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<p>The roots of the quadratic equation $x^2+4x-1=0$ are $\alpha$ and $\beta$.</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 values of $\alpha+\beta$ and of $\alpha\beta$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Without solving the equation, find the value of each of the following.</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">$\dfrac{1}{\alpha^2}+\dfrac{1}{\beta^2}$</div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-paragraph">$\left(\alpha-\dfrac{2}{\beta^2}\right)\left(\beta-\dfrac{2}{\alpha^2}\right)$</div></li></ol></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Determine the quadratic equation with integer coefficients which has roots $\alpha-\dfrac{2}{\beta^2}$ and $\beta-\dfrac{2}{\alpha^2}$.</div></li></ol>

The roots of the quadratic equation $x^2+4x-1=0$ are $\alpha$ and $\beta$.

Write down the values of $\alpha+\beta$ and of $\alpha\beta$.

Without solving the equation, find the value of each of the following.

Determine the quadratic equation with integer coefficients which has roots $\alpha-\dfrac{2}{\beta^2}$ and $\beta-\dfrac{2}{\alpha^2}$.

Sum and Product of Roots

Structured chain to an equation with adjusted roots

Quadratic Functions, Equations and Inequalities

<p>Consider $h(x)=(ax)^2\cos x,\ x\in\mathbb{R}$. It is given that&nbsp;$a$ is a real constant.</p>
<ol class="abc-list"><li data-list-item-id="item-1783668286410-82w6cr0ks" data-list-item-path="item-1783668286410-82w6cr0ks"><div class="list-item-paragraph">Show that&nbsp;$h(x)$ is an even function.</div></li><li data-list-item-id="item-1783668286410-8qbyb8vq5" data-list-item-path="item-1783668286410-8qbyb8vq5"><div class="list-item-paragraph">It is given that&nbsp;$h(\pi)=-4$. Find the possible value(s) of $a$.</div></li></ol>

Consider $h(x)=(ax)^2\cos x,\ x\in\mathbb{R}$. It is given that $a$ is a real constant.

Odd and Even Functions

Functions

<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

Transformations of Graphs

<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>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.

Transformations of Graphs - General

<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

<p>$\displaystyle\int{\frac{1}{\sqrt{7+10x-{{x}^{2}}}}}\mathrm{ d}x$</p>

$\displaystyle\int{\frac{1}{\sqrt{7+10x-{{x}^{2}}}}}\mathrm{ d}x$

Integration of ${\frac{1}{\sqrt{a^2 - x^2}}, \frac{1}{a^2 + x^2}, \frac{1}{a^2 - x^2}, \frac{1}{x^2 - a^2}}$

Techniques of  Integration

<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>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>

A curve, $y=f(x)$, undergoes the transformations $A$, $B$, $C$ and $D$ in succession:

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>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.

<p>Consider a function $g$ defined as&nbsp;$g:x\mapsto (x+2)^2-3,\ x\in\mathbb{R}$.</p>
<ol class="abc-list"><li data-list-item-id="inverse-restrict-q3a" data-list-item-path="inverse-restrict-q3a"><div class="list-item-paragraph">Sketch the graph of $g$ and hence explain why $g$ cannot have an inverse.</div></li><li data-list-item-id="inverse-restrict-q3b" data-list-item-path="inverse-restrict-q3b"><div class="list-item-paragraph">If&nbsp;$g:x\mapsto (x+2)^2-3,\ x\ge q$ has an inverse, write down the smallest value $q$ can take.</div></li><li data-list-item-id="inverse-restrict-q3c" data-list-item-path="inverse-restrict-q3c"><div class="list-item-paragraph">For this value of $q$, find the function $g^{-1}$.</div></li><li data-list-item-id="inverse-restrict-q3d" data-list-item-path="inverse-restrict-q3d"><div class="list-item-paragraph">Sketch the graphs of $g$ and $g^{-1}$ on the same axes, clearly showing the relationship between them.</div></li><li data-list-item-id="inverse-restrict-q3e" data-list-item-path="inverse-restrict-q3e"><div class="list-item-paragraph">Write down the range of $g^{-1}$.</div></li><li data-list-item-id="item-1783676614199-a1m0ed6xb" data-list-item-path="item-1783676614199-a1m0ed6xb"><div class="list-item-paragraph">Find the exact solution of the equation $g (x) =g^{-1}(x)$.</div></li></ol>

Consider a function $g$ defined as $g:x\mapsto (x+2)^2-3,\ x\in\mathbb{R}$.

<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>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>The roots of the equation $2x^4-4x^3-5x^2+6x-10=0$ are $\alpha$, $\beta$, $\gamma$ and $\delta$.</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 values of $\sum\alpha$, $\sum\alpha\beta$, $\sum\alpha\beta\gamma$ and $\alpha\beta\gamma\delta$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Hence find the value of $\alpha^2+\beta^2+\gamma^2+\delta^2$.</div></li></ol>

The roots of the equation $2x^4-4x^3-5x^2+6x-10=0$ are $\alpha$, $\beta$, $\gamma$ and $\delta$.

Root sums and products of a quartic

Polynomials, Cubic Equations and Partial Fractions

<p>Sand is poured onto a horizontal surface to form a right circular conical pile. The radius of the base of the pile is always $\frac{3}{4}$ of its vertical height. The volume of the pile is increasing at a constant rate of $10\text{ cm}^3\text{/s}$. Find the rate at which the curved surface area of the pile is increasing at the instant when its height is $4\text{ cm}$.<span class="inline-marks">[5]</span></p>

Sand is poured onto a horizontal surface to form a right circular conical pile. The radius of the base of the pile is always $\frac{3}{4}$ of its vertical height. The volume of the pile is increasing at a constant rate of $10\text{ cm}^3\text{/s}$. Find the rate at which the curved surface area of the pile is increasing at the instant when its height is $4\text{ cm}$.

Connected Rates of Changes

Connected Rates for a Conical Sand Pile

Applications of Differentiation

<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>

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>$ \displaystyle\int{x\sqrt{2x-1}\mathrm{d}x}$</p>

$ \displaystyle\int{x\sqrt{2x-1}\mathrm{d}x}$

IB HL Math Question Bank