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      <text>2.2.2.1 Parametric Representation of a Function</text>
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      <text><![CDATA[<h3><span style="color: inherit; font-family: inherit;">Parametric Representation of a Function</span><br></h3>
<p>Suppose we write \(x\) and \(y\) in terms of \(t\) in the form \[x=4t \qquad y=2t^2, \qquad   \hbox{ for } -1\le t\le 1\]
 For different values of \(t\) between \(-1\) and \(1\), we can calculate pairs of values of \(x\) and \(y\). For example when \(t=1\) we see that \(x=4(1)=4\) and \(y=2\times 1^2=2\). That is, \(t=1\) corresponds to the point with \((x,y)\) coordinates\((4,2)\).</p>
<p>A table of values is given below.</p>
<p style="text-align: center;"><strong>Table </strong></p>
<table class="HELM_table">
<tbody>
<tr style="border-bottom: 1px solid #000;">
<td style="text-align: right;">\(t\)</td>
<td style="text-align: right;">\(-1\)</td>
<td style="text-align: right;">\(-0.5\)</td>
<td style="text-align: right;">\(0\)</td>
<td style="text-align: right;">\(0.5\)</td>
<td style="text-align: right;">\(1\)</td>
</tr>
<tr style="border-bottom: 1px solid #000;">
<td style="text-align: right;">\(x\)</td>
<td style="text-align: right;">\(-4\)</td>
<td style="text-align: right;">\(-2\)</td>
<td style="text-align: right;">\(0\)</td>
<td style="text-align: right;">\(2\)</td>
<td style="text-align: right;">\(4\)</td>
</tr>
<tr>
<td style="text-align: right;">\(y\)</td>
<td style="text-align: right;">\(2\)</td>
<td style="text-align: right;">\(0.5\)</td>
<td style="text-align: right;">\(0\)</td>
<td style="text-align: right;">\(0.5\)</td>
<td style="text-align: right;">\(2\)</td>
</tr>
</tbody>
</table><br>
<p>If the resulting points are plotted on a graph then different values of <span class="math inline">\(t\)</span> correspond to different points on the graph. The graph of is plotted in below.</p>
<figure class="HELM_figure"><img src="@@PLUGINFILE@@/2.2-fig3-quadratic%20.svg" alt="Graph of the function defined parametrically by x=4t, y=2t^2" width="225" height="159" class="img-responsive atto_image_button_text-bottom">
<figcaption>Graph of the function defined parametrically by \(x=4t\), \(y=2t^2\), \(-1\le t\le1\)</figcaption>
</figure>
<p>It is often possible to convert a parametric representation of a function into the more usual form by combining the two expressions to eliminate the parameter. Thus if \(x=4t\) we can write \(\displaystyle{t={x\over 4}}\) and so</p>
\[\begin{aligned}
y=2t^2 &amp;= 2\left({x\over 4}\right)^2 \\
&amp;= {2x^2\over 16}\\
&amp;= {x^2\over 8}
\end{aligned}\]
<p>Using \(y=\dfrac{x^2}{8}\) we can, by giving \(x\) values, find corresponding values of \(y\). Plotting these \((x,y)\) values gives, of course, exactly the same curve as in the figure above.</p>]]></text>
<file name="2.2-fig3-quadratic .svg" path="/" encoding="base64">[encoded data, 671472 characters]</file>
<file name="fig11.svg" path="/" encoding="base64">[encoded data, 55412 characters]</file>
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