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<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$y=\frac{{{x}^{2}}}{3}+\frac{2}{x}-1$.  

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

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

Estimating Gradients from Tangents

Graphical and Analytical Solutions

Plotting Curves from Tables of Values

Graphing a Rational Function and Solving Equations Graphically

Functions and Graphs

<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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