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PSLE Foundation Mathematics

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">\[\mathrm{f}(x)=x^2+x+3,\ x\geq a\]</p>
<p>where&nbsp;$a$ is a real constant.</p>
<ol class="abc-list"><li data-list-item-id="q9-a" data-list-item-path="q9-a"><div class="list-item-paragraph">State the least value of $a$ for which $\mathrm{f}^{-1}$ exists.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="q9-b" data-list-item-path="q9-b"><div class="list-item-paragraph">Using this value of $a$, find $\mathrm{f}^{-1}(x)$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="q9-c" data-list-item-path="q9-c"><div class="list-item-paragraph">Find the domain of $\mathrm{f}^{-1}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="q9-d" data-list-item-path="q9-d"><div class="list-item-paragraph">Sketch, on the same axes, the graphs of $\mathrm{f}$ and $\mathrm{f}^{-1}$, labelling clearly the coordinates of any endpoints.<span class="inline-marks">[3]</span></div></li></ol>
<p>The function $\mathrm{g}$ is defined by $\mathrm{g}(x)=x^2+x+3$.</p>
<ol class="abc-list" start="5"><li data-list-item-id="q9-e" data-list-item-path="q9-e"><div class="list-item-paragraph">Find two possible linear functions $\mathrm{h}$ such that&nbsp;$(\mathrm{g}\circ\mathrm{h})(x)=4x^2-14x+15$.<span class="inline-marks">[6]</span></div></li></ol>

The function $\mathrm{f}$ is defined by

State the least value of $a$ for which $\mathrm{f}^{-1}$ exists.

Using this value of $a$, find $\mathrm{f}^{-1}(x)$.

Sketch, on the same axes, the graphs of $\mathrm{f}$ and $\mathrm{f}^{-1}$, labelling clearly the coordinates of any endpoints.

The function $\mathrm{g}$ is defined by $\mathrm{g}(x)=x^2+x+3$.

Find two possible linear functions $\mathrm{h}$ such that $(\mathrm{g}\circ\mathrm{h})(x)=4x^2-14x+15$.

Composite Functions

Function Notation, Domain and Range

Inverse Functions

Sketching Quadratic Graphs

IB AA HL Q9: Quadratic inverse and composition (paper unspecified)

Quadratic Functions, Equations and Inequalities

Functions

<p>The cross-section of a skateboard half-pipe is modelled by the quadratic function $y =  \frac{1}{2}x^2 - 4x + 9$, where&nbsp;$x$ metres represents the horizontal distance from left end of the half-pipe, and $y$ metres represents the height above the ground.</p>
<p>The graph of the function models the curved surface of the half-pipe.</p>
<ol class="i-list"><li data-list-item-id="item-1779863970301-8i7w7kccu" data-list-item-path="item-1779863970301-8i7w7kccu"><div class="list-item-paragraph"><p>Show that the equation of the curve can be written in the form $y = \frac{1}{2}(x - 4)^2 + 1.$</p><p>Hence, state the coordinates of the turning point.</p></div></li><li data-list-item-id="item-1779864013798-i6efiat6a" data-list-item-path="item-1779864013798-i6efiat6a"><div class="list-item-paragraph">Explain what the turning point represents in the context of the half-pipe.</div></li><li data-list-item-id="item-1779864053174-je0mqtrit" data-list-item-path="item-1779864053174-je0mqtrit"><div class="list-item-paragraph">Sketch the graph of the quadratic function clearly showing:</div><ul class="bullet-list"><li data-list-item-id="subitem-1779864068279-6voek6d7l" data-list-item-path="item-1779864053174-je0mqtrit/subitem-1779864068279-6voek6d7l"><div class="list-item-paragraph">the turning point,</div></li><li data-list-item-id="subitem-1779864074574-hmuf5w09s" data-list-item-path="item-1779864053174-je0mqtrit/subitem-1779864074574-hmuf5w09s"><div class="list-item-paragraph"><p>the intercept with the $y$-axis,</p></div></li><li data-list-item-id="subitem-1779864074994-t5ucj8kvb" data-list-item-path="item-1779864053174-je0mqtrit/subitem-1779864074994-t5ucj8kvb"><div class="list-item-paragraph">the axis of symmetry.</div></li></ul></li><li data-list-item-id="item-1779864093967-t3qlykxyu" data-list-item-path="item-1779864093967-t3qlykxyu"><div class="list-item-paragraph">If the right end of the half pipe has a height of $13.5$ m from the ground, find the width of the half pipe from one end to the other.</div></li></ol>

The cross-section of a skateboard half-pipe is modelled by the quadratic function $y =  \frac{1}{2}x^2 - 4x + 9$, where $x$ metres represents the horizontal distance from left end of the half-pipe, and $y$ metres represents the height above the ground.

The graph of the function models the curved surface of the half-pipe.

Completing the Square and Vertex Form

Maximum and Minimum Values of General Quadratic Functions

Applications of Quadratic Functions and Equations

<p>Sketch the graph of $y = (x - 1)(x + 5)$.</p>
<p>Indicate clearly the coordinates of points where the graph crosses the $x$- and $y$-axes, the coordinates of the turning point and write down the equation of the line of symmetry of the curve.<span class="inline-marks">[5]</span></p>
<p>Hence, or otherwise, solve the inequality $x^2 + 4x - 5 \le 0$.<span class="inline-marks">[1]</span></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/1769517826384-pn2smbhjeen.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk1MTc4MjYzODQtcG4yc21iaGplZW4ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg2Nzk1MywiZXhwIjoxNzkyODY4MjUzfQ.X7qyb4I3XOb1c8OScbFMKX7yDw_HOxGIeiOE785iVaY" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

Sketch the graph of $y = (x - 1)(x + 5)$.

Indicate clearly the coordinates of points where the graph crosses the $x$- and $y$-axes, the coordinates of the turning point and write down the equation of the line of symmetry of the curve.

Hence, or otherwise, solve the inequality $x^2 + 4x - 5 \le 0$.

Quadratic Inequalities

IP2 Class Test (May 2025) Q12

<p>Express $x^2 - 2x - 1$ in the form $(x - a)^2 - b$, where $a$ and $b$ are integers. Hence,</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the minimum value of $x^2 - 2x - 1$ and the value of $x$ at which the minimum value occurs.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y = x^2 - 2x - 1$, stating clearly the turning point, $x$-intercepts and the $y$-intercept.</span></div></li></ol>

Express $x^2 - 2x - 1$ in the form $(x - a)^2 - b$, where $a$ and $b$ are integers. Hence,

Find the minimum value of $x^2 - 2x - 1$ and the value of $x$ at which the minimum value occurs.

Sketch the graph of $y = x^2 - 2x - 1$, stating clearly the turning point, $x$-intercepts and the $y$-intercept.

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Express $4x^2 + 4x - 15$ in the form of $a(x + b)^2 + c$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, sketch the graph of $y = 4x^2 + 4x - 15$, indicating clearly the coordinates of the turning point and $y$-intercept.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain why another graph $y = -\left(x + \frac{1}{2}\right)^2 - 19$ will not intersect the graph of $y = 4x^2 + 4x - 15$.</span><span class="inline-marks">[2]</span></div></li></ol>

Express $4x^2 + 4x - 15$ in the form of $a(x + b)^2 + c$.

Hence, sketch the graph of $y = 4x^2 + 4x - 15$, indicating clearly the coordinates of the turning point and $y$-intercept.

Explain why another graph $y = -\left(x + \frac{1}{2}\right)^2 - 19$ will not intersect the graph of $y = 4x^2 + 4x - 15$.

2023 CSS AM S3 EOY Q5

<p>Sketch the graph of $y={{x}^{2}}-3x-10$, indicate clearly the turning point and cutting point(s) on the axes.</p>
<p>Hence, find</p>
<ol class="i-list"><li><div class="list-item-paragraph">the maximum or minimum point of the function and the value of $x$ where it occurs, </div></li><li><div class="list-item-paragraph">the equation of the line of symmetry of the graph.</div></li></ol>

Sketch the graph of $y={{x}^{2}}-3x-10$, indicate clearly the turning point and cutting point(s) on the axes.

<p>Express $y=-3{{x}^{2}}+6x-11$ in the form $y=a{{(x+h)}^{2}}+k$ where&nbsp;$a$, $h$ and $k$ are constants. State the maximum value of $y$ and the corresponding value of $x$.  Hence sketch the graph of $y=-3{{x}^{2}}+6x-11$ and state the coordinates of the maximum point of the curve.</p>

Express $y=-3{{x}^{2}}+6x-11$ in the form $y=a{{(x+h)}^{2}}+k$ where $a$, $h$ and $k$ are constants. State the maximum value of $y$ and the corresponding value of $x$.  Hence sketch the graph of $y=-3{{x}^{2}}+6x-11$ and state the coordinates of the maximum point of the curve.

<p>A tank is firing rounds during a target practice.  The height, $h$ metres, of a projectile above the ground during flight is governed by the equation $h=5t-\frac{1}{5}{{t}^{2}}$ where $t$ is the time in seconds for the projectile in flight.</p>
<ol class="i-list"><li><div class="list-item-paragraph">What is the maximum height achieved by a projectile?</div></li><li><div class="list-item-paragraph">Sketch the curve $h=5t-\frac{1}{5}{{t}^{2}}$ for $h\ge 0$. </div></li><li><div class="list-item-paragraph">Find the range of values of $t$ for which a round is at least $10$ metres above the ground.</div></li></ol>

A tank is firing rounds during a target practice.  The height, $h$ metres, of a projectile above the ground during flight is governed by the equation $h=5t-\frac{1}{5}{{t}^{2}}$ where $t$ is the time in seconds for the projectile in flight.

<p>Express $y=2{{x}^{2}}+4x-6$ in the form $a{{(x+b)}^{2}}+k$, where $a$, $b$ and $k$ are constants.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">State the minimum value of the function and the value of $x$ when this occurs.</div></li><li><div class="list-item-paragraph">Hence, sketch the graph of $y=2{{x}^{2}}+4x-6$ and state the coordinates of $x$ and $y$-intercepts and turning points.</div></li></ol>

Express $y=2{{x}^{2}}+4x-6$ in the form $a{{(x+b)}^{2}}+k$, where $a$, $b$ and $k$ are constants.

<p>Express $y={{x}^{2}}-8x+17$ in the form of $y={{(x-h)}^{2}}+k$. Hence, state</p>
<ol class="i-list"><li><div class="list-item-paragraph">the maximum or minimum point of the function and the value of $x$ where it occurs </div></li><li><div class="list-item-paragraph">the equation of the line of symmetry of the graph. </div></li></ol>
<p>Sketch the graph and indicate clearly the turning point and cutting point(s) on the axes.</p>

Express $y={{x}^{2}}-8x+17$ in the form of $y={{(x-h)}^{2}}+k$. Hence, state

Sketch the graph and indicate clearly the turning point and cutting point(s) on the axes.

Completing the Square for $a=1$

<p><strong>Answer the whole of this question on a sheet of graph paper.</strong></p>
<p>Copy and complete the following table of values for&nbsp;$y=25+4x-2{{x}^{2}}$.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 500px;"><colgroup><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p><em>$x$</em></p></th><th style="text-align: center; --cell-justify: center"><p>$-3$</p></th><th style="text-align: center; --cell-justify: center"><p>$-2$</p></th><th style="text-align: center; --cell-justify: center"><p>$-1$</p></th><th style="text-align: center; --cell-justify: center"><p>$0$</p></th><th style="text-align: center; --cell-justify: center"><p>$1$</p></th><th style="text-align: center; --cell-justify: center"><p>$2$</p></th><th style="text-align: center; --cell-justify: center"><p>$3$</p></th><th style="text-align: center; --cell-justify: center"><p>$4$</p></th><th style="text-align: center; --cell-justify: center"><p>$5$</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p><em>$y$</em></p></td><td style="text-align: center; --cell-justify: center"><p>$-5$</p></td><td style="text-align: center; --cell-justify: center"><p>$9$</p></td><td style="text-align: center; --cell-justify: center"><p>$a$</p></td><td style="text-align: center; --cell-justify: center"><p>$25$</p></td><td style="text-align: center; --cell-justify: center"><p>$27$</p></td><td style="text-align: center; --cell-justify: center"><p>$25$</p></td><td style="text-align: center; --cell-justify: center"><p>$19$</p></td><td style="text-align: center; --cell-justify: center"><p>$9$</p></td><td style="text-align: center; --cell-justify: center"><p>$-5$</p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the value of $a$.</div></li><li><div class="list-item-paragraph">Using a scale of $2$ cm to represent $1$ unit on the $x$-axis and $2$ cm to represent $5$ units on the $y$-axis, draw the graph of $y=25+4x-2{{x}^{2}}$. <br></div></li><li><div class="list-item-paragraph">Use your graph to estimate the <strong>solutions</strong> for the equation $25+4x-2{{x}^{2}}=0$.</div></li><li><div class="list-item-paragraph">By drawing a suitable straight line, estimate the <strong>solutions</strong> for the equation</div><ol class="i-list"><li><div class="list-item-paragraph">$25+4x-2{{x}^{2}}=17.5$,</div></li><li><div class="list-item-paragraph">$25+4x-2{{x}^{2}}=-3$.</div></li></ol></li></ol>

**Answer the whole of this question on a sheet of graph paper.**

Copy and complete the following table of values for $y=25+4x-2{{x}^{2}}$.

Solving Quadratic Equations Graphically

Graphical Solution of Quadratic Function

Functions and Graphs

<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y=-{{\left( x-5 \right)}^{2}}$, showing the intercepts and coordinates of the turning point clearly.</div></li><li><div class="list-item-paragraph">Write down the equation of the line of symmetry.</div></li></ol>

Sketching a Quadratic Graph with a Maximum Turning Point and Line of Symmetry

<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y={{\left( x+2 \right)}^{2}}-1$. </div><div class="list-item-paragraph">Indicate clearly the values where the graph crosses the $x$- and $y$-axes and the turning point</div></li><li><div class="list-item-paragraph">Write down the equation of the line of symmetry.</div></li></ol>

Sketching a Quadratic Graph in Vertex Form and Finding Its Line of Symmetry

<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y=3-{{(x-1)}^{2}}$, showing the intercepts and coordinates of the turning point clearly.</div></li><li><div class="list-item-paragraph">State the coordinates of the turning point.</div></li></ol>

Sketching a Quadratic Graph from Vertex Form and Identifying Key Features

<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y={{x}^{2}}-1$.  </div><div class="list-item-paragraph">Indicate clearly the values where the graph crosses the $x$- and $y$-axes and the turning point.</div></li><li><div class="list-item-paragraph">State the coordinates of the turning point.</div></li></ol>

Sketching a Basic Quadratic Graph and Identifying the Turning Point

<ol class="i-list"><li><div class="list-item-paragraph">Express $y={{x}^{2}}-4x+7$ in the form $y={{(x-a)}^{2}}+b$.</div></li><li><div class="list-item-paragraph">Sketch the graph of $y={{x}^{2}}-4x+7$ in the space provided below, showing clearly the $y$-intercept, the $x$-intercepts (if any) and the coordinates of the turning point.</div></li></ol>

Completing the Square and Graphing a Quadratic Function

<ol class="i-list"><li><div class="list-item-paragraph">Express ${{x}^{2}}-2x-5$  in the form of ${{\left( x+a \right)}^{2}}+b$.</div></li><li><div class="list-item-paragraph">Sketch the graph of $y={{x}^{2}}-2x-5$ on the axes below. </div><div class="list-item-paragraph">Indicate clearly the values where the graph crosses $y$-axes and the turning point.</div></li></ol>

Completing the Square and Sketching a Quadratic Graph

<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y=(3-2x)(1+2x)$.  </div><div class="list-item-paragraph">Indicate clearly the values where the graph crosses the $x$- and $y$-axes. </div></li><li><div class="list-item-paragraph">Write down the equation of the line of symmetry of the curve.</div></li></ol>

Sketching a Quadratic Graph in Factorised Form with Non-Standard Coefficients

<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y=2x(5-x)$.</div><div class="list-item-paragraph">Indicate clearly the values where the graph crosses the $x$- and $y$-axes. </div></li><li><div class="list-item-paragraph">Write down the equation of the line of symmetry of the curve.</div></li></ol>

Sketching a Quadratic Graph and Finding Its Line of Symmetry

<ol class="i-list"><li data-list-item-id="item-1765517777698-zjlx184ys" data-list-item-path="item-1765517777698-zjlx184ys"><div class="list-item-paragraph">Sketch the graph of $y=(3+x)(x-2)$. </div><div class="list-item-paragraph">Indicate clearly the values where the graph crosses the $x$- and $y$-axes. </div></li><li data-list-item-id="item-1765517778411-amipmdh5a" data-list-item-path="item-1765517778411-amipmdh5a"><div class="list-item-paragraph">Write down the equation of the line of symmetry of the curve.</div></li><li data-list-item-id="item-1765517778936-hgsboa3lj" data-list-item-path="item-1765517778936-hgsboa3lj"><div class="list-item-paragraph">Write down the value of $h$, given that the range of the graph is $y\ge h$.</div></li></ol>

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