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PSLE

PSLE Foundation Mathematics

- Express x2-6x+8 in the form a(x+b)2+c. - The curve y=x2-6x+8 is drawn. Using your answer in (a), write down - the coordinates of the turning point, - the equation of the line of symmetry of the curve.

Express $x^2-6x+8$ in the form $a(x+b)^2+c$.

The curve $y=x^2-6x+8$ is drawn. Using your answer in (a), write down

Completing the Square and Vertex Form

Applications of Graphs

2025 MGS(S) PRELIMS P1 Q12

Quadratic Functions, Equations and Inequalities

Algebraic Expressions, Identities and Formulae

Functions and Graphs

<p>Express $y=2x^2-12x+11$ in completed-square form and state its minimum value.</p>

Express $y=2x^2-12x+11$ in completed-square form and state its minimum value.

Maximum and Minimum Values of General Quadratic Functions

Minimum value by completing the square

<p>The equation of a curve is $y = -2x^2 + 3x + 5$.</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Write $y = -2x^2 + 3x + 5$ in the form $y = a(x - h)^2 + k$, where&nbsp;$a$, $h$ and $k$ are constants.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Using your results in part <strong>(a)</strong>, explain clearly why the maximum value of $y$ is $k$ when&nbsp;$x = h$.<span class="inline-marks">[2]</span></div></li></ol>

The equation of a curve is $y = -2x^2 + 3x + 5$.

2024 GESS PRELIMS P1 Q1

<ol class="i-list"><li data-list-item-id="item-1"><div class="list-item-paragraph">Express $4x^2 + 8x - 5$ in the form $p(x + q)^2 + r$, where&nbsp;$p$, $q$ and $r$ are constants to be found.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2"><div class="list-item-paragraph">Hence, state the coordinates of the turning point of the curve $y = 4x^2 + 8x - 5$.<span class="inline-marks">[1]</span></div></li></ol>

2024 FMSS PRELIMS P1 Q1

<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.

Applications of Quadratic Functions and Equations

Sketching Quadratic Graphs

<p>Given the curve $y = kx^2 + 2kx - 3$, where&nbsp;$k$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="item-a-9"><div class="list-item-content"><span class="list-item-text">By expressing $y$ in the form $k(x+b)^2 - 1$, determine the value of $k$ and of $b$.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b-9"><div class="list-item-content"><span class="list-item-text">Hence, show that the curve is always below the $x$-axis.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1772957050027-bldwukdfi"><div class="list-item-paragraph">State the turning point of the curve and determine the nature of this point.<span class="inline-marks">[2]</span></div></li></ol>

Given the curve $y=kx^2+2kx-3$, where $k$ is a constant.

By expressing $y$ in the form $k(x+b)^2-1$, determine the value of $k$ and of $b$.

Hence, show that the curve is always below the $x$-axis.

State the turning point of the curve and determine the nature of this point.

Quadratic Expressions Positive or Negative for All Real Values

2025 AHS PRELIMS P1 Q5

<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>A golfer hits a ball such that the height, $h$ metres, of the ball above the ground $t$ seconds later is given by $h = 0.2t(40 - t)$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">By expressing the function in the form $h = a(t - m)^2 + n$, where $a$, $m$ and $n$ are constants, explain whether the ball can reach a height of $100$ metres.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the range of values of $t$ for which the ball is at most $75$ metres above the ground.</span><span class="inline-marks">[4]</span></div></li></ol>

A golfer hits a ball such that the height, $h$ metres, of the ball above the ground $t$ seconds later is given by $h = 0.2t(40 - t)$.

By expressing the function in the form $h = a(t - m)^2 + n$, where $a$, $m$ and $n$ are constants, explain whether the ball can reach a height of $100$ metres.

Find the range of values of $t$ for which the ball is at most $75$ metres above the ground.

Quadratic Inequalities

2023 Ahmad Ibrahim Sec AM S3 EOY Q4

<p>Express $y={{x}^{2}}+6x+5$ in the form of $y={{(x-h)}^{2}}+k$.</p>

Express $y={{x}^{2}}+6x+5$ in the form of $y={{(x-h)}^{2}}+k$.

<p>Express $y={{x}^{2}}+2x-4$ in the form of $y={{(x-h)}^{2}}+k$.</p>

Express $y={{x}^{2}}+2x-4$ in the form of $y={{(x-h)}^{2}}+k$.

when $a=1$

<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>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 $a=1$

<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

<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 640px;"><colgroup><col style="width: 160px;"><col style="width: 160px;"><col style="width: 160px;"><col style="width: 160px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Quadratic Function</p></th><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"><p>State whether the turning point is Maximum or Minimum</p></th><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"><p>Coordinates of the turning point</p></th><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"><p>Equation of line of symmetry</p></th></tr></thead><tbody><tr style="height: 50px;"><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"><p>(a)&nbsp;&nbsp;&nbsp; $y={{\left( x-1 \right)}^{2}}+2$</p></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 50px;"><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"><p>(b)&nbsp;&nbsp;&nbsp; $y={{\left( x+2 \right)}^{2}}+5$</p></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 50px;"><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"><p>(c)&nbsp;&nbsp;&nbsp; $y=-{{\left( x+3 \right)}^{2}}-3$</p></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 50px;"><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"><p>(d)&nbsp;&nbsp;&nbsp; $y=-{{\left( x-4 \right)}^{2}}+7$</p></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 50px;"><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"><p>(e)&nbsp;&nbsp;&nbsp; $y=1-{{\left( x+1 \right)}^{2}}$</p></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 50px;"><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"><p>(f)&nbsp;&nbsp;&nbsp; $y=2+{{\left( x+5 \right)}^{2}}$</p></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td><td style="height: 50px; min-height: 50px; text-align: left; --cell-justify: flex-start"></td></tr></tbody></table></div></div>

Identifying Turning Points and Lines of Symmetry for Quadratic Functions in Vertex Form

<p>Let&nbsp;$a,b,c$ be the sides of a triangle such that</p>
<p style="text-align: center">$a+b-2\sqrt{a-5}-4\sqrt{b-4}=3\sqrt{c-3}-\frac{1}{2}c+1$.</p>
<p>Justify with clear reasons, the type of the triangle.</p>

Let $a,b,c$ be the sides of a triangle such that

$a+b-2\sqrt{a-5}-4\sqrt{b-4}=3\sqrt{c-3}-\frac{1}{2}c+1$.

Justify with clear reasons, the type of the triangle.

Equations Involving Surds

IP1U13 SWII (Pythagoras) Q23

Surds

<p>Express each of the following expressions in the form ${{\left( x+a \right)}^{2}}+b$ .</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{x}^{2}}+6x$</div></li><li><div class="list-item-paragraph">${{x}^{2}}-3x$</div></li><li><div class="list-item-paragraph">${{x}^{2}}+\frac{4}{5}x$</div></li><li><div class="list-item-paragraph">${{x}^{2}}-2\frac{1}{2}x$</div></li></ol>

Express each of the following expressions in the form ${{\left( x+a \right)}^{2}}+b$ .

Completing the Square (Express in the Form ${{\left( x+a \right)}^{2}}+b$)

<p>Express each of the following expressions in the form ${{\left( x+a \right)}^{2}}+b$ .</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{x}^{2}}+6x$ </div></li><li><div class="list-item-paragraph">${{x}^{2}}-5x$</div></li></ol>

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