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<ol class="abc-list"><li data-list-item-id="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-0" data-list-item-path="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-0"><div class="list-item-paragraph">Given that&nbsp;$a$ is a positive constant, describe a sequence of transformations which transform the graph of $y=e^x$ to the graph of $y=ae^{-x+6}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1" data-list-item-path="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1"><div class="list-item-paragraph"><p>The function g and h are defined by</p><p>\[\begin{align*} \mathrm{g} &amp;: x \mapsto ae^{-x+6}, \quad x \in \mathbb{R}, \quad a &gt; 0, \\ \mathrm{h} &amp;: x \mapsto 6 + \ln x, \quad x &gt; 0 \end{align*}\]</p></div><ol class="i-list"><li data-list-item-id="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1-sub-0" data-list-item-path="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1/h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1-sub-0"><div class="list-item-paragraph"><p>Find the range of $g$.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1-sub-1" data-list-item-path="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1/h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1-sub-1"><div class="list-item-paragraph">Hence determine whether the function hg exists.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1-sub-2" data-list-item-path="h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1/h2-2025-mi-promo-q7-mi-q7-item-block-0-item-1-sub-2"><div class="list-item-paragraph"><p>Given instead that&nbsp;$a=3$.</p></div><ol class="num-list"><li data-list-item-id="item-1789110199479-w9h9bsfum" data-list-item-path="item-1789110199479-w9h9bsfum"><div class="list-item-paragraph"><p>Find $\mathrm{hg}(x)$ and state the domain of $\mathrm{hg}$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1789110210271-d4buk4zf3" data-list-item-path="item-1789110210271-d4buk4zf3"><div class="list-item-paragraph"><p>Find the exact range of $\mathrm{hg}$.</p><span class="inline-marks">[2]</span></div></li></ol></li></ol></li></ol>

Given that $a$ is a positive constant, describe a sequence of transformations which transform the graph of $y=e^x$ to the graph of $y=ae^{-x+6}$.

Composite Functions

Composite Transformations

Range of a Function

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Functions

Transformations of Graphs

<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.

Graphical Method

Reciprocal Graph, y=1⁄f(x)

Inequalities involving Modulus Functions

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Inequalities

<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

<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).

<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 $f(x)=(x+1)(x-2)(x-5)$. 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 $f(x)=\dfrac{3x-2}{x+4}$, 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="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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>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 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODM1MzE4MzEzOTktcnZ6a3AyOGwzOS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjE5OTUwLCJleHAiOjE3ODk2MjAyNTB9.U5u30YTO8vWn8u_KqXMd7P-7N2LvZpEKVuCQbTft8oU" 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.

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>

Reciprocal and Gradient Graphs Q3

<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

<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.

<p>The diagram shows the graph of $y = \mathrm{f}(x)$ with asymptotes $x = 0$ and $y = 2x - 1$. The graph has a turning point at $\left(2, \frac{21}{4}\right)$ and it cuts the $x$-axis at $x = a$ where&nbsp;$a &lt; 0$.</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/1779465259176-jks9ys3gxbb.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Nzk0NjUyNTkxNzYtamtzOXlzM2d4YmIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYxOTk1MCwiZXhwIjoxNzg5NjIwMjUwfQ.HQdofHg6gMThhlpE0gfwZVWbjPCbuuGbWug3nztTvH4" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Sketch the graph of $y = \mathrm{f}'(x)$, stating the equations of any asymptotes and the coordinates of any turning points and points of intersection with the axes.<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">The curve $y = \mathrm{f}(x)$ undergoes the transformations $A$, $B$ and $C$ in succession.<br><br>$\qquad A$: A translation of $2$ units in the negative $x$-direction.<br>$\qquad B$: A stretch parallel to the $y$-axis with scale factor $2$.<br>$\qquad C$: A translation of $1$ unit in the positive $y$-direction.<br><br>Find the equations of the asymptotes of the resulting curve.<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows the graph of $y = \mathrm{f}(x)$ with asymptotes $x = 0$ and $y = 2x - 1$. The graph has a turning point at $\left(2, \frac{21}{4}\right)$ and it cuts the $x$-axis at $x = a$ where $a < 0$.

2024 MI PRU2 P1 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="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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>

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