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PSLE

PSLE Foundation Mathematics

<p>The curve with equation $y = \mathrm{f}(x)$ for&nbsp;$x \leq b$, where&nbsp;$\mathrm{f}(x)$ is a <strong>quadratic</strong> function, intersects the $x$-axis at the points $(a,0)$ and $(b,0)$, and the $y$-axis at the point $(0,c)$. The graph of $y = \mathrm{f}(x)$ is shown in Figure 1. The scales on the $x$- and $y$-axes are the same.</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/pasted/1783531831399-rvzkp28l39.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODM1MzE4MzEzOTktcnZ6a3AyOGwzOS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODM5MTM0LCJleHAiOjE3OTI4Mzk0MzR9.B7KMznNzGseyIZ1o-S2P5dZgiBLK17Gdbf8pk5BJmes" style="max-width: min(300px, 100%); height: auto; display: block;" alt="Figure 1" class="aligncenter custom-width" /></figure><div class="image-caption" style="text-align: center;">Figure 1</div></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-paragraph">Sketch the graph of $y = \mathrm{f}(\left| x \right|)$, labelling the points where the graph intersects or touches the axes.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-paragraph">Sketch the graph of $y = \frac{1}{\mathrm{f}(x)}$, labelling the points where the graph intersects or touches the axes, as well as the equations of any asymptotes.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-a-iii" data-list-item-path="item-a/item-a-iii"><div class="list-item-paragraph">Describe fully a sequence of transformations which transforms the graph of $y = \mathrm{f}(2x+1)$ onto the graph of $y = \mathrm{f}(x)$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The function $y = \mathrm{g}(x)$ is such that&nbsp;$\mathrm{g}(x) = \mathrm{f}(x)$ for&nbsp;$x \leq k$, and $y = \mathrm{g}^{-1}(x)$ exists.</div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b/item-b-i"><div class="list-item-paragraph">Find the largest possible value of $k$ in terms of $a$ and $b$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-paragraph">Using the value of $k$ found in <strong>(b)(i)</strong>, sketch the graph of $y = \mathrm{g}^{-1}(x)$ on Figure 1 in the Answer Booklet, labelling the intersections with the axes. Write down the range of $\mathrm{g}^{-1}$ in terms of $a$ and $b$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-iii" data-list-item-path="item-b/item-b-iii"><div class="list-item-paragraph">Explain why the solution to $\mathrm{g}^{-1}(x) = x$ satisfies the equation $\mathrm{g}^{-1}(x) = \mathrm{g}(x)$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

The curve with equation $y = \mathrm{f}(x)$ for $x \leq b$, where $\mathrm{f}(x)$ is a **quadratic** function, intersects the $x$-axis at the points $(a,0)$ and $(b,0)$, and the $y$-axis at the point $(0,c)$. The graph of $y = \mathrm{f}(x)$ is shown in Figure 1. The scales on the $x$- and $y$-axes are the same.

Sketching a Function and Its Inverse

Intersections of a Function and Its Inverse

Restricting a Domain to Obtain an Inverse

Inverse Functions

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

Graph Transformations of $f(ax+b)$

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

2025 ACJC P1 Q8

Transformations of Graphs

Functions

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The graph of $y=\mathrm{f}(x)$ has turning points at $A(-a, 0)$, $B(a, b)$ and $C(c, -b)$ and intersects the $x$-axis at the points $D(d, 0)$ and $E(e, 0)$. The graph of $y=\mathrm{f}(x)$ is as shown in the diagram below.<span class="inline-marks">[3]</span></div></li></ol>
<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/pasted/1769223141925-ir1uwhrzoth.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkyMjMxNDE5MjUtaXIxdXdocnpvdGgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTEzNCwiZXhwIjoxNzkyODM5NDM0fQ.pL3l7J-DRKhPszPBEiIdRlpMay0xOcGKQxnYxpmIXx0" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Sketch the graph of $y=\mathrm{f}(a-x)$, labelling the coordinates of the corresponding points of $A$, $B$, $C$, $D$, and $E$ clearly.<span class="inline-marks">[3]</span></p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The graph of $y=\mathrm{g}(x)$ has asymptotes $y=0$, $y=-m$ and $x=-n$, and intersects the $y$-axis at $P(0, p)$, where&nbsp;$p &lt; m$. The graph of $y=\mathrm{g}(x)$ is as shown in the diagram below.</div></li></ol>
<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/pasted/1769223158996-vh24fdfkh2.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkyMjMxNTg5OTYtdmgyNGZkZmtoMi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODM5MTM0LCJleHAiOjE3OTI4Mzk0MzR9.oOcnQ1fc1ylD064yxowwskmXy3QHHy_QAI2UBnLXLtg" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Sketch on separate diagrams the graphs of</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">$y=|\mathrm{g}(x)|$ and<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">$y=\mathrm{g}(|x|)$,<span class="inline-marks">[2]</span></div></li></ol>
<p>labelling clearly the asymptote(s) and coordinates of $P$ on each graph.</p>

The graph of $y=\mathrm{f}(x)$ has turning points at $A(-a, 0)$, $B(a, b)$ and $C(c, -b)$ and intersects the $x$-axis at the points $D(d, 0)$ and $E(e, 0)$. The graph of $y=\mathrm{f}(x)$ is as shown in the diagram below.

Sketch the graph of $y=\mathrm{f}(a-x)$, labelling the coordinates of the corresponding points of $A$, $B$, $C$, $D$, and $E$ clearly.

The graph of $y=\mathrm{g}(x)$ has asymptotes $y=0$, $y=-m$ and $x=-n$, and intersects the $y$-axis at $P(0, p)$, where $p < m$. The graph of $y=\mathrm{g}(x)$ is as shown in the diagram below.

Sketch on separate diagrams the graphs of

labelling clearly the asymptote(s) and coordinates of $P$ on each graph.

Graphs of Modulus Function

2025 NYJC P1 Q4

Graphing Techniques

<p>The curve $C$ has equation given by $y = \frac{1 + 3x - 18x^2}{9x + 3}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-content"><span class="list-item-text">Without the use of a calculator, find the range of values that&nbsp;$y$ can take.</span><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b"><div class="list-item-content"><span class="list-item-text">Sketch the graph of $C$ indicating clearly its asymptotes and stationary points.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c"><div class="list-item-content"><span class="list-item-text">State a sequence of transformations that will transform the curve $C$ to the curve with equation $y = \frac{1 + 3(2x - 1) - 18(2x - 1)^2}{9(2x - 1) + 3}$.</span><span class="inline-marks">[2]</span></div></li></ol>

The curve $C$ has equation given by $y = \frac{1 + 3x - 18x^2}{9x + 3}$.

Without the use of a calculator, find the range of values that $y$ can take.

Sketch the graph of $C$ indicating clearly its asymptotes and stationary points.

State a sequence of transformations that will transform the curve $C$ to the curve with equation $y = \frac{1 + 3(2x - 1) - 18(2x - 1)^2}{9(2x - 1) + 3}$.

Sketching Rational Functions

Asymptotes and Discontinuities

Finding Possible y-values of a Curve

2025 ASRJC P1 Q5

<div class="graph-block" data-graph-config="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" data-width="333" data-height="237" style="width:333px;height:237px;max-width:100%;display:block;margin:0 auto;"></div>
<p>Sketch $y = 3f(2x) - 1$</p>
<p>Sketch $y = f(1 - 2x)$</p>
<p>Sketch $3y = f(-|x|)$</p>

Composite Transformations

<ol class="abc-list"><li><div class="list-item-paragraph">Describe a sequence of transformations which would transform a curve $y=\mathrm{f}\left( x \right)$ onto the curve $y=\mathrm{f}\left( 1-2x \right)$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Sketch the graph of $y=\left| 2-3{{\mathrm{e}}^{x}} \right|$, showing the relevant features.<span class="inline-marks">[2]</span></div></li></ol>
<p>Without the use of a calculator, hence find the exact range of values of $x$ that satisfy the inequality $\left| 2-3{{\mathrm{e}}^{x}} \right|&gt;{{\mathrm{e}}^{2x}}$, for&nbsp;$x\ge 0$.<span class="inline-marks">[4]</span></p>

Without the use of a calculator, hence find the exact range of values of $x$ that satisfy the inequality $\left| 2-3{{\mathrm{e}}^{x}} \right|>{{\mathrm{e}}^{2x}}$, for $x\ge 0$.

Inequalities involving Substitution

Graphical Method

Inequalities involving Modulus Functions

2021 TJC Promo Q1

Inequalities

<ol class="abc-list"><li><div class="list-item-paragraph">Describe a sequence of two transformations that will transform the graph of $y=\mathrm{f}\left( x \right)$ to the graph of $y=\mathrm{f}\left( \alpha x+\beta \right)$, where $\alpha $ and $\beta $ are positive constants.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Diagram 1 and Diagram 2 show the graphs of $y=\mathrm{f}\left( x \right)$ and $y=\mathrm{f}\left( \alpha x+\beta \right)$ respectively. The turning points with coordinates $\left( 0,0 \right)$ and $\left( 2,4 \right)$ on $y=\mathrm{f}\left( x \right)$ correspond to the points with coordinates $\left( -2,0 \right)$ and $(2,4)$ respectively on $y=\mathrm{f}\left( \alpha x+\beta \right)$. The asymptotes $x=1$ and $y=x+\lambda $ on $y=\mathrm{f}\left( x \right)$ correspond to the asymptotes $x=0$ and $y=\alpha x+\beta +\lambda $ respectively on $y=\mathrm{f}\left( \alpha x+\beta \right)$.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760961224319-fqe5qqizi4a.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA5NjEyMjQzMTktZnFlNXFxaXppNGEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTEzNCwiZXhwIjoxNzkyODM5NDM0fQ.FjLFVGOfMS9nSk8u4-YhjUAdpIaKSQgAQL0M0JWuwYY" width="400" class="aligncenter" /></figure>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find the value of $\alpha $ and of $\beta $.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">On a separate diagram, sketch the graph of $y=\mathrm{f}\,\mathrm{ }\!\!'\!\!\mathrm{ }\left( x \right)$, giving the coordinates of the points where the graph crosses the axes and the equations of the asymptotes, if any.</div><p style="text-align: right;">[3]</p></li></ol>

Diagram 1 and Diagram 2 show the graphs of $y=\mathrm{f}\left( x \right)$ and $y=\mathrm{f}\left( \alpha x+\beta \right)$ respectively. The turning points with coordinates $\left( 0,0 \right)$ and $\left( 2,4 \right)$ on $y=\mathrm{f}\left( x \right)$ correspond to the points with coordinates $\left( -2,0 \right)$ and $(2,4)$ respectively on $y=\mathrm{f}\left( \alpha x+\beta \right)$. The asymptotes $x=1$ and $y=x+\lambda $ on $y=\mathrm{f}\left( x \right)$ correspond to the asymptotes $x=0$ and $y=\alpha x+\beta +\lambda $ respectively on $y=\mathrm{f}\left( \alpha x+\beta \right)$.

Determining Graph Transformation Parameters

Sketching Derivative Graphs

2024 HCI Promo Q7

Applications of Differentiation

<p>The diagram below shows the graph of the curve $y=\mathrm{f}\left( x \right)$, with turning point $\left( \frac{k}{2},2 \right)$ and asymptotes with equations $x=k$ and $y=0$.</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/pasted/1770970904281-sp4wx5vdqrp.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA5NzA5MDQyODEtc3A0d3g1dmRxcnAucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTEzNCwiZXhwIjoxNzkyODM5NDM0fQ.mx82Cp7i6ax-K-a11Wf-ppt_ujKqF17qtx_PXq84PYs" style="max-width: min(380px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The curve undergoes the following sequence of transformations in succession</p>
<p style="margin-left: 3rem">A: Translate $k$ units in the negative $x$-direction</p>
<p style="margin-left: 3rem">B: Reflection about the $y$-axis</p>
<p style="margin-left: 3rem">C: Translate $k$ units in the positive $x$-direction</p>
<ol class="i-list"><li data-list-item-id="item-1760863367027-aq5562mju" data-list-item-path="item-1760863367027-aq5562mju"><div class="list-item-paragraph">Sketch the curve after the transformations, indicating clearly the equations of the asymptotes and the coordinates of the turning point.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760863367850-mzuntb0pr" data-list-item-path="item-1760863367850-mzuntb0pr"><div class="list-item-paragraph">Find the equation of the curve after the transformations in the form $y=\mathrm{f}\left( ax+\mathrm{b} \right)$ where&nbsp;${a}$ and ${b}$ are constants to be determined, in terms of $k$ where appropriate.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760863369125-ka5zekfv8" data-list-item-path="item-1760863369125-ka5zekfv8"><div class="list-item-paragraph">Another curve $y=\mathrm{g}\left( x \right)$ is such that&nbsp;$\mathrm{g}\left( x \right)=\mathrm{g}\left( 4-x \right)$ for all $x$. State the equation of a line of symmetry of this curve.<span class="inline-marks">[1]</span></div></li></ol>

The diagram below shows the graph of the curve $y=\mathrm{f}\left( x \right)$, with turning point $\left( \frac{k}{2},2 \right)$ and asymptotes with equations $x=k$ and $y=0$.

The curve undergoes the following sequence of transformations in succession

A: Translate $k$ units in the negative $x$-direction

C: Translate $k$ units in the positive $x$-direction

2024 ACJC Promo Q3

<ol class="abc-list"><li><div class="list-item-paragraph">The curve $y=\mathrm{f}\left( x \right)$ has a horizontal asymptote $y=k$ and cuts the axes at $\left( a,0 \right)$ and $\left( 0,b \right)$, where $a$, $b$ and $k$ are non-zero constants. It is given that ${{\mathrm{f}}^{-1}}\left( x \right)$ exists. State, if possible, the coordinates of the points where the following curves cut the axes and the equations of their asymptotes.</div><ol class="i-list"><li><div class="list-item-paragraph">$y=\mathrm{f}\left( 2x-3 \right)$</div></li><li><div class="list-item-paragraph">$y={{\mathrm{f}}^{-1}}\left( x \right)$</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

2021 NYJC P1 Q5

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAyNDc1OTMzNDYtcDZ5Yzd0a2JjanEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTEzNCwiZXhwIjoxNzkyODM5NDM0fQ.rptmXbf5r0iCSCz-qHIwavWxNIaWVA5l6FxoHV5_Ol8" 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.

Recovering Original Graphs from Transformations

2009 AJC P1 Q11

<p>Sigmoid functions are used to model many natural processes such as population growth of virus. One example of a Sigmoid function $\mathrm{f}$ is given by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \frac{1}{1+{{\mathrm{e}}^{-x}}}$, $x\in \mathbb{R}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760244734514-9vpp0u6r3" data-list-item-path="item-1760244734514-9vpp0u6r3"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$, indicating clearly the equation(s) of any asymptote(s) and the coordinates of any points where the curve crosses the axes.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760244735053-5fgp6y18q" data-list-item-path="item-1760244735053-5fgp6y18q"><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}\left( x \right)$ in similar form.<span class="inline-marks">[3]</span></div></li></ol>
<p>Another function $\mathrm{g}$ is given by $g:x\mapsto 3x-1$, $x\in \mathbb{R}$, $0\le x\le 2$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760244762000-66i8i3qnm" data-list-item-path="item-1760244762000-66i8i3qnm"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{fg}$ exists and find the range of $\mathrm{fg}$, expressing your answer in terms of $\mathrm{e}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760244762668-aa5zc39ca" data-list-item-path="item-1760244762668-aa5zc39ca"><div class="list-item-paragraph">Describe a sequence of transformations which transform the graph of $y=\mathrm{f}\left( x \right)$ onto the graph of $y=\mathrm{fg}\left( x \right)$.<span class="inline-marks">[2]</span></div></li></ol>

Sigmoid functions are used to model many natural processes such as population growth of virus. One example of a Sigmoid function $\mathrm{f}$ is given by

$\mathrm{f}:x\mapsto \frac{1}{1+{{\mathrm{e}}^{-x}}}$, $x\in \mathbb{R}$.

Another function $\mathrm{g}$ is given by $g:x\mapsto 3x-1$, $x\in \mathbb{R}$, $0\le x\le 2$.

Sketching Exponential Functions and Solving Equations Graphically

Composite Functions

Existence of Composite Functions

Range of a Composite Function

2021 CJC Promo Q11

<p>Given that ${{3}^{x}}={{2}^{kx}}$ for all values of $x$, find $k$, giving your answer in terms of natural logarithms. Describe a geometrical transformation which maps the graph of $y={{2}^{x}}$ onto the graph of $y={{3}^{x}}$.</p>

Given that ${{3}^{x}}={{2}^{kx}}$ for all values of $x$, find $k$, giving your answer in terms of natural logarithms. Describe a geometrical transformation which maps the graph of $y={{2}^{x}}$ onto the graph of $y={{3}^{x}}$.

Translations, Reflections and Scalings of Graphs

ACJC Tutorial 3 Question 10

<ol class="abc-list"><li><div class="list-item-paragraph">The diagram below shows the graph of $y=\mathrm{f}\left( x \right)$. The curve passes through the points $\left( -2,0 \right)$, $\left( 0,\frac{6}{5} \right)$ and $\left( 4,0 \right)$, and has turning points at $\left( -1,2 \right)$, $\left( 1,0 \right)$ and $\left( 3,4 \right)$.</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/1764583542339-zw084geqg1.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjQ1ODM1NDIzMzktencwODRnZXFnMS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODM5MTM0LCJleHAiOjE3OTI4Mzk0MzR9.SkKzMhoHZe45ZBfe1lU2DND3ASTZ0CPJPmjIlgPQZwU" alt="List item image" class="aligncenter custom-width" style="max-width: min(200px, 100%); height: auto;" /></div><ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( 2x+3 \right)$, indicating clearly the axial intercepts, stationary points and asymptotes (if any).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Sketch the graph of $y=\frac{1}{\mathrm{f}\left( x \right)}$, indicating clearly the axial intercepts, stationary points and asymptotes (if any).<span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-paragraph">Describe the sequence of transformations that will transform the graph of $y={{x}^{3}}$ to the graph of $y=1-8{{x}^{3}}$.<span class="inline-marks">[3]</span></div></li></ol>

2022 ASRJC Promo Q8

<p>A curve $C$ has equation $y=\frac{1}{\sqrt{4ax-{{x}^{2}}}}$, where&nbsp;$a &gt; 0$.</p>
<ol class="abc-list"><li data-list-item-id="item-1759587358176-rixxmhe9w" data-list-item-path="item-1759587358176-rixxmhe9w"><div class="list-item-paragraph">Sketch $C$ and give the equations of any asymptotes, in terms of $a$ where appropriate.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759587358709-q4nslamko" data-list-item-path="item-1759587358709-q4nslamko"><div class="list-item-paragraph">Find the smallest possible value of $y$ in terms of $a$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1759587359526-8wai8eyd6" data-list-item-path="item-1759587359526-8wai8eyd6"><div class="list-item-paragraph">Describe the transformation that maps the graph of $C$ onto the graph of $y=\frac{1}{\sqrt{4{{a}^{2}}-{{x}^{2}}}}$.<span class="inline-marks">[3]</span></div></li></ol>

A curve $C$ has equation $y=\frac{1}{\sqrt{4ax-{{x}^{2}}}}$, where $a > 0$.

Sketch $C$ and give the equations of any asymptotes, in terms of $a$ where appropriate.

Find the smallest possible value of $y$ in terms of $a$.

Describe the transformation that maps the graph of $C$ onto the graph of $y=\frac{1}{\sqrt{4{{a}^{2}}-{{x}^{2}}}}$.

N2021 P1 Q6

<ol class="abc-list"><li data-list-item-id="item-1759478762749-ocwaugm1w" data-list-item-path="item-1759478762749-ocwaugm1w"><div class="list-item-paragraph">Describe a sequence of two transformations that maps the graph of $y=\ln \left( \frac{{{x}^{2}}}{x+1} \right)$ onto the graph of $y=\ln \left( \frac{2x+1}{4{{x}^{2}}} \right)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759478763372-u2v7r38ei" data-list-item-path="item-1759478763372-u2v7r38ei"><div class="list-item-paragraph">The diagram below shows the graph of $y=\mathrm{f}\left( x \right)$. It has a maximum point at $A\left( -1,-1 \right)$ and a minimum point at $B\left( 1,\frac{1}{4} \right)$. The graph has asymptotes $y=\frac{1}{2}$, $x=0$ and $x=-2$.</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/1764583940719-p7wd5ghyqtg.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjQ1ODM5NDA3MTktcDd3ZDVnaHlxdGcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTEzNCwiZXhwIjoxNzkyODM5NDM0fQ.CFUHTvcZydI-sQRuFaZx-Ntxnc8859U2I-WOUgk_5Tg" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div><div class="list-item-paragraph">Sketch, on separate diagrams, the graphs of.</div><ol class="i-list"><li data-list-item-id="subitem-1759478800872-4fcwac156" data-list-item-path="item-1759478763372-u2v7r38ei/subitem-1759478800872-4fcwac156"><div class="list-item-paragraph">$y=\mathrm{f}\left( 2-x \right)$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1759478801022-2bpua3sy2" data-list-item-path="item-1759478763372-u2v7r38ei/subitem-1759478801022-2bpua3sy2"><div class="list-item-paragraph">$y=\frac{1}{\mathrm{f}\left( x \right)}$,<span class="inline-marks">[3]</span></div></li></ol><div class="list-item-after-content">stating clearly the equations of any asymptotes, coordinates of any points of intersection with both axes and the points corresponding to $A$ and $B$.</div></li></ol>

Describe a sequence of two transformations that maps the graph of $y=\ln \left( \frac{{{x}^{2}}}{x+1} \right)$ onto the graph of $y=\ln \left( \frac{2x+1}{4{{x}^{2}}} \right)$.

2019 DHS P1 Q5

<ol class="abc-list"><li data-list-item-id="item-1759474022783-i154g9y7j" data-list-item-path="item-1759474022783-i154g9y7j"><div class="list-item-paragraph">The graph of $y=\mathrm{f}\left( x \right)$ is shown below.</div><div class="list-item-paragraph">On separate diagrams, sketch the following graphs, indicating clearly the key features</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/1764584513541-ddn44zkg8vl.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjQ1ODQ1MTM1NDEtZGRuNDR6a2c4dmwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTEzNCwiZXhwIjoxNzkyODM5NDM0fQ.P7MpLgYMm_7qfnUwfSaFNQncFo7vd-MnRm1MMqh7K9k" alt="List item image" class="aligncenter custom-width" style="max-width: min(300px, 100%); height: auto;" /></div><ol class="i-list"><li data-list-item-id="subitem-1759474041248-os9c71l7u" data-list-item-path="item-1759474022783-i154g9y7j/subitem-1759474041248-os9c71l7u"><div class="list-item-paragraph">$y=\mathrm{f}\left( 1-x \right)$,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1759474056106-8xq02z8g1" data-list-item-path="item-1759474022783-i154g9y7j/subitem-1759474056106-8xq02z8g1"><div class="list-item-paragraph">$y=\mathrm{f}'\left( x \right)$.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-1759474069013-vzkukwwgb" data-list-item-path="item-1759474069013-vzkukwwgb"><div class="list-item-paragraph">State a sequence of transformations which transform the graph of $y=\ln \left( 1-\frac{x}{2} \right)$ onto the graph of $y=\ln \left( \frac{2}{1-x} \right)$.<span class="inline-marks">[3]</span></div></li></ol>

State a sequence of transformations which transform the graph of $y=\ln \left( 1-\frac{x}{2} \right)$ onto the graph of $y=\ln \left( \frac{2}{1-x} \right)$.

2021 RI P1 Q5

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTkyMjYwOTgyNTYtbHdmZWhuZGRqei5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODM5MTM0LCJleHAiOjE3OTI4Mzk0MzR9.CbD-USIOvprMvMmK0R6qpt_FTwjBl-IYuMP_gZWPkiI" 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

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

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

2021 ACJC Promo Q4

<p>The diagram shows the curve $y=\mathrm{f}\left( x \right)$. The curve has maximum points at $\left( -5,4 \right)$ and the origin, and crosses the $x$-axis at $\left( -4,0 \right)$. The lines $y=0$, $x=-3$ and $y=-x+2$ are the horizontal, vertical and oblique asymptotes to the curve 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/1759132149707-cciqofelaba.jpg?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTkxMzIxNDk3MDctY2NpcW9mZWxhYmEuanBnIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTEzNCwiZXhwIjoxNzkyODM5NDM0fQ.KMEysbFQzklH76gCewbNP8NNQzJba2xOUGG96DTq_Qs" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-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">$y=\mathrm{f}\left( \frac{x+1}{2} \right)$<span class="inline-marks">[3]</span></div></li></ol>
<p>labelling clearly the equation(s) of any asymptote(s), coordinates of any axial intercept(s) and turning point(s) where applicable.</p>

The diagram shows the curve $y=\mathrm{f}\left( x \right)$. The curve has maximum points at $\left( -5,4 \right)$ and the origin, and crosses the $x$-axis at $\left( -4,0 \right)$. The lines $y=0$, $x=-3$ and $y=-x+2$ are the horizontal, vertical and oblique asymptotes to the curve respectively.

$y=\mathrm{f}\left( \frac{x+1}{2} \right)$

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

2017 RI P1 Q5

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjQ1ODM5MTQ0NjgtemlhMG05MGJ5OS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODM5MTM0LCJleHAiOjE3OTI4Mzk0MzR9.zX9kBXvbOu-IFwRzPPDHZ-Notuu-hxjmoTYUAwXipsk" 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 $y=-2\left( x+2 \right)+\frac{1}{x}$.

Math Question Bank