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PSLE Foundation Mathematics

<p>Given&nbsp;\(h(x)=1-|x-2|\), where&nbsp;\(0\le x\le4\), and \(H(x)=\dfrac{1}{h(x)}\), the range of \(H(x)\) is the set&nbsp;\(R\) such that&nbsp;\(R=\{y:y\ge a\text{ or}\hspace{0.5em}y\le b\}\) where&nbsp;\(a,b\in\mathbb Z\).</p>
<p>Determine the values of \(a\) and \(b\).</p>

Given \(h(x)=1-|x-2|\), where \(0\le x\le4\), and \(H(x)=\dfrac{1}{h(x)}\), the range of \(H(x)\) is the set \(R\) such that \(R=\{y:y\ge a\text{ or }y\le b\}\) where \(a,b\in\mathbb Z\).

Modulus Functions

Range of a Function

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

Graphs of Modulus Function

Two branches in a reciprocal modulus range

Functions

Modulus Function

Transformations of Graphs

<p>For&nbsp;\(-2\le x\le3\), define \(p(x)=|2x-1|-3\) and \(q(x)=\dfrac{1}{p(x)}\) wherever \(p(x)\ne0\). The range of \(q\) is \(\{y:y\le a\text{ or}\hspace{0.5em}y\ge b\}\). Find the exact values of \(a\) and \(b\).<span class="inline-marks">[4]</span></p>

For \(-2\le x\le3\), define \(p(x)=|2x-1|-3\) and \(q(x)=\dfrac{1}{p(x)}\) wherever \(p(x)\ne0\). The range of \(q\) is \(\{y:y\le a\text{ or }y\ge b\}\). Find the exact values of \(a\) and \(b\).

Reciprocal range of a stretched modulus trough

<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

Recovering Original Graphs from Transformations

Inequalities involving Modulus Functions

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Graphing Techniques

Inequalities

<p>The diagram below shows the graph of \(y = \mathrm{f}(x)\).</p>
<div class="graph-block" data-graph-config="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" data-width="520" data-height="310" style="width:520px;height:310px;max-width:100%;display:block;margin:0 auto;"></div>
<p>Sketch each of the following on a separate diagram, clearly indicating the \(y\)-intercept, turning point(s) and asymptote(s).</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">\(y = 2\mathrm{f}(x - 2) + 1\)<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">\(y = -\mathrm{f}(x - 1) + 2\)<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">\(y = \mathrm{f}(-x + 1) - 2\)<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">\(y = \mathrm{f}(2x - 2) + 1\)<span class="inline-marks">[3]</span></div></li></ol>

The diagram below shows the graph of \(y = \mathrm{f}(x)\).

Sketch each of the following on a separate diagram, clearly indicating the \(y\)-intercept, turning point(s) and asymptote(s).

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODM1MzE4MzEzOTktcnZ6a3AyOGwzOS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODM5MTkxLCJleHAiOjE3OTI4Mzk0OTF9.IsSnr3hZpOs-mZXkpXw4DvyVRXNkNDVPt1p059XFncc" 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

Graph Transformations of \(f(ax+b)\)

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

2025 ACJC P1 Q8

<div class="content-block content-block--columns"><div class="question-columns question-columns--3" data-layout="3"><div class="question-column" style="flex: 1; align-self: flex-start;"><p style="text-align: center">\(y=\mathrm{f}\left( x \right)\)</p><div class="graph-block" data-graph-config="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" data-width="300" data-height="250" style="width:300px;height:250px;max-width:100%;display:block;margin:0 auto;"></div></div><div class="question-column-divider"></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p style="text-align: center">\(y=\frac{1}{\mathrm{f}\left( x \right)}\) </p></div><div class="question-column-divider"></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p style="text-align: center">\(y=\mathrm{f}'\left( x \right)\)<br><br></p></div></div></div>

Sketching Derivative Graphs

Reciprocal and Gradient Graphs Q3

Applications of Differentiation

<div class="content-block content-block--columns"><div class="question-columns question-columns--3" data-layout="3"><div class="question-column" style="flex: 1; align-self: flex-start;"><p style="text-align: center">\(y=\mathrm{f}\left( x \right)\)</p><div class="graph-block" data-graph-config="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" data-width="236" data-height="200" style="width:236px;height:200px;max-width:100%;display:block;margin:0 auto;"></div></div><div class="question-column-divider"></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p style="text-align: center">\(y=\frac{1}{\mathrm{f}\left( x \right)}\) </p></div><div class="question-column-divider"></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p style="text-align: center">\(y=\mathrm{f}'\left( x \right)\)<br><br></p></div></div></div>

Reciprocal and Gradient Graphs Q2

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Describe a sequence of \(2\) transformations that transforms the curve \(C\) with equation \(\frac{x^2}{9} + y^2 = 1\) onto the curve \(D\) with equation \((x - 2)^2 + y^2 = 1\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The diagram shows the curve with equation \(y = \mathrm{f}(x)\). The curve passes through the origin and the point \((2, 0)\) and has a minimum point at \((3, -4)\). The asymptotes of the curve are the lines \(x = 1\), \(y = 0\) and \(y = 2\).</div><div class="graph-block" data-graph-config="eyJheGlzIjp0cnVlLCJncmlkIjpmYWxzZSwid2lkdGgiOjQwMCwiaGVpZ2h0IjoyNTAsImVsZW1lbnRzIjpbeyJpZCI6IjI4NWRkY2Y4LTVjNTctNDgyNy1iYzUwLTA2NWM3ODhlZjZiNiIsInR5cGUiOiJpbWFnZSIsImZpeGVkIjp0cnVlLCJwb2ludDEiOlstNi43MiwtMy4yNV0sInZpc2libGUiOmZhbHNlLCJpbWFnZVVybCI6Imh0dHBzOi8vdm9jd2twZ2FwdmtzYXJ0eGhob3Iuc3VwYWJhc2UuY28vc3RvcmFnZS92MS9vYmplY3QvcHVibGljL3F1ZXN0aW9uLWltYWdlcy9wYXN0ZWQvMTc3OTQ2NDEwMDgzNy11bDllMW91YmdnOC5wbmciLCJpbWFnZVdpZHRoIjoxNC45NSwiaW1hZ2VIZWlnaHQiOjguODQsImltYWdlQXNwZWN0UmF0aW8iOjEuNjkyNjgyOTI2ODI5MjY4Mn0seyJpZCI6IjFmMTE0ZDgzLWY4NzEtNDc3OS04MjQzLWNkMDJhNDAxMTZmMSIsImRhc2giOjIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbLTYuNDIsMS4zMV0sInBvaW50MiI6WzcuNDIsMS4zXSwic3Ryb2tlV2lkdGgiOjEuNX0seyJpZCI6IjM5NDFmOGIwLTI4YjItNDY2Mi1hZmU4LTk5ZDA0YTZjOThiMCIsImRhc2giOjIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbMS4xNCw0LjgzXSwicG9pbnQyIjpbMS4xMywtNC4zNl0sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiI2MGNkMzgxNy02ZjZiLTQwYjQtYjNmMi02YmRmYjBlMmYxYWIiLCJ0eXBlIjoiYmV6aWVyIiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnRzIjpbWy02LjI5LDEuMV0sWy0wLjgsMS4wM10sWzAuODEsMS41NF0sWzEsLTQuMzJdXSwic3Ryb2tlV2lkdGgiOjEuNX0seyJpZCI6IjU4ZWMwYTVjLTk2YjItNDk3Ny05NzI2LWU2MTc3MjE5ZWE5NyIsInR5cGUiOiJiZXppZXIiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludHMiOltbMS4zLDQuODZdLFsxLjcsLTAuMjVdLFsyLjM4LC0yLjhdLFszLjI0LC0yLjY1XSxbNC4wMSwtMi4yOF0sWzMuODMsLTAuMDJdLFs3LjMzLC0wLjExXV0sInN0cm9rZVdpZHRoIjoxLjV9LHsieCI6LTUuOTcsInkiOjIuNCwiaWQiOiJkNjVlYWE4NS1kY2IzLTQ2ZjAtYjI0Mi02MzkyOWZmMDk4YTgiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoeT0yXFwpIiwidGV4dFJvdGF0aW9uIjowfSx7IngiOjMuMjQsInkiOjIuNzksImlkIjoiZTEwN2Q4N2UtNWEzMS00NGVhLTk0YzItN2Q3NThiZmFmZmMyIiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKHk9XFxtYXRocm17Zn0oeClcXCkiLCJ0ZXh0Um90YXRpb24iOjB9LHsieCI6Mi4yOCwieSI6LTAuMzIsImlkIjoiM2U3NWI4MGEtODJiMy00YTdmLWJlZWEtOTU3YzJjZmQxMWNkIiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKDJcXCkiLCJ0ZXh0Um90YXRpb24iOjB9LHsieCI6Ni4xMywieSI6MC41MSwiaWQiOiI5YjYyMDJjZS00ZTkxLTQ0YzUtYjc5YS1kMDA0MzBkOWMzYjkiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoeT0wXFwpIiwidGV4dFJvdGF0aW9uIjowfSx7IngiOjMuMiwieSI6LTIuOTIsImlkIjoiNGZkZDhjODktYzhiMC00Y2JhLTg3MTEtYzU5M2IwYjBlYmZjIiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKCgzLC00KVxcKSIsInRleHRSb3RhdGlvbiI6MH0seyJ4IjozLjExLCJ5IjotMi42NSwiaWQiOiJiYmZiNDExZC1lMWUyLTRlY2QtYjk0NS03NTIzODY5ZWYyNDkiLCJ0eXBlIjoicG9pbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IiIsInBvaW50U2l6ZSI6MS41LCJwb2ludFN0eWxlIjoiY2lyY2xlIn0seyJ4IjowLjc5LCJ5IjotNC41NiwiaWQiOiJiNDllMDI1Yy05OWRkLTQxNGEtODQzMS1kNmFjYzczMTVlMmIiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoeD0xXFwpIiwidGV4dFJvdGF0aW9uIjowfV0sInhBeGlzTGFiZWwiOiJcXCh4XFwpIiwieUF4aXNMYWJlbCI6IlxcKHlcXCkiLCJib3VuZGluZ0JveCI6Wy01LDUsNSwtNV0sIm9yaWdpbkxhYmVsIjoiXFwoT1xcKSIsImdyaWRTZXR0aW5ncyI6eyJtYWpvckNvbG9yIjoiIzk0YTNiOCIsIm1ham9yU3RlcFgiOjEsIm1ham9yU3RlcFkiOjEsIm1pbm9yQ29sb3IiOiIjY2JkNWUxIiwibWFqb3JPcGFjaXR5IjowLjQ1LCJtaW5vck9wYWNpdHkiOjAuNDUsIm1pbm9yRWxlbWVudHNYIjo0LCJtaW5vckVsZW1lbnRzWSI6NCwibWFqb3JTdHJva2VXaWR0aCI6MC44LCJtaW5vclN0cm9rZVdpZHRoIjowLjV9LCJzaG93QXhpc0xhYmVscyI6ZmFsc2UsInNob3dOYXZpZ2F0aW9uIjpmYWxzZSwia2VlcEFzcGVjdFJhdGlvIjp0cnVlLCJ4QXhpc0xhYmVsT2Zmc2V0IjpbMiwtMTFdLCJ5QXhpc0xhYmVsT2Zmc2V0IjpbLTE0LDVdLCJvcmlnaW5MYWJlbE9mZnNldCI6Wy01LC0yXX0=" data-width="400" data-height="250" style="width:400px;height:250px;max-width:100%;display:block;margin:0 auto;"></div><div class="list-item-paragraph">Sketch the graph of \(y = \frac{1}{\mathrm{f}(x)}\), indicating the equations of any asymptotes and the coordinates of any points where it crosses the axes and of any turning points, where applicable.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">The graph of \(y = \mathrm{g}(x)\) is a semi-circle with its centre at the point \((a, 0)\), where&nbsp;\(a &lt; 0\). The diagram shows the graph of \(y = \mathrm{g}(|x|)\), which intersects the axes at the points \((-b, 0)\), \((b, 0)\) and \((0, c)\), where&nbsp;\(a \neq -b\).<span class="inline-marks">[3]</span></div><div class="graph-block" data-graph-config="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" data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div><div class="list-item-paragraph">Sketch the graph of \(y = \mathrm{g}(x)\), labelling the coordinates of the centre and the radius of the semi-circle, in terms of \(a\) and/or \(b\).</div></li></ol>

Composite Transformations

2024 MI PRU1 Q5

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The diagram below shows the graph of \(y = \mathrm{f}(x)\). The curve passes through the points \(A(-a,0)\) and \(B(0,-a)\) and has a minimum point at \(C(2a,-2a)\), where&nbsp;\(a &gt; 0\). The lines \(x = -2a\) and \(y = -a\) are asymptotes of the curve.</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/1779453402163-dw91vlxqcgf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Nzk0NTM0MDIxNjMtZHc5MXZseHFjZ2YucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTE5MSwiZXhwIjoxNzkyODM5NDkxfQ._uCHDvlaly9WBVgDtGBOtXQ2Fs2O5oW7CpmqKN2xLok" alt="List item image" class="aligncenter custom-width" style="max-width: min(350px, 100%); height: auto;" /></div><div class="list-item-paragraph">On separate diagrams, sketch the graph of </div><ol class="i-list"><li data-list-item-id="subitem-i" data-list-item-path="item-a/subitem-i"><div class="list-item-paragraph">\(y = \mathrm{f}(x-a) + a\),<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-ii" data-list-item-path="item-a/subitem-ii"><div class="list-item-paragraph">\(y = \frac{1}{\mathrm{f}(x)}\),<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The curve whose equation is \(y = \sin x\), undergoes in succession, the following transformations:<br><br>\(\mathbf{A}\): A translation by \(\pi\) units in the negative \(x\)-direction.<br>\(\mathbf{B}\): A scaling parallel to the \(x\)-axis by a factor of \(3\).<br>\(\mathbf{C}\): A translation by \(2\) units in the positive \(y\)-direction.<br><br>Find the equation of the resulting curve.<span class="inline-marks">[3]</span></div></li></ol>

2024 JPJC Promo Q8

<ol class="abc-list"><li data-list-item-id="item-1769597978425-ylmgvub6x" data-list-item-path="item-1769597978425-ylmgvub6x"><div class="list-item-paragraph">The diagram below shows the graph of  \(y=\mathrm{f}\left( x \right)\). The graph has a minimum point at \(\left( 1,3 \right)\) and intersects the axes at \(x=-2\)and \(y=4\).  The equations of the asymptotes are \(y=2x\) and \(x=-1\).</div><div class="graph-block" data-graph-config="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" data-width="275" data-height="225" style="width:275px;height:225px;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="subitem-1769598014440-g82ujpqa0" data-list-item-path="item-1769597978425-ylmgvub6x/subitem-1769598014440-g82ujpqa0"><div class="list-item-paragraph">\(y=\mathrm{f}'\left( x \right)\),<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1769598034127-wnklyqesr" data-list-item-path="item-1769597978425-ylmgvub6x/subitem-1769598034127-wnklyqesr"><div class="list-item-paragraph">\(y=\frac{1}{\mathrm{f}(x)}\),<span class="inline-marks">[2]</span></div></li></ol></li></ol>

2013 TJC P1 Q7

<div class="graph-block" data-graph-config="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" data-width="300" data-height="250" style="width:300px;height:250px;max-width:100%;display:block;margin:0 auto;"></div>
<p>On separate diagrams, sketch the following curves, labelling clearly any asymptotes, axial intercepts and turning points.</p>
<ol class="i-list"><li data-list-item-id="item-1769597893630-krt420k34" data-list-item-path="item-1769597893630-krt420k34"><div class="list-item-paragraph">\(y=\frac{1}{\mathrm{f}(x)}\)<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1769597895573-siwp5u06x" data-list-item-path="item-1769597895573-siwp5u06x"><div class="list-item-paragraph">\(y=\mathrm{{f}'}(x)\)<span class="inline-marks">[3]</span></div></li></ol>

On separate diagrams, sketch the following curves, labelling clearly any asymptotes, axial intercepts and turning points.

2015 NJC P1 Q2

<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>

2015 IJC P1 Q8

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">On the same diagram, sketch the graphs of \(y = |x - a|\) and \(y = \frac{1}{|x - a|}\), where&nbsp;\(a\) is a real constant such that&nbsp;\(a &gt; 1\). You should show clearly the equations of any asymptotes and axial intercepts of both graphs.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, or otherwise, solve the inequality \(|x - a| &gt; \frac{1}{|x - a|}\).</span><span class="inline-marks">[3]</span></div></li></ol>

On the same diagram, sketch the graphs of \(y = |x - a|\) and \(y = \frac{1}{|x - a|}\), where \(a\) is a real constant such that \(a > 1\). You should show clearly the equations of any asymptotes and axial intercepts of both graphs.

Hence, or otherwise, solve the inequality \(|x - a| > \frac{1}{|x - a|}\).

Inequalities involving Rational Functions

Asymptotes and Discontinuities

2025 TMJC P1 Q1

<p>The diagram below shows a sketch of the graph of \(y = \mathrm{f}(x)\). The curve passes through the points with coordinates \((1, 0)\) and \((3, 0)\), and has turning points at \((1, 0)\) and \((4, 4)\). The asymptotes are \(x = 0\), \(x = 2\) and \(y = 2\).</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="250" style="width:300px;height:250px;max-width:100%;display:block;margin:0 auto;"></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-content"><span class="list-item-text">Sketch on separate diagrams, the graphs of \(y = \mathrm{f}'(x)\),</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-content"><span class="list-item-text">Sketch on separate diagrams, the graphs of \(y = \frac{1}{\mathrm{f}(x)}\),</span><span class="inline-marks">[3]</span></div></li></ol></li></ol>
<p>showing clearly the main features of the graphs.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">Describe a sequence of transformations which transform the graph of \(y = \frac{x^2 + 1}{x}\) to the graph of \(y = \frac{2x^2 - 5x + 4}{x - 2}\).</span><span class="inline-marks">[3]</span></div></li></ol>

The diagram below shows a sketch of the graph of \(y = \mathrm{f}(x)\). The curve passes through the points with coordinates \((1, 0)\) and \((3, 0)\), and has turning points at \((1, 0)\) and \((4, 4)\). The asymptotes are \(x = 0\), \(x = 2\) and \(y = 2\).

showing clearly the main features of the graphs.

Describe a sequence of transformations which transform the graph of \(y = \frac{x^2 + 1}{x}\) to the graph of \(y = \frac{2x^2 - 5x + 4}{x - 2}\).

2025 RI P1 Q5

<p>A curve has equation \(y = \mathrm{f}(x)\), where&nbsp;\(\mathrm{f}(x) = 1 - \sqrt{q^2 - x^2}\) for&nbsp;\(q &gt; 1\). State the shape of \(y = \mathrm{f}(x)\).<span class="inline-marks">[1]</span></p>
<ol class="i-list"><li data-list-item-id="item-3-i" data-list-item-path="item-3-i"><div class="list-item-paragraph">Sketch the curve \(y = \frac{1}{\mathrm{f}(x)}\), giving the equations of any asymptotes and the coordinates of the end-points.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-3-ii" data-list-item-path="item-3-ii"><div class="list-item-paragraph">Describe the transformations that map the graph of \(y = \mathrm{f}(x)\) to \(y = -\sqrt{1-x^2}\).<span class="inline-marks">[3]</span></div></li></ol>

A curve has equation \(y = \mathrm{f}(x)\), where \(\mathrm{f}(x) = 1 - \sqrt{q^2 - x^2}\) for \(q > 1\). State the shape of \(y = \mathrm{f}(x)\).

Sketch the curve \(y = \frac{1}{\mathrm{f}(x)}\), giving the equations of any asymptotes and the coordinates of the end-points.

Describe the transformations that map the graph of \(y = \mathrm{f}(x)\) to \(y = -\sqrt{1-x^2}\).

Graphs of Conic Sections

2025 NJC P1 Q4

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;\(y = \ln \left( \frac{e^2}{3x} \right)\) can be written in the form \(y = a + b \ln (cx)\), where&nbsp;\(a\), \(b\) and \(c\) are integers to be found. Hence, state a sequence of transformations which transform the graph of \(y = \ln x\) onto the graph of \(y = \ln \left( \frac{e^2}{3x} \right)\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The curve \(y = \mathrm{f}(x)\) passes through the point \(P\) with coordinates \((a, b)\), where&nbsp;\(b \neq 0\). The tangent to the curve at \(P\) has gradient \(5\). When&nbsp;\(y = \mathrm{f}(x)\) is transformed onto the curve \(y = \mathrm{g}(x)\), \(P\) corresponds to the point \(R\) on \(y = \mathrm{g}(x)\). For each of the following curves, state the coordinates of \(R\) and find the gradient of the curve at \(R\).</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">\(\mathrm{g}(x) = 2\mathrm{f}(x - 1)\)<span class="inline-marks">[3]</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">\(\mathrm{g}(x) = \frac{1}{\mathrm{f}(x)}\)<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Show that \(y = \ln \left( \frac{e^2}{3x} \right)\) can be written in the form \(y = a + b \ln (cx)\), where \(a\), \(b\) and \(c\) are integers to be found. Hence, state a sequence of transformations which transform the graph of \(y = \ln x\) onto the graph of \(y = \ln \left( \frac{e^2}{3x} \right)\).

The curve \(y = \mathrm{f}(x)\) passes through the point \(P\) with coordinates \((a, b)\), where \(b \neq 0\). The tangent to the curve at \(P\) has gradient \(5\). When \(y = \mathrm{f}(x)\) is transformed onto the curve \(y = \mathrm{g}(x)\), \(P\) corresponds to the point \(R\) on \(y = \mathrm{g}(x)\). For each of the following curves, state the coordinates of \(R\) and find the gradient of the curve at \(R\).

Derivative as the Gradient of a Curve

Chain Rule

2025 JPJC P1 Q10

Techniques of Differentiation

<p>Given that&nbsp;\(b\) is a real constant such that&nbsp;\(0 &lt; b &lt; 4\), describe fully a sequence of transformations that transforms the curve \(y=x^2\) to the curve \(y=4x^2+bx+1\).<span class="inline-marks">[4]</span></p>
<p>Sketch the curve \(y=\frac{1}{4x^2+bx+1}\). Give the equation of any asymptotes and the coordinates of any axial intercepts and turning points, in terms of \(b\) where appropriate.<span class="inline-marks">[3]</span></p>

Given that \(b\) is a real constant such that \(0 < b < 4\), describe fully a sequence of transformations that transforms the curve \(y=x^2\) to the curve \(y=4x^2+bx+1\).

Sketch the curve \(y=\frac{1}{4x^2+bx+1}\). Give the equation of any asymptotes and the coordinates of any axial intercepts and turning points, in terms of \(b\) where appropriate.

Sketching Rational Functions

2025 HCI P1 Q6

<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 550px;"><colgroup><col style="width: 275px;"><col style="width: 275px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p>\({y = \mathrm{f}(x)}\)</p></th><th style="text-align: center; --cell-justify: center"><p>\({y = \frac{1}{\mathrm{f}(x)}}\)</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><div class="graph-block" data-graph-config="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" data-width="250" data-height="200" style="width:250px;height:200px;max-width:100%;display:block;margin:0 auto;"></div></td><td style="text-align: left; 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<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 550px;"><colgroup><col style="width: 275px;"><col style="width: 275px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p>\({y = \mathrm{f}(x)}\)</p></th><th style="text-align: center; --cell-justify: center"><p>\({y = \frac{1}{\mathrm{f}(x)}}\)</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><div class="graph-block" data-graph-config="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" data-width="250" data-height="200" style="width:250px;height:200px;max-width:100%;display:block;margin:0 auto;"></div></td><td style="text-align: left; 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