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PSLE Foundation Mathematics

<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.

Sketching Exponential Functions and Solving Equations Graphically

Estimating Gradients from Tangents

Plotting Curves from Tables of Values

Exploring Exponential Graphs and Their Applications

Functions and Graphs

<p>The variables $x$ and $y$ are connected by the equation$y=\frac{{{x}^{2}}}{3}+\frac{2}{x}-1$.  </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>$x$</p></th><th style="text-align: center; --cell-justify: center"><p>$0.5$</p></th><th style="text-align: center; --cell-justify: center"><p>$0.6$</p></th><th style="text-align: center; --cell-justify: center"><p>$0.8$</p></th><th style="text-align: center; --cell-justify: center"><p>$1.0$</p></th><th style="text-align: center; --cell-justify: center"><p>$1.5$</p></th><th style="text-align: center; --cell-justify: center"><p>$2.0$</p></th><th style="text-align: center; --cell-justify: center"><p>$2.5$</p></th><th style="text-align: center; --cell-justify: center"><p>$3.0$</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>$y$</p></td><td style="text-align: center; --cell-justify: center"><p>$3.08$</p></td><td style="text-align: center; --cell-justify: center"><p>$2.45$</p></td><td style="text-align: center; --cell-justify: center"><p>$p$</p></td><td style="text-align: center; --cell-justify: center"><p>$1.33$</p></td><td style="text-align: center; --cell-justify: center"><p>$1.08$</p></td><td style="text-align: center; --cell-justify: center"><p>$1.33$</p></td><td style="text-align: center; --cell-justify: center"><p>$1.88$</p></td><td style="text-align: center; --cell-justify: center"><p>$2.67$</p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the value of $p$.</div></li><li><div class="list-item-paragraph">Using the scale of $4$ cm to $1$ unit on each axis, draw a horizontal $x$-axis for&nbsp;$0\le x\le 3$and a vertical $y$-axis for&nbsp;$0\le y\le 4$.  </div></li></ol>
<p>On your axes, plot the points given in the table and join them with a smooth curve.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Use your graph to find two solutions of $\frac{{{x}^{2}}}{3}+\frac{2}{x}-3=0$ in the range $0\le x\le 3$.</div></li><li><div class="list-item-paragraph">By drawing a tangent, find the gradient of the curve at the point $\left( 1,1.33 \right)$.</div></li></ol>

The variables $x$ and $y$ are connected by the equation$y=\frac{{{x}^{2}}}{3}+\frac{2}{x}-1$.  

On your axes, plot the points given in the table and join them with a smooth curve.

Graphical and Analytical Solutions

Graphing a Rational Function and Solving Equations Graphically

<p>The table gives some values of $x$ and the corresponding values of $y$, correct to one decimal place, where&nbsp;$y=\frac{{{x}^{3}}}{6}-x+2$.</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>$x$</p></th><th style="text-align: center; --cell-justify: center"><p>$-4$</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></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>$y$</p></td><td style="text-align: center; --cell-justify: center"><p><em>$-4.7$</em></p></td><td style="text-align: center; --cell-justify: center"><p>$p$</p></td><td style="text-align: center; --cell-justify: center"><p>$2.7$</p></td><td style="text-align: center; --cell-justify: center"><p>$2.8$</p></td><td style="text-align: center; --cell-justify: center"><p>$2$</p></td><td style="text-align: center; --cell-justify: center"><p>$1.2$</p></td><td style="text-align: center; --cell-justify: center"><p>$1.3$</p></td><td style="text-align: center; --cell-justify: center"><p>$3.5$</p></td><td style="text-align: center; --cell-justify: center"><p>$8.7$</p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the value of $p$.</div></li><li><div class="list-item-paragraph">Using a scale of $2$ cm to represent $1$ unit, draw a horizontal $x$-axis for&nbsp;$-4\le x\le 4$.</div><div class="list-item-paragraph">Using a scale of $1$ cm to represent $1$ unit, draw a vertical $y$-axis for$-5\le y\le 9$.</div><div class="list-item-paragraph">On your axes, plot the points given in the table and join them with a smooth curve.</div></li><li><div class="list-item-paragraph">Use your graph to find the solution of $\frac{{{x}^{3}}}{6}-x=-5$.</div></li><li><div class="list-item-paragraph">By drawing a tangent, find the gradient of the curve at the point where&nbsp;$x=3$.</div></li><li><div class="list-item-paragraph">On the axes used in part <strong>(b)</strong>, draw the graph of the straight line $y+2x=-1\mathrm{ }$ for&nbsp;$-4\le x\le 4$.</div></li><li><ol class="i-list"><li><div class="list-item-paragraph">Write down the $x$ coordinate of the point at which the two graphs intersect.</div></li><li><div class="list-item-paragraph">Find the equation, in the form $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$, which is satisfied by the values of $x$ found in part <strong>(f)(i).</strong></div></li></ol></li></ol>

The table gives some values of $x$ and the corresponding values of $y$, correct to one decimal place, where $y=\frac{{{x}^{3}}}{6}-x+2$.

Graphical Solution of a Cubic Equation and Its Applications

<p>The following table gives the corresponding values of $x$ and $y$ which are connected by the equation $y={{x}^{3}}-3x-1$.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center"><p>${x}$</p></th><th style="text-align: center; --cell-justify: center"><p>${-1.75}$</p></th><th style="text-align: center; --cell-justify: center"><p>${-1.5}$</p></th><th style="text-align: center; --cell-justify: center"><p>${-1}$</p></th><th style="text-align: center; --cell-justify: center"><p>${-0.5}$</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>${2.25}$</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>${y}$</p></td><td style="text-align: center; --cell-justify: center"><p>${-1.1}$</p></td><td style="text-align: center; --cell-justify: center"><p>${0.1}$</p></td><td style="text-align: center; --cell-justify: center"><p>${1}$</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>${-2.4}$</p></td><td style="text-align: center; --cell-justify: center"><p>${-3}$</p></td><td style="text-align: center; --cell-justify: center"><p>${a}$</p></td><td style="text-align: center; --cell-justify: center"><p>${1}$</p></td><td style="text-align: center; --cell-justify: center"><p>${3.6}$</p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-1765916962717-nzpw8209i" data-list-item-path="item-1765916962717-nzpw8209i"><div class="list-item-paragraph">Find the value of $a$, <strong>giving your answer correct to 1 decimal place.</strong></div></li><li data-list-item-id="item-1765916963265-1613r40at" data-list-item-path="item-1765916963265-1613r40at"><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, draw the graph of $y=x^{3}-3x-1$ in the range $-1.75\le x\le 2.25$.</div></li><li data-list-item-id="item-1765916964375-2uoo8o1t5" data-list-item-path="item-1765916964375-2uoo8o1t5"><div class="list-item-paragraph">Use your graph to solve the equation</div><ol class="i-list"><li data-list-item-id="subitem-1765916967024-5wb3c846d" data-list-item-path="item-1765916964375-2uoo8o1t5/subitem-1765916967024-5wb3c846d"><div class="list-item-paragraph">$x^{3}-3x=1$</div></li><li data-list-item-id="subitem-1765916968178-hku2x0mfu" data-list-item-path="item-1765916964375-2uoo8o1t5/subitem-1765916968178-hku2x0mfu"><div class="list-item-paragraph">$x^{3}-5x=2$</div></li></ol></li><li data-list-item-id="item-1765916965433-8qrdpyn7e" data-list-item-path="item-1765916965433-8qrdpyn7e"><div class="list-item-paragraph">By drawing a suitable straight line, estimate the gradient of the curve when</div><ol class="i-list"><li data-list-item-id="subitem-1765916970262-9w1vnzgye" data-list-item-path="item-1765916965433-8qrdpyn7e/subitem-1765916970262-9w1vnzgye"><div class="list-item-paragraph">$\left(-0.5,0.1\right)$,</div></li><li data-list-item-id="subitem-1765916971154-4f2cc6dk1" data-list-item-path="item-1765916965433-8qrdpyn7e/subitem-1765916971154-4f2cc6dk1"><div class="list-item-paragraph">$x=1.5$.</div></li></ol></li></ol>
<p>Source correction for (d)(i): at $x=-0.5$, the curve has&nbsp;$y=0.375$, approximately $0.4$, not $0.1$. Draw the tangent at the point on the curve with $x=-0.5$.</p>

The following table gives the corresponding values of $x$ and $y$ which are connected by the equation $y={{x}^{3}}-3x-1$.

Source correction for (d)(i): at $x=-0.5$, the curve has $y=0.375$, approximately $0.4$, not $0.1$. Draw the tangent at the point on the curve with $x=-0.5$.

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