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PSLE Foundation Mathematics

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA5NjEyMjQzMTktZnFlNXFxaXppNGEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTI1NywiZXhwIjoxNzkyODM5NTU3fQ.R6yrQ8-Ck9bD4X_WK1dmgYKEUu8hc0W-VG3RjNPAeFo" 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

Graph Transformations of $f(ax+b)$

Sketching Derivative Graphs

2024 HCI Promo Q7

Transformations of Graphs

Applications of Differentiation

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">If $\mathrm{f}\left( x \right)=\sin 2x$, describe the transformation which would transform the graph of $y=\mathrm{f}\left( x \right)$ to the graph of $y=-\mathrm{f}''\left( x \right)$.</div></li><li><div class="list-item-paragraph">If $\mathrm{g}\left( x \right)={{2}^{kx+3}}$, $k>0$, describe a sequence of two transformations which would transform the graph of $y=\mathrm{g}\left( x \right)$ to the graph of $y=\frac{1}{\mathrm{g}\left( x \right)}$.</div></li></ol></li><li><div class="list-item-paragraph">The diagram shows the graph of $y=\mathrm{h}\left( x \right)$ with $\left( -2,4 \right)$ as its maximum point. The curve passes through the points $\left( 1,0 \right)$, $\left( 3,0 \right)$ and $\left( 0,2 \right)$, and the lines $x=2$ and $y=2$ are asymptotes of the curve.</div><div class="list-item-image"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759819740071-zjcqi4u4rk.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTk4MTk3NDAwNzEtempjcWk0dTRyay5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODM5MjU3LCJleHAiOjE3OTI4Mzk1NTd9.4CmGzu7QKeTEezKfTGinIxj6WX7gSW7VG5yp4NigA9c" alt="List item image" /></div><div class="list-item-paragraph">The following properties are known about the graph of $y=a\mathrm{h}\left( x+b \right)$, where $a$ and $b$ are real constants.</div><ol class="i-list"><li><div class="list-item-paragraph">$y=a\mathrm{h}\left( x+b \right)\ge -1$ for all values of $x$ where it is defined.</div></li><li><div class="list-item-paragraph">The two asymptotes of the graph of $y=a\mathrm{h}\left( x+b \right)$ intersect in the region where $x<0$.</div></li></ol></li></ol>
<p style="margin-left: 3rem">Write down the range of values of $a$ and $b$.</p>

Write down the range of values of $a$ and $b$.

Composite Transformations

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

Translations, Reflections and Scalings of Graphs

2015 RI P1 Q8

<p>The ellipse ${{C}_{1}}$ with the equation $\frac{{{x}^{2}}}{{{6}^{2}}}+\frac{{{y}^{2}}}{{{3}^{2}}}=1$ is transformed by a stretch with scale factor $k$ parallel to the $x$-axis, followed by a translation of $m$ units in the positive $x$-direction, followed by a translation of $n$ units in the negative $y$-direction, to form ${{C}_{2}}$.</p>
<ol class="i-list"><li data-list-item-id="item-1759232476631-6zs1rq5e7" data-list-item-path="item-1759232476631-6zs1rq5e7"><div class="list-item-paragraph">Find the equation of ${{C}_{2}}$ in terms of $k$, $m$ and $n$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759232477336-4jdv779oc" data-list-item-path="item-1759232477336-4jdv779oc"><div class="list-item-paragraph">Given that&nbsp;${{C}_{2}}$ is a circle centred at $\left( 4,-7 \right)$, find the values of $k$, $m$ and $n$.<span class="inline-marks">[2]</span></div></li></ol>

The ellipse ${{C}_{1}}$ with the equation $\frac{{{x}^{2}}}{{{6}^{2}}}+\frac{{{y}^{2}}}{{{3}^{2}}}=1$ is transformed by a stretch with scale factor $k$ parallel to the $x$-axis, followed by a translation of $m$ units in the positive $x$-direction, followed by a translation of $n$ units in the negative $y$-direction, to form ${{C}_{2}}$.

Find the equation of ${{C}_{2}}$ in terms of $k$, $m$ and $n$.

Given that ${{C}_{2}}$ is a circle centred at $\left( 4,-7 \right)$, find the values of $k$, $m$ and $n$.

Graphs of Conic Sections

2015 SAJC P1 Q5

Graphing Techniques

<p>The curve $y={{x}^{4}}$ is transformed onto the curves with equation $y=\mathrm{f}\left( x \right)$. The turning point on $y={{x}^{4}}$ corresponds to the point with coordinates $\left( a,b \right)$ on $y=\mathrm{f}\left( x \right)$. The curve $y=\mathrm{f}\left( x \right)$ also passes through the point with coordinate $\left( 0,c \right)$. Given that&nbsp;$\mathrm{f}\left( x \right)$ has the form $k{{\left( x-l \right)}^{4}}+m$ and that&nbsp;$a$, $b$ and $c$ are positive constants with $c&gt;b$, express $k$, $l$ and $m$ in terms of $a$, $b$ and $c$.<span class="inline-marks">[2]</span></p>
<p>By sketching the curve $y=\mathrm{f}\left( x \right)$, or otherwise, sketch the curve $y=\frac{1}{\mathrm{f}\left( x \right)}$. State, in terms of $a$, $b$ and $c$, the coordinates of any points where&nbsp;$y=\frac{1}{\mathrm{f}\left( x \right)}$ crosses the axes and of any turning points.<span class="inline-marks">[4]</span></p>

The curve $y={{x}^{4}}$ is transformed onto the curves with equation $y=\mathrm{f}\left( x \right)$. The turning point on $y={{x}^{4}}$ corresponds to the point with coordinates $\left( a,b \right)$ on $y=\mathrm{f}\left( x \right)$. The curve $y=\mathrm{f}\left( x \right)$ also passes through the point with coordinate $\left( 0,c \right)$. Given that $\mathrm{f}\left( x \right)$ has the form $k{{\left( x-l \right)}^{4}}+m$ and that $a$, $b$ and $c$ are positive constants with $c>b$, express $k$, $l$ and $m$ in terms of $a$, $b$ and $c$.

By sketching the curve $y=\mathrm{f}\left( x \right)$, or otherwise, sketch the curve $y=\frac{1}{\mathrm{f}\left( x \right)}$. State, in terms of $a$, $b$ and $c$, the coordinates of any points where $y=\frac{1}{\mathrm{f}\left( x \right)}$ crosses the axes and of any turning points.

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