y=-2\left( x+2 \right)+\frac{1}{x}$.

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