A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The curve $C$ has equation</p>
<p>\[x=(y+1)e^{-\frac{y}{2}},\qquad y\ge0.\]</p>
<p>The line $x=1$ intersects $C$ at $A(1,0)$ and at a second point $B$.</p>
<ol class="i-list"><li data-list-item-id="b818afb7-41c3-4440-814f-cebde95ce4c2" data-list-item-path="b818afb7-41c3-4440-814f-cebde95ce4c2"><div class="list-item-paragraph">Using differentiation, find the exact coordinates of the point on $C$ where&nbsp;$x$ has its greatest value.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="35fabeb6-60ee-401c-9944-333d45b502e4" data-list-item-path="35fabeb6-60ee-401c-9944-333d45b502e4"><div class="list-item-paragraph">Sketch $C$ and the line $x=1$ on the same axes. State the equation of the asymptote of $C$ and find the coordinates of $B$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="aa9e4784-84aa-43c0-bc7a-f6b2d9a58f92" data-list-item-path="aa9e4784-84aa-43c0-bc7a-f6b2d9a58f92"><div class="list-item-paragraph">Find the area of the finite region enclosed by $C$ and the line $x=1$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="9a5b99e2-d202-43e6-9ba3-93603dc40d0d" data-list-item-path="9a5b99e2-d202-43e6-9ba3-93603dc40d0d"><div class="list-item-paragraph">Find the volume generated when this region is rotated through $2\pi$ radians about the $y$-axis.<span class="inline-marks">[2]</span></div></li></ol>

\[x=(y+1)e^{-\frac{y}{2}},\qquad y\ge0.\]

The line $x=1$ intersects $C$ at $A(1,0)$ and at a second point $B$.

Using differentiation, find the exact coordinates of the point on $C$ where $x$ has its greatest value.

Sketch $C$ and the line $x=1$ on the same axes. State the equation of the asymptote of $C$ and find the coordinates of $B$.

Sketching Exponential Functions and Solving Equations Graphically

Volume of Revolution about the y-axis

Asymptotes and Discontinuities

Area bounded by 2 curves

Stationary Points and Their Nature

An enclosed region beside a sideways exponential curve

Applications of Integration

Graphing Techniques

Applications of Differentiation

<p>Sketch the graph represented by each of the following equations, clearly indicating the equations of asymptote(s), axial intercept(s), and the coordinates of the turning point(s), if any.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$y = 2x^3 - 5x^2 + x - 2$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = 4x^2 e^x$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = \frac{\ln x}{2x}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = \frac{x^2 + 2x - 8}{x^2 + 2x - 3}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = |2x - x^2|$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = \frac{ax + 1}{x - b}$, where constants $a$, $b$ are given by</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$-1 &lt; a &lt; 0$ and $0 &lt; b &lt; 1$;</span></div></li><li><div class="list-item-content"><span class="list-item-text">$a &lt; -1$ and $b &gt; 1$.</span></div></li></ol></li></ol>
<p>What could be said about the graph if&nbsp;$ab + 1 = 0$?</p>

Sketch the graph represented by each of the following equations, clearly indicating the equations of asymptote(s), axial intercept(s), and the coordinates of the turning point(s), if any.

$y = \frac{x^2 + 2x - 8}{x^2 + 2x - 3}$

$y = \frac{ax + 1}{x - b}$, where constants $a$, $b$ are given by

What could be said about the graph if $ab + 1 = 0$?

Sketching Rational Functions

Sketching Logarithmic Functions and Solving Equations Graphically

Graphs of Modulus Function

Polynomial Functions and Key Features

Transformations of Graphs

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Solve the equation $\log_2 (x^2 + 4x - 1) = 2 + 2\log_4 (3x + 2)$.</span><span class="inline-marks">[5]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch the graphs of $y = \log_2 x$ and $y = e^{2x}$ on the same axes. Use your graphs to explain why $\log_2 x + e^{2x} &gt; 0$ for $x &gt; 1$.</span><span class="inline-marks">[4]</span></div></li></ol>

Solve the equation $\log_2 (x^2 + 4x - 1) = 2 + 2\log_4 (3x + 2)$.

Sketch the graphs of $y = \log_2 x$ and $y = e^{2x}$ on the same axes. Use your graphs to explain why $\log_2 x + e^{2x} > 0$ for $x > 1$.

Solving Logarithm Equations

2023 Ahmad Ibrahim Sec AM S3 EOY Q8

Exponential and Logarithmic Functions

<p>The variables $w$, $x$ and $y$ are connected by the following differential equations:</p>
<p style="text-align: center">$\frac{\mathrm{d}w}{\mathrm{d}x} = -\frac{3}{2}w - 2 \quad\quad (A)$</p>
<p style="text-align: center">$\frac{\mathrm{d}y}{\mathrm{d}x} = w \qquad\qquad\quad\,\,\, (B)$</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Solve equation (A) to find $w$ in terms of $x$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find $y$ in terms of $x$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Different arbitrary constants found in part <strong>(ii)</strong> yields different solution curves. Some of the solution curves are straight lines. Write down the equation of one of these lines. On a single diagram, sketch your line together with a non-linear solution curve that has your line as an asymptote, indicating clearly any axes intercepts.<span class="inline-marks">[3]</span></div></li></ol>

The variables $w$, $x$ and $y$ are connected by the following differential equations:

$\frac{\mathrm{d}w}{\mathrm{d}x} = -\frac{3}{2}w - 2 \quad\quad (A)$

$\frac{\mathrm{d}y}{\mathrm{d}x} = w \qquad\qquad\quad\,\,\, (B)$

Solve equation (A) to find $w$ in terms of $x$.

Solving Differential Equations by Separation of Variables

Solving Differential Equations by Direct Integration

Sketching Solution Curves of Differential Equations

Order, General and Particular Solutions of Differential Equations

2016 SAJC P1 Q8 [Modified]

Differential Equations

<p>A swimming pool contains $375000$ litres of pure water. Water containing $s$ milligrams of free chlorine per litre flows into the pool at a rate of $10$ litres per minute. The pool is also draining at a rate of $10$ litres per minute, such that the volume of the water in the pool remains constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the rate at which the mass of the free chlorine, $x$ grams, is changing in the pool over time, $t$ minutes, can be modelled by the differential equation<br>\[\frac{\mathrm{d}x}{\mathrm{d}t} = \frac{375s - x}{37500},\]State a necessary assumption for the above model to be valid.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that no free chlorine is present in the pool initially, find $x$ in terms of $t$ and $s$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find, in terms of $s$, the mass of free chlorine present in the pool after one hour.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $x$ against $t$ which is relevant to the context, labelling the point(s) where the curve crosses the axes and the equation(s) of the asymptote(s). Hence determine the mass of free chlorine present in the pool after a long period in terms of $s$.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>To prevent health complications, the recommended safe level of free chlorine to be used is between $375$ grams and $1125$ grams in a pool which contains $375000$ litres of water. Find the range of values of $s$ that should be used.<span class="inline-marks">[2]</span></p>

A swimming pool contains $375000$ litres of pure water. Water containing $s$ milligrams of free chlorine per litre flows into the pool at a rate of $10$ litres per minute. The pool is also draining at a rate of $10$ litres per minute, such that the volume of the water in the pool remains constant.

Given that no free chlorine is present in the pool initially, find $x$ in terms of $t$ and $s$.

Find, in terms of $s$, the mass of free chlorine present in the pool after one hour.

Sketch the graph of $x$ against $t$ which is relevant to the context, labelling the point(s) where the curve crosses the axes and the equation(s) of the asymptote(s). Hence determine the mass of free chlorine present in the pool after a long period in terms of $s$.

To prevent health complications, the recommended safe level of free chlorine to be used is between $375$ grams and $1125$ grams in a pool which contains $375000$ litres of water. Find the range of values of $s$ that should be used.

Equilibrium Solutions and Long-Term Behaviour

Mixing and Concentration Models

Assumptions and Limitations of Differential Equation Models

<p>A particular species reproduce at a rate of $100k\%$, where $k$ is a positive constant. Due to natural causes, members of the population die at a rate that is proportional to the square of its population size with constant of proportionality $\frac{k}{n}$, where $n$ is a positive constant. It may be assumed that there are no other external factors that affect the growth of the population, whose size is represented by $x$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write, in terms of $k$ and $n$, a differential equation for the population growth rate.</span></div></li><li><div class="list-item-paragraph">Based on your differential equation in part <strong>(i)</strong>, find, in terms of $k$ and/or $n$, the size of the population when its growth is the fastest, and the growth rate at this instant.</div></li><li><div class="list-item-paragraph">Solve the differential equation in part <strong>(i)</strong>, given that the population is growing the fastest initially. Express $x$ in terms of $t$.</div></li><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $x$ against $t$, stating the equation of the asymptote in terms of $n$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">With reference to your graph, explain the significance of $n$ in context.</span></div></li></ol>

A particular species reproduce at a rate of $100k\%$, where $k$ is a positive constant. Due to natural causes, members of the population die at a rate that is proportional to the square of its population size with constant of proportionality $\frac{k}{n}$, where $n$ is a positive constant. It may be assumed that there are no other external factors that affect the growth of the population, whose size is represented by $x$.

Write, in terms of $k$ and $n$, a differential equation for the population growth rate.

Sketch the graph of $x$ against $t$, stating the equation of the asymptote in terms of $n$.

With reference to your graph, explain the significance of $n$ in context.

Problems Involving Maximisation and Minimisation

Logistic Growth Models

Modelling with Multiple Rates of Change

<p>At the beginning of $1990$, the population, $P$, of a certain species of insects after a period of $n$ years is given by $P = 40000e^{-0.05n}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the population of the insects</div><ol class="i-list"><li><div class="list-item-paragraph">at the beginning of $1990$.</div></li><li><div class="list-item-paragraph">at the beginning of $2015$, correct to $3$ significant figures.</div></li></ol></li><li><div class="list-item-paragraph">Sketch the graph of $P$ against $n$.</div></li></ol>

At the beginning of $1990$, the population, $P$, of a certain species of insects after a period of $n$ years is given by $P = 40000e^{-0.05n}$.

Applications of Exponential and Logarithmic Functions

<ol class="abc-list"><li><div class="list-item-paragraph">Solve the following integral.</div><ol class="i-list"><li><div class="list-item-paragraph">$\displaystyle \displaystyle\int \frac{1-2x}{\sqrt{3+2x-x^2}} \, \mathrm{d}x$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">$\displaystyle \displaystyle\int x \ln(2-x^2) \, \mathrm{d}x$, where&nbsp;$-\sqrt{2} &lt; x &lt; \ \sqrt{2}$.<span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-paragraph">A function $\mathrm{f}$ is defined by $\mathrm{f}(x) = \mathrm{e}^x - 2$.</div><ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y = |\mathrm{f}(x)|$, indicating clearly the equation(s) of asymptote(s), if any.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence find $\displaystyle \displaystyle\int_0^{\ln 3} |\mathrm{f}(x)| \, \mathrm{d}x$, giving your answer in exact form.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

Integration by Splitting the Numerator

Integration by Parts

Modulus Functions

Area Bounded by Curves

Integration by Completing the Square

Standard Integrals with Quadratic Denominators

2025 ASRJC P2 Q5

Techniques of Integration

Functions

<p>Sketch $y=2{{\mathrm{e}}^{2x+1}}-3$. Hence find the equation of the straight line which must be added on the graph of $y=2{{\mathrm{e}}^{2x+1}}-3$ to solve the equation $x+\ln \sqrt{2}=\frac{1}{2}\left[ \ln \left( 5-x \right)-1 \right]$.</p>

Sketch $y=2{{\mathrm{e}}^{2x+1}}-3$. Hence find the equation of the straight line which must be added on the graph of $y=2{{\mathrm{e}}^{2x+1}}-3$ to solve the equation $x+\ln \sqrt{2}=\frac{1}{2}\left[ \ln \left( 5-x \right)-1 \right]$.

<p>Sketch the curve $y=2-{{\mathrm{e}}^{3x}}$ for all real values of $x$, showing clearly on your diagram the coordinates of any points of intersection with the $x$- and $y$- axes.</p>
<p>Find the equation of a suitable straight line that can be drawn on the graph of $y=2-{{\mathrm{e}}^{3x}}$ to obtain a solution of the equation $x=\ln \sqrt[3]{4-x}$.</p>

Sketch the curve $y=2-{{\mathrm{e}}^{3x}}$ for all real values of $x$, showing clearly on your diagram the coordinates of any points of intersection with the $x$- and $y$- axes.

Find the equation of a suitable straight line that can be drawn on the graph of $y=2-{{\mathrm{e}}^{3x}}$ to obtain a solution of the equation $x=\ln \sqrt[3]{4-x}$.

<p>The diagram below shows the graphs of $y={{\mathrm{e}}^{2x}}$ and $y=2{{\mathrm{e}}^{x}}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Identify each graph  and explain your answer. </div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1766120106338-do7yit90lot.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjYxMjAxMDYzMzgtZG83eWl0OTBsb3QucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0NjI0NiwiZXhwIjoxNzkzNTQ2NTQ2fQ.szXnJiCybvgmb_UugOXqkQdR7HYxDDe6Er0slJzS_6o" alt="List item image" class="aligncenter custom-width" style="max-width: min(300px, 100%); height: auto;" /></div></li><li><div class="list-item-paragraph">By drawing a suitable line on the diagram, state the number of solutions to each equation .</div><ol class="abc-list"><li><div class="list-item-paragraph">$2{{\mathrm{e}}^{x}}=-1$</div></li><li><div class="list-item-paragraph">$2x-\ln 5=\ln \left( 2x+1 \right)$</div></li></ol></li></ol>

The diagram below shows the graphs of $y={{\mathrm{e}}^{2x}}$ and $y=2{{\mathrm{e}}^{x}}$.

<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y={{\operatorname{e}}^{-x}}$.</div></li><li><div class="list-item-paragraph">In order to solve the equation $\frac{1}{3}x=\ln \frac{1}{\sqrt[3]{x-3}}$, a suitable straight line has to be drawn on the same diagram as the graph of $y={{\operatorname{e}}^{-x}}$. Find the equation of the straight line and the number of solutions to the equation.</div></li></ol>

<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y={{2}^{x}}$.</div></li><li><div class="list-item-paragraph">In order to solve the equation $\frac{1}{2}x={{\log }_{2}}\sqrt{3-2x}$, a suitable straight line has to be drawn on the same diagram as the graph of $y={{2}^{x}}$. Find the equation of the straight line and the number of solutions to the equation.<br><br></div></li></ol>

Solving Equation Graphically

<p>The curve of $y={{a}^{3x+b}}+3$ passes through $\left( 0,\frac{7}{2} \right)$ and $\left( -\frac{2}{3},5 \right)$. Find the values of $a$ and $b$.</p>
<p>Hence, sketch the graphs of $y={{a}^{3x+b}}+3$, labelling the coordinates of axial intercepts and equations of asymptotes.</p>

The curve of $y={{a}^{3x+b}}+3$ passes through $\left( 0,\frac{7}{2} \right)$ and $\left( -\frac{2}{3},5 \right)$. Find the values of $a$ and $b$.

Hence, sketch the graphs of $y={{a}^{3x+b}}+3$, labelling the coordinates of axial intercepts and equations of asymptotes.

<p>The curve of $y=a{{3}^{bx+2}}-3$ passes through $\left( -1,3 \right)$ and $\left( 1,51 \right)$. Find the values of $a$ and $b$.</p>
<p>Hence, sketch the graphs of $y=a{{3}^{bx+2}}-3$, labelling the coordinates of axial intercepts and equations of asymptotes.<br></p>

The curve of $y=a{{3}^{bx+2}}-3$ passes through $\left( -1,3 \right)$ and $\left( 1,51 \right)$. Find the values of $a$ and $b$.

Hence, sketch the graphs of $y=a{{3}^{bx+2}}-3$, labelling the coordinates of axial intercepts and equations of asymptotes.


Finding Unknowns of an Exponential Function

<p>Sketch the graphs of each of the following functions, labelling the coordinates of axial intercepts and equations of asymptotes.</p>
<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;"><ol class="abc-list"><li><div class="list-item-paragraph">$y={{\left( \frac{1}{3} \right)}^{2x-2}}+1$</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list" start="2"><li><div class="list-item-paragraph">$y=2{{\operatorname{e}}^{1-2x}}$</div></li></ol></div></div></div>

Sketch the graphs of each of the following functions, labelling the coordinates of axial intercepts and equations of asymptotes.

<p>Sketch the graphs of each of the following functions, labelling the coordinates of axial intercepts and equations of asymptotes.</p>
<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;"><ol class="abc-list"><li><div class="list-item-paragraph">$y={{2}^{2x+1}}-1$</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list" start="2"><li><div class="list-item-paragraph">$y={{\left( \frac{1}{\mathrm{e}} \right)}^{2x}}-2$</div></li></ol></div></div></div>

<p>Sketch the graphs of each of the following functions, labelling the coordinates of axial intercepts and equations of asymptotes.</p>
<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;"><ol class="abc-list"><li><div class="list-item-paragraph">$y={{\operatorname{e}}^{x+1}}+3$</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list" start="2"><li><div class="list-item-paragraph">$y={{\left( \frac{1}{2} \right)}^{x+3}}$</div></li></ol></div></div></div>

Graphs of Exponential Functions

<p>Variables $x$ and $y$ are related by the equation $y=1-{{2}^{x}}$.  </p>
<p>Some corresponding values of $x$ and $y$ are given in the following table.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 450px;"><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;"></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>$-2$</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>$0.5$</p></th><th style="text-align: center; --cell-justify: center"><p>$1$</p></th><th style="text-align: center; --cell-justify: center"><p>$1.5$</p></th><th style="text-align: center; --cell-justify: center"><p>$2$</p></th><th style="text-align: center; --cell-justify: center"><p>$2.5$</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>$0.75$</p></td><td style="text-align: center; --cell-justify: center"><p>$0.5$</p></td><td style="text-align: center; --cell-justify: center"><p>$0$</p></td><td style="text-align: center; --cell-justify: center"><p>$-0.4$</p></td><td style="text-align: center; --cell-justify: center"><p>$-1$</p></td><td style="text-align: center; --cell-justify: center"><p>$-1.8$</p></td><td style="text-align: center; --cell-justify: center"><p>$-3$</p></td><td style="text-align: center; --cell-justify: center"><p><em>$p$</em></p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Using a scale of $4$ cm to represent $1$ unit on the $x$-axis, and $2$ cm to represent $1$ unit on the $y$-axis.</div><div class="list-item-paragraph">Draw the graph of $y=1-{{2}^{x}}$ for&nbsp;$-2\le x\le 2.5$.</div></li><li><div class="list-item-paragraph">Use the graph and solve the equation ${{2}^{x}}=5$.</div></li><li><div class="list-item-paragraph">By drawing a tangent at the point where&nbsp;$x=2$, find the gradient of the curve at the point $\left( 2,-3 \right)$.</div></li><li><div class="list-item-paragraph">Solve the equation ${{2}^{x}}+2x-2=0$.</div></li></ol>

Variables $x$ and $y$ are related by the equation $y=1-{{2}^{x}}$.  

Some corresponding values of $x$ and $y$ are given in the following table.

Estimating Gradients from Tangents

Plotting Curves from Tables of Values

Exploring Exponential Graphs and Their Applications

Functions and Graphs

<p>The diagrams below show the graph of four curves.</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/1769910996678-detzi0zpyy.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk5MTA5OTY2NzgtZGV0emkwenB5eS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTQ2MjQ2LCJleHAiOjE3OTM1NDY1NDZ9.vY73ITvIxRBCT208BHSb41q6odT_dQhdx6aoJm789Uc" style="max-width: min(450px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down the figure number that represents the graph of the equation:</div><ol class="i-list"><li><div class="list-item-paragraph">$y=-{{x}^{3}}$</div></li><li><div class="list-item-paragraph">$y=-\frac{2}{{{x}^{2}}}$</div></li></ol></li><li><div class="list-item-paragraph">In the answer space below, sketch the graph of $y={{2}^{x}}$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769911050920-z401smnfypo.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk5MTEwNTA5MjAtejQwMXNtbmZ5cG8ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0NjI0NiwiZXhwIjoxNzkzNTQ2NTQ2fQ.LEQBHR02_gWQl7OInaa4a5Y5rQslyuaOlhTWGQPKUl4" alt="List item image" class="aligncenter custom-width" style="max-width: min(300px, 100%); height: auto;" /></div></li></ol>

The diagrams below show the graph of four curves.

Math Question Bank