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<p>Given that&nbsp;$\mathrm{f}(x) = 3^x$, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}(-|x|)$ on separate diagrams. Clearly indicate the intercept(s) on the graph.<span class="inline-marks">[4]</span></p>

Given that $\mathrm{f}(x) = 3^x$, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}(-|x|)$ on separate diagrams. Clearly indicate the intercept(s) on the graph.

Transformations of Graphs - General

Transformations of Graphs

<p>Given that&nbsp;$\mathrm{f}(x) = x(x + 3)(x - 2)$, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}(|x|)$ on separate diagrams. Clearly indicate the intercepts and asymptotes on the graph.<span class="inline-marks">[4]</span></p>

Given that $\mathrm{f}(x) = x(x + 3)(x - 2)$, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}(|x|)$ on separate diagrams. Clearly indicate the intercepts and asymptotes on the graph.

<p>Given that&nbsp;$\mathrm{f}(x) = \dfrac{x + 3}{x - 2}$, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}(|x|)$ on separate diagrams. Clearly indicate the intercepts and asymptotes on the graph.<span class="inline-marks">[4]</span></p>

Given that $\mathrm{f}(x) = \dfrac{x + 3}{x - 2}$, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}(|x|)$ on separate diagrams. Clearly indicate the intercepts and asymptotes on the graph.

<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).

Reciprocal Graph, y=1⁄f(x)

<p>The curve whose equation is $y=3-e^x$ undergoes, in succession, the following transformations:</p>
<p>A: Translation by $2$ units in the positive $x$-direction.</p>
<p>B: Scaling by a scale factor of $\frac{1}{2}$ parallel to the $x$-axis.</p>
<p>C: Translation by $1$ unit in the positive $y$-direction.</p>
<p>D: Scaling by a scale factor of $2$ parallel to the $y$-axis.</p>
<p>Find the equation of the resulting curve.<span class="inline-marks">[5]</span></p>

The curve whose equation is $y=3-e^x$ undergoes, in succession, the following transformations:

A: Translation by $2$ units in the positive $x$-direction.

B: Scaling by a scale factor of $\frac{1}{2}$ parallel to the $x$-axis.

C: Translation by $1$ unit in the positive $y$-direction.

D: Scaling by a scale factor of $2$ parallel to the $y$-axis.

Find the equation of the resulting curve.

Successive transformations of an exponential curve

<p>The curve whose equation is $y=x^3+2$ undergoes, in succession, the following transformations:</p>
<p>A: Translation by $3$ units in the positive $x$-direction.</p>
<p>B: Scaling by a scale factor of $2$ parallel to the $x$-axis.</p>
<p>C: Translation by $4$ units in the negative $y$-direction.</p>
<p>D: Scaling by a scale factor of $3$ parallel to the $y$-axis.</p>
<p>Find the equation of the resulting curve.<span class="inline-marks">[5]</span></p>

The curve whose equation is $y=x^3+2$ undergoes, in succession, the following transformations:

A: Translation by $3$ units in the positive $x$-direction.

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

C: Translation by $4$ units in the negative $y$-direction.

D: Scaling by a scale factor of $3$ parallel to the $y$-axis.

Successive transformations of a cubic curve

<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.

Recovering a logarithmic function after graph transformations

<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

<p>A function is defined by $f(x)=\ln(x-2)$, where&nbsp;$x\ne 2$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph"><p>Sketch the graph of $y=f(x)$, clearly indicating the asymptotes and intercepts.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Hence, sketch the graph of $y=f(|x|)$ on a separate set of axes, clearly indicating any asymptotes and axial intercepts.<span class="inline-marks">[3]</span></div></li></ol>

A function is defined by $f(x)=\ln(x-2)$, where $x\ne 2$.

<p>A function is defined by $f(x)=(x+1)(x-2)(x-5)$. Sketch the graphs of $y=f(x)$ and $y=f(-|x|)$ on separate diagrams, clearly indicating their axial intercepts and any asymptotes.<span class="inline-marks">[8]</span></p>

A function is defined by $f(x)=(x+1)(x-2)(x-5)$. Sketch the graphs of $y=f(x)$ and $y=f(-|x|)$ on separate diagrams, clearly indicating their axial intercepts and any asymptotes.

Cubic graph transformation with negative modulus

<p>A function is defined by $f(x)=\dfrac{5x-2}{2x-3}$, where&nbsp;$x\ne \frac{3}{2}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Write down the equations of the vertical and horizontal asymptotes of the graph of $y=f(x)$.<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">State the $x$- and $y$-intercepts of the graph of $y=f(x)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Sketch the graph of $y=f(x)$, clearly indicating the asymptotes and intercepts.<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">Hence, sketch the graph of $y=f(|x|)$ on a separate set of axes, clearly indicating any asymptotes and axial intercepts.<span class="inline-marks">[3]</span></div></li></ol>

A function is defined by $f(x)=\dfrac{5x-2}{2x-3}$, where $x\ne \frac{3}{2}$.

<p>A function is defined by $f(x)=\dfrac{3x-2}{x+4}$, where&nbsp;$x\ne -4$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Write down the equations of the vertical and horizontal asymptotes of the graph of $y=f(x)$.<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">State the $x$- and $y$-intercepts of the graph of $y=f(x)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Sketch the graph of $y=f(x)$, clearly indicating the asymptotes and intercepts.<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">Hence, sketch the graph of $y=f(|x|)$ on a separate set of axes, clearly indicating any asymptotes and axial intercepts.<span class="inline-marks">[3]</span></div></li></ol>

A function is defined by $f(x)=\dfrac{3x-2}{x+4}$, where $x\ne -4$.

Transformations of a rational function and modulus

<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="eyJheGlzIjp0cnVlLCJncmlkIjpmYWxzZSwid2lkdGgiOjI1MCwiaGVpZ2h0IjoyNTAsImVsZW1lbnRzIjpbeyJ4IjoxLCJ5IjowLCJpZCI6IjZiODUwNTNhLTJhNTgtNDQxZS04ZTdhLTBlNTI0YzM4MDc3NiIsInR5cGUiOiJwb2ludCIsImNvbG9yIjoiIzAwMDAwMCIsImZpeGVkIjpmYWxzZSwibGFiZWwiOiJcXCgxXFwpIiwidmlzaWJsZSI6dHJ1ZSwicG9pbnRTaXplIjoxLjUsInBvaW50U3R5bGUiOiJjaXJjbGUiLCJsYWJlbE9mZnNldCI6Wy05LjMsLTEzLjNdfSx7IngiOjMsInkiOjAsImlkIjoiNjgwMWE2YTUtNDdiYi00NTcwLThjZDEtMTc5NWFlMGEwYWFhIiwidHlwZSI6InBvaW50IiwiY29sb3IiOiIjMDAwMDAwIiwiZml4ZWQiOmZhbHNlLCJsYWJlbCI6IlxcKDNcXCkiLCJ2aXNpYmxlIjpmYWxzZSwicG9pbnRTaXplIjoxLCJwb2ludFN0eWxlIjoiY2lyY2xlIiwibGFiZWxPZmZzZXQiOls2LjksLTEzLjNdfSx7IngiOjQsInkiOjIsImlkIjoiYTY1M2JhMWEtMWNjZi00ZGNjLWE3ODMtMWE3NDJkMzQ3MmE1IiwidHlwZSI6InBvaW50IiwiY29sb3IiOiIjMDAwMDAwIiwiZml4ZWQiOnRydWUsImxhYmVsIjoiXFwoKDQsMilcXCkiLCJwb2ludFNpemUiOjEsInBvaW50U3R5bGUiOiJjaXJjbGUiLCJsYWJlbE9mZnNldCI6Wy05LjYsMTMuMl19LHsieCI6NSwieSI6LTAuMDMsImlkIjoiYWQxNmVmNTgtNTk3MS00MzJlLTkxMTctMGE4MmE4M2ExMzM4IiwidHlwZSI6InBvaW50IiwiY29sb3IiOiIjMDAwMDAwIiwiZml4ZWQiOmZhbHNlLCJsYWJlbCI6IlxcKDVcXCkiLCJwb2ludFNpemUiOjEsInBvaW50U3R5bGUiOiJjaXJjbGUiLCJsYWJlbE9mZnNldCI6Wy04LjcsLTEyLjRdfSx7IngiOjIsInkiOi0yLCJpZCI6IjEwMzNhMzg4LTE0ZjMtNDFmZi1hZjk0LWIwZmE5YjE0NWIyZCIsInR5cGUiOiJwb2ludCIsImNvbG9yIjoiIzAwMDAwMCIsImZpeGVkIjp0cnVlLCJsYWJlbCI6IlxcKCgyLC0yKVxcKSIsInBvaW50U2l6ZSI6MSwicG9pbnRTdHlsZSI6ImNpcmNsZSIsImxhYmVsT2Zmc2V0IjpbLTIwLjQsLTEzLjRdfSx7ImlkIjoiN2YxNmI0MzctOTU5Zi00MmQzLTgwZTctMmRlMGJjMjhjODk0IiwiZGFzaCI6MiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludDEiOlstOS4yOCwtNF0sInBvaW50MiI6WzkuMjQsLTRdLCJ2aXNpYmxlIjp0cnVlLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoiOGMyNTY4MDAtODU1Mi00ZWM0LTlhMjgtYjlhZTZmZTg4ZjZiIiwidHlwZSI6InNwbGluZSIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50cyI6W1swLjIyLDEwLjE2XSxbMC41NywzLjE1XSxbMS4wMSwwXSxbMS40LC0xLjQyXSxbMi4wMSwtMl0sWzIuNiwtMS41NF0sWzMsMF0sWzMuMzEsMS4yMV0sWzQuMDEsMl0sWzQuNjEsMS4zMV0sWzUuMDEsMF0sWzUuMjQsLTAuOTldLFs2LjYzLC0xMC45N11dLCJ0ZW5zaW9uIjoxLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoiZWYzNzUzN2QtZjM2Ny00MDY1LTg1MDAtY2FmOTc2NjhhNTY0IiwidHlwZSI6InNwbGluZSIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50cyI6W1stOS42MywtNC4zOV0sWy00LjI4LC01LjI4XSxbLTEuMTYsLTcuMjldLFstMC4zMywtMTIuMDRdXSwidGVuc2lvbiI6MSwic3Ryb2tlV2lkdGgiOjEuNX0seyJ4IjowLjU3LCJ5IjotNy4yOSwiaWQiOiI5ZDkyZTNlNS1jNmM3LTQyYmUtOTA5My03NjRjMjgwMDEzMDciLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoeD0wXFwpIiwidGV4dFJvdGF0aW9uIjowfSx7IngiOi02Ljc0LCJ5IjotMy4xMiwiaWQiOiJhMTc0NWRhZi01NTJmLTQ0NzMtYjhmYi1iZWViZTU3NDc2NGIiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoeT00XFwpIiwidGV4dFJvdGF0aW9uIjowfSx7IngiOjEuNCwieSI6NS42MSwiaWQiOiI5ZjBiMmY0Yy1iOGE5LTQ0ZDYtYmZjZS0wMzc4NjcwMDlmMjgiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoeT1cXG1hdGhybXtmfSh4KVxcKSIsInRleHRSb3RhdGlvbiI6MH0seyJpZCI6ImRiY2JkZWNjLWVmMmQtNGMwNi05OTAzLWU3MDMxNjliMDIyNSIsImRhc2giOjMsInR5cGUiOiJwcm9qZWN0aW9uIiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiIiLCJwb2ludDEiOlsxLDBdLCJsYWJlbE9mZnNldCI6WzAsLTE0XSwic3Ryb2tlV2lkdGgiOjEsInByb2plY3Rpb25BeGlzIjoieCIsInByb2plY3Rpb25Tb3VyY2VJZCI6ImE5NTY1N2ZjLWI5MWUtNGQ5Ni1iNmE4LTI1YjFlNTZmODhiZSIsInByb2plY3Rpb25Tb3VyY2VGaWVsZCI6Inh5In0seyJ4IjozLCJ5IjowLCJpZCI6ImJhODcyNjc5LTM0YjAtNGNhNy05ZjI4LTEzYzgwNDFmY2UxYyIsInR5cGUiOiJwb2ludCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiIiwicG9pbnRTaXplIjoxLCJwb2ludFN0eWxlIjoiY2lyY2xlIn0seyJ4IjozLCJ5IjotMC42LCJpZCI6IjFiYzgzNmQ2LTNmOGItNDZhMy04YTFiLTEyZDlmNTVhMmI0ZSIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXCgzXFwpIiwidGV4dFJvdGF0aW9uIjowfV0sInNob3dYQXhpcyI6dHJ1ZSwic2hvd1lBeGlzIjp0cnVlLCJ4QXhpc0xhYmVsIjoiXFwoeFxcKSIsInlBeGlzTGFiZWwiOiJcXCh5XFwpIiwiYm91bmRpbmdCb3giOlstOCw4LjUsNywtOV0sIm9yaWdpbkxhYmVsIjoiXFwoT1xcKSIsImdyaWRTZXR0aW5ncyI6eyJtYWpvckNvbG9yIjoiIzk0YTNiOCIsIm1ham9yU3RlcFgiOjEsIm1ham9yU3RlcFkiOjEsIm1pbm9yQ29sb3IiOiIjY2JkNWUxIiwibWFqb3JPcGFjaXR5IjowLjU1LCJtaW5vck9wYWNpdHkiOjAuMzUsIm1pbm9yRWxlbWVudHNYIjo0LCJtaW5vckVsZW1lbnRzWSI6NCwibWFqb3JTdHJva2VXaWR0aCI6MC44LCJtaW5vclN0cm9rZVdpZHRoIjowLjV9LCJzaG93QXhpc0xhYmVscyI6ZmFsc2UsInNob3dOYXZpZ2F0aW9uIjpmYWxzZSwia2VlcEFzcGVjdFJhdGlvIjpmYWxzZSwieEF4aXNMYWJlbE9mZnNldCI6WzIsLTExXSwieUF4aXNMYWJlbE9mZnNldCI6Wy0xMy4zLDMuOV0sIm9yaWdpbkxhYmVsT2Zmc2V0IjpbLTUsLTJdfQ==" 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;"><p>Given that&nbsp;$(3,0)$ is a point of inflection of $y=\mathrm{f}\left( x \right)$. State the maximal range of values of $x$ for which the graph of $y=\mathrm{f}\left( x \right)$ is </p><ol class="abc-list"><li data-list-item-id="item-1782721008576-2k1azs74n" data-list-item-path="item-1782721008576-2k1azs74n"><div class="list-item-paragraph">strictly decreasing,</div></li><li data-list-item-id="item-1782721009392-41xoixgky" data-list-item-path="item-1782721009392-41xoixgky"><div class="list-item-paragraph">decreasing,</div></li><li data-list-item-id="item-1782721009848-wlrux4odu" data-list-item-path="item-1782721009848-wlrux4odu"><div class="list-item-paragraph">concave upwards,</div></li><li data-list-item-id="item-1782721010480-idm7y5b1r" data-list-item-path="item-1782721010480-idm7y5b1r"><div class="list-item-paragraph">concave downwards.</div></li></ol></div></div></div>

Concavity and Gradients

<p>The curve whose equation is ${y = \ln(2-x)}$ undergoes, in succession the following transformations:<br>A: Translation by ${5}$ units in the positive ${x}$-direction.<br>B: Scaling by a scale factor of ${\frac{1}{3}}$ parallel to the ${x}$-axis.<br>C: Translation of ${5}$ units in the negative ${y}$-direction.<br>D: Scaling by a scale factor of ${\frac{1}{2}}$ parallel to the ${y}$-axis.<br>Find the equation of the resulting curve.</p>

The curve whose equation is ${y = \ln(2-x)}$ undergoes, in succession the following transformations:
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 ${\frac{1}{2}}$ parallel to the ${y}$-axis.
Find the equation of the resulting curve.

<p>The curve whose equation is ${y = \frac{1}{x-1}}$ undergoes, in succession the following transformations:<br>A: Translation by ${2}$ units in the negative ${x}$ direction.<br>B: Scaling by a scale factor of ${3}$ parallel to the ${x}$-axis.<br>C: Translation of ${6}$ units in the positive ${y}$-direction.<br>D: Scaling by a scale factor of ${5}$ parallel to the ${y}$-axis.<br>Find the equation of the resulting curve.</p>

The curve whose equation is ${y = \frac{1}{x-1}}$ undergoes, in succession the following transformations:
A: Translation by ${2}$ units in the negative ${x}$ direction.
B: Scaling by a scale factor of ${3}$ parallel to the ${x}$-axis.
C: Translation of ${6}$ units in the positive ${y}$-direction.
D: Scaling by a scale factor of ${5}$ parallel to the ${y}$-axis.
Find the equation of the resulting curve.

<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="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" 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>

2024 MI PRU1 Q5

<p>It is given that&nbsp;$\mathrm{f}\left( x \right)={{\left( 1-2x \right)}^{3}}-2$ is the final graph after $\mathrm{g}\left( x \right)={{x}^{3}}$ undergoes exactly $4$ transformations. Describe the transformations involved and state the coordinates in which $\left( 0,0 \right)$ is mapped onto the graph of $\mathrm{f}\left( x \right)$.</p>

It is given that $\mathrm{f}\left( x \right)={{\left( 1-2x \right)}^{3}}-2$ is the final graph after $\mathrm{g}\left( x \right)={{x}^{3}}$ undergoes exactly $4$ transformations. Describe the transformations involved and state the coordinates in which $\left( 0,0 \right)$ is mapped onto the graph of $\mathrm{f}\left( x \right)$.

<p>Describe two different transformations, each of which would map</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-content"><span class="list-item-text">the graph of $y = \frac{1}{x}$ onto the graph of $y = \frac{1}{2x}$,</span></div></li><li data-list-item-id="item-b"><div class="list-item-content"><span class="list-item-text">the graph of $y = x^2$ onto the graph of $y = 4x^2$.</span></div></li></ol>

Describe two different transformations, each of which would map

the graph of $y = \frac{1}{x}$ onto the graph of $y = \frac{1}{2x}$,

the graph of $y = x^2$ onto the graph of $y = 4x^2$.

<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;"><p>State the maximal range of values of $x$ for which the graph of $y=\mathrm{f}\left( x \right)$ is </p><ol class="abc-list"><li data-list-item-id="item-1782721448758-9v2vlclnm" data-list-item-path="item-1782721448758-9v2vlclnm"><div class="list-item-paragraph">strictly decreasing,</div></li><li data-list-item-id="item-1782721449230-9q3crfsy1" data-list-item-path="item-1782721449230-9q3crfsy1"><div class="list-item-paragraph">decreasing,</div></li><li data-list-item-id="item-1782721449750-4vvlkis5t" data-list-item-path="item-1782721449750-4vvlkis5t"><div class="list-item-paragraph">concave upwards,</div></li><li data-list-item-id="item-1782721450214-fmkc17s80" data-list-item-path="item-1782721450214-fmkc17s80"><div class="list-item-paragraph">concave downwards</div></li></ol></div></div></div>

<p></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="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1771834571585-e61ojqrom1v.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzE4MzQ1NzE1ODUtZTYxb2pxcm9tMXYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcwOTk4NSwiZXhwIjoxNzg1NzEwMjg1fQ.ht0WRG7WaD6l_KBJ9dp5FZ53s2DLGDlmNuWY-Cz3JxA" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p>The diagram below shows the curve $y=\mathrm{f}'(x)$.</p><ol class="abc-list"><li data-list-item-id="item-1782722076759-ajsrwfjnl" data-list-item-path="item-1782722076759-ajsrwfjnl"><div class="list-item-paragraph">State the $x$-coordinates of the stationary points for the curve $y=\mathrm{f}(x)$. Hence determine the nature of these stationary points.</div></li><li data-list-item-id="item-1782722077575-ubahx25n0" data-list-item-path="item-1782722077575-ubahx25n0"><div class="list-item-paragraph">State the maximal range of values of $x$ such that the curve $y=\mathrm{f}(x)$ is,</div><ol class="i-list"><li data-list-item-id="subitem-1782722080063-mrocqlaob" data-list-item-path="item-1782722077575-ubahx25n0/subitem-1782722080063-mrocqlaob"><div class="list-item-paragraph">concave downwards,</div></li><li data-list-item-id="subitem-1782722080615-72ftoujks" data-list-item-path="item-1782722077575-ubahx25n0/subitem-1782722080615-72ftoujks"><div class="list-item-paragraph">decreasing.</div></li></ol></li></ol></div></div></div>

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